tag:blogger.com,1999:blog-6724976506726565752024-03-17T20:04:25.280-07:00Math MutationShow notes for the Math Mutation podcast.erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.comBlogger161125tag:blogger.com,1999:blog-672497650672656575.post-86227956586832762022023-12-27T06:15:00.000-08:002023-12-27T06:15:23.219-08:00288: One Stone To Rule Them All<p><a href="http://www.erikseligman.com/mm/mm288.mp3"> Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">With the end of 2023 approaching, I spent some time browsing the internet for interesting math news from this year, and realized there was one big unexpected breakthrough by an amateur that really deserves discussion:<span class="Apple-converted-space"> </span>the aperiodic monotile.<span class="Apple-converted-space"> </span>While that term sounds exotic, it actually covers a very simple concept. <span class="Apple-converted-space"> </span>We’ve talked about planar tilings before:<span class="Apple-converted-space"> </span>basically, this just means a shape of tile that can cover an infinite version of your bathroom floor. <span class="Apple-converted-space"> </span>Your current bathroom probably has a bunch of square or hexagonal tiles, which completely fill the available space. <span class="Apple-converted-space"> </span>Well, maybe there are a few practical issues at the edges, but if you imagine the floor to be infinite, those shapes could fill it forever. <span class="Apple-converted-space"> </span>If you want to fill your floor with a set of more interesting shapes, you could imagine drawing arbitrary squiggly lines in a hexagon to divide it into several smaller tiles of arbitrary irregularity, and thus these smaller tiles would cover your floor as well. <span class="Apple-converted-space"> </span>But all these coverings are periodic:<span class="Apple-converted-space"> </span>they consist of an infinite repetition of some small pattern at known intervals. <span class="Apple-converted-space"> </span>In contrast, an aperiodic tiling would still cover your floor with copies of a few basic shapes, but the pattern would not show this kind of regular repetition.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">When the idea first came up in the early 1960s, logician Hao Wang had hypothesized that no aperiodic tiling existed. <span class="Apple-converted-space"> </span>But he was proven wrong within a few years by his student Robert Berger, who discovered a set of tiles that would indeed cover the plane aperiodically. <span class="Apple-converted-space"> </span>His initial set was very complicated, consisting of over 20,000 tiles. <span class="Apple-converted-space"> </span>But this led to a surge of interest in the topic, and by 1974 Roger Penrose had published a simple example of a pair of 4-sided tiles, known as the “kite” and “dart”, that could cover the plane aperiodically with just their two shapes. <span class="Apple-converted-space"> </span>These were shown to be examples of an infinite variety of 2-tile sets that enabled such tilings. <span class="Apple-converted-space"> </span>And that is essentially where the problem stood, for almost 50 years, with nobody knowing whether a single tile could cover a plane aperiodically. <span class="Apple-converted-space"> </span>Bizarrely, during this long wait, chemists discovered that this mathematical game actually had an application in the real world, where these aperiodic tilings formed the basis of “quasicrystals”, real crystalline structures. <span class="Apple-converted-space"> </span>Quasicrystals have been used for applications like nonstick cookware and reinforcing steel, and led to the 2011 Nobel Prize in Chemistry awarded to Israeli scientist Dan Shechtman<span class="s1" style="color: #18191a;">. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Throughout this half century, many mathematicians wondered if there might be a single tile, rather than a set, that could cover the plane aperiodically. <span class="Apple-converted-space"> </span>The problem was nicknamed the “Einstein” problem, not because of any relation to the famous physicist, but as a pun, with the words “Ein Stein” being German for “One Stone”.<span class="Apple-converted-space"> </span>In 2010 Socolar and Taylor got close, publishing a single tile that could solve this problem— but their tile was unconnected, consisting of a central bumpy hexagon and six floating double-square shapes that orbited it in known positions. <span class="Apple-converted-space"> </span>A nice breakthrough, but arguably not a single tile, depending on how you define the concept. <span class="Apple-converted-space"> </span>Other researchers found a different solution that was contiguous, but only worked if some small overlaps were allowed; also an interesting result, but not really a satisfying solution to the Einstein problem.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">The real solution finally came earlier this year, published by Smith, Myers, Kaplan, and Goodman-Strauss:<span class="Apple-converted-space"> </span>a single tile that would actually cover the plane without regularly repeating a pattern. <span class="Apple-converted-space"> </span>It was formed by stitching together eight kite shapes into something that looked like an odd-shaped hat:<span class="Apple-converted-space"> </span>not trivial, but surprisingly simple for something that had evaded discovery for almost fifty years. <span class="Apple-converted-space"> </span>They also proved that this was just a representative of an infinite family of shapes with this property.<span class="Apple-converted-space"> </span>You can see it illustrated in some of the articles linked in the show notes at mathmutation.com.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">The most surprising thing about this discovery was the fact that it was discovered by an amateur— it didn’t stem from efforts by a professional mathematician. <span class="Apple-converted-space"> </span>David Smith is a retired print technician who enjoys jigsaw puzzles and playing with shapes. <span class="Apple-converted-space"> </span>He was using a commonly available software package called the PolyForm puzzle solver, seeing what kind of interesting patterns he could make with different shaped tiles. <span class="Apple-converted-space"> </span>When he saw that the patterns with his experimental “hat” tile looked especially unusual, he decided to cut out a bunch of this tile out of cardboard and start experimenting by hand.<span class="Apple-converted-space"> </span>Convincing himself that he had found something significant, he contacted a professor he knew, Craig Kaplan of the University of Waterloo.<span class="Apple-converted-space"> </span>Together, they investigated further, and after Kaplan recruited two more colleagues to assist with the proof, they confirmed that Smith really had, after all these years, found an Einstein tile. <span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">As a result, if you’ve been thinking of remodeling your bathroom, there is now a nice new floor decoration option you should seriously consider.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://cs.uwaterloo.ca/~csk/hat/">https://cs.uwaterloo.ca/~csk/hat/</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/">https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://hedraweb.wordpress.com/2023/03/23/its-a-shape-jim-but-not-as-we-know-it/">https://hedraweb.wordpress.com/2023/03/23/its-a-shape-jim-but-not-as-we-know-it/</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Aperiodic_tiling">https://en.wikipedia.org/wiki/Aperiodic_tiling</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Penrose_tiling">https://en.wikipedia.org/wiki/Penrose_tiling</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Quasicrystal">https://en.wikipedia.org/wiki/Quasicrystal</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/">https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/</a></span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"><span class="s2" style="text-decoration-line: underline;"><a href="https://maxwelldemon.com/2010/04/01/socolar_taylor_aperiodic_tile/">https://maxwelldemon.com/2010/04/01/socolar_taylor_aperiodic_tile/</a></span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variant-position: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-63747282280797781512023-10-09T16:23:00.000-07:002023-10-09T16:23:33.931-07:00287: The Grim State of Modern Pizza<p><a href="http://www.erikseligman.com/mm/mm287.mp3"> Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">You’ve probably read or heard at some point about the “replication crisis”, and the related epidemic of scientific fraud, discovered over the past few decades.<span class="Apple-converted-space"> </span>Researchers at many institutions and universities have been accused of modifying or making up data to support their desired results, after detailed analysis determined that the reported numbers were inconsistent is subtle ways. <span class="Apple-converted-space"> </span>Or maybe you haven’t heard about this— the major media have been sadly deficient in paying attention to these stories. <span class="Apple-converted-space"> </span>Most amusingly, earlier this year, such allegations were made against famous Harvard professor Francesca Gino, who has written endless papers on honesty and ethics.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Usually these allegations come examining past papers and their associated datasets, and performing various statistical tests to figure out if the numbers have a very low probability of being reasonable. <span class="Apple-converted-space"> </span>For example, in an earlier podcast we discussed Benford’s Law, a subtle property of leading digits in large datasets, which has been known for many years now. <span class="Apple-converted-space"> </span>But all these complex tests have overlooked a basic property of data that, in hindsight, seems so simple a grade-school child could have invented it. <span class="Apple-converted-space"> </span>Finally published in 2016 by Brown and Heathers, this is known as the “granularity-related inconsistency of means” test, or GRIM test for short.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Here’s the basic idea behind the GRIM test. <span class="Apple-converted-space"> </span>If you are averaging a bunch of numbers, there are only certain values possible in the final digits, based on the value you are dividing into the sum of the numbers.<span class="Apple-converted-space"> </span>For example, suppose I tell you I asked 10 people to rate Math Mutation on a scale of 1-100, with only whole number values allowed, no decimals. <span class="Apple-converted-space"> </span>Then I report the average result as “95.337”, indicating the incredible public appreciation of my podcast. <span class="Apple-converted-space"> </span>Sounds great, doesn’t it?<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">But if you think about it, something is fishy here. <span class="Apple-converted-space"> </span>I supposedly got some integer total from those 10 people, and divided it by 10, and got 95.337. <span class="Apple-converted-space"> </span>Exactly what is the integer you can divide by 10 to get 95.337?<span class="Apple-converted-space"> </span>Of course there is none— there should be at most one digit past the decimal when you divide by 10! <span class="Apple-converted-space"> </span>For other numbers, there are wider selections of decimals possible; but in general, if you know you got a bunch of whole numbers and divided by a specific whole number, you can determine the possible averages. <span class="Apple-converted-space"> </span>That’s the GRIM test— checking if the digits in an average (or similar calculation) are consistent with the claimed data. <span class="Apple-converted-space"> </span>What’s really cool about this test is that, unlike the many statistical tests that check for low probabilities of given results, the GRIM test is absolute:<span class="Apple-converted-space"> </span>if a paper fails it, there’s a 100% chance that its reported numbers are inconsistent.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Now you would think this issue is so obvious that nobody would be dumb enough to publish results that fail the GRIM test. <span class="Apple-converted-space"> </span>But there you would be wrong:<span class="Apple-converted-space"> </span>when they first published about this test, Brown and Heathers applied it to 71 recent papers from major psychology journals, and 36 showed at least one GRIM failure. <span class="Apple-converted-space"> </span>The GRIM test also played a major role in exposing problems with the famous “pizza studies” at Cornell’s Food and Brand Lab, which claimed to discover surprising relationships between variables such as the price of pizza, size of slices, male or female accompaniment, and the amount eaten. <span class="Apple-converted-space"> </span>Sounds like a silly topic, but this research had real-world effects, leading to lab director Brian Wansink’s appointment to major USDA committees & helping to shape US military “healthy eating” programs.<span class="Apple-converted-space"> </span>Wansink ended up retracting 15 papers, though insisting that all the issues were honest mistakes or sloppiness rather than fraud. <span class="Apple-converted-space"> </span>Tragically, humanity then had to revert to our primitive 20th-century understanding of the nature of pizza consumption.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Brown and Heathers are careful to point out that a GRIM test failure doesn’t necessarily indicate fraud. <span class="Apple-converted-space"> </span>Perhaps the descriptions of research methods in a particular paper ignore some detail, like a certain number of participants being disqualified for some legitimate reason and reducing the actual divisor, that would change the GRIM analysis. <span class="Apple-converted-space"> </span>In other cases the authors have offered excuses like mistakes by low-paid assistants, simple accidents with the notes, typos, or other similar boo-boos short of intentional fraud.<span class="Apple-converted-space"> </span>But the whole point of scientific papers is to convincingly describe the results in a way that others could reproduce it— so I don’t think these explanations fully let the authors off the hook. <span class="Apple-converted-space"> </span>And these tests don’t even include the many cases of uncooperative authors who refuse to send researchers like Brown and Heathers their original data. <span class="Apple-converted-space"> </span>Thus it seems clear that the large number of GRIM failures, and retracted papers as a result of this and similar tests, indicate a serious problem with the way research is coordinated, published, and rewarded in modern academia.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/GRIM_test">https://en.wikipedia.org/wiki/GRIM_test</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://jamesheathers.medium.com/the-grim-test-a-method-for-evaluating-published-research-9a4e5f05e870">https://jamesheathers.medium.com/the-grim-test-a-method-for-evaluating-published-research-9a4e5f05e870</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://jamesheathers.medium.com/the-grim-test-further-points-follow-ups-and-future-directions-afd55ff67bb0#.vmgjvdvkf">https://jamesheathers.medium.com/the-grim-test-further-points-follow-ups-and-future-directions-afd55ff67bb0#.vmgjvdvkf</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.npr.org/2023/06/26/1184289296/harvard-professor-dishonesty-francesca-gino">https://www.npr.org/2023/06/26/1184289296/harvard-professor-dishonesty-francesca-gino</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Benford%27s_law">https://en.wikipedia.org/wiki/Benford%27s_law</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.vox.com/science-and-health/2018/9/19/17879102/brian-wansink-cornell-food-brand-lab-retractions-jama">https://www.vox.com/science-and-health/2018/9/19/17879102/brian-wansink-cornell-food-brand-lab-retractions-jama</a></span></li></ul><p></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-50431121616456525162023-08-27T07:15:00.000-07:002023-08-27T07:15:13.625-07:00286: Adventures In Translation<p><a href="http://www.erikseligman.com/mm/mm286.mp3"> Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">You may recall that in several earlier podcasts, 252 and 125, I discussed the famous math writer Douglas Hofstadter’s surprising venture into translating poetry from French and Russian into English. <span class="Apple-converted-space"> </span>Hofstadter is best known for his popular books that explore the boundaries of math and cognitive science, such as “Godel Escher Bach” and “Metamagical Themas”. <span class="Apple-converted-space"> </span>I found his discussions of translation challenges fascinating:<span class="Apple-converted-space"> </span>you would think that translation should be a simple mathematical mapping, a 1-1 correspondence of words from one language into another, but it’s actually much more than that. <span class="Apple-converted-space"> </span>Aside from different typical structures and speech patterns between languages, every word sits in a cloud of miscellaneous connotations and relationships with other concepts, so you need to really think about the right way to convey the author’s original intentions. <span class="Apple-converted-space"> </span>This is why tools like Google Translate can give you the general idea of what a passage says, but very rarely create a translated sentence that sounds real and natural. <span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">After reading Hofstadter’s description of his translation work, I thought it might be interesting to translate something myself, though no obvious candidates came to mind, <span class="Apple-converted-space"> </span>But in the past few years, I became acquainted with a recently escaped Cuban dissident, Nelson Rodriguez Chartrand, who had just published a memoir in Spanish. <span class="Apple-converted-space"> </span>Curious to read this memoir, and with no English translation having been planned by the author, I volunteered to take this on myself.<span class="Apple-converted-space"> </span>Now I’ll be the first to admit I’m not fluent in Spanish, though I spent four years studying it in high school, but I had the advantage of direct access to the author to help clarify areas of confusion, as well as the vast general resources of the Internet. <span class="Apple-converted-space"> </span>This also was probably a much easier translation challenge than Hofstadter’s in general, since this memoir was in prose, so no need to worry about things like rhyme and meter.<span class="Apple-converted-space"> </span>I also had some decent knowledge of context, having read numerous memoirs by emigres from Communist countries in the past, as well as interviewing several including Nelson.<span class="Apple-converted-space"> </span>Thus, I went ahead and began translating.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Now, some of you might think this is trivially easy these days due to Google Translate, but as I mentioned before, that tool does not create very good text on its own. <span class="Apple-converted-space"> </span>You may recall the popular amusement when it first came out, where you translate a small passage across several languages and back to English, and chuckle at how ridiculous it ends up looking. <span class="Apple-converted-space"> </span>It is useful, however, as a first step:<span class="Apple-converted-space"> </span>as I approached each paragraph, I used that tool to create a “gloss”, an initial awkward translation to use as a starting point. <span class="Apple-converted-space"> </span>From there, I would try to understand the core concepts being expressed, and try to rewrite the sentences in more natural sounding English. <span class="Apple-converted-space"> </span>I would figure this out with a combination of reviewing the original Spanish text, researching some alternate word translations online, and consulting with Nelson in the harder cases.<span class="Apple-converted-space"> </span>The most entertaining challenges were the cases where the initial version simply didn’t make sense at all.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">One simple example was a children’s cheer, “Fidel, Fidel, que tiene Fidel, que los imperialistas no pueden con él”, which Google literally translates as “Fidel, Fidel, that Fidel has, that the imperialists cannot with him”. <span class="Apple-converted-space"> </span>That doesn’t make much sense initially. <span class="Apple-converted-space"> </span>After consulting a few sources, I settled on the translation “Fidel, Fidel, what Fidel has, the imperialists cannot overcome”, which probably expresses the core intention a bit better. <span class="Apple-converted-space"> </span>It could also be a question, asking “Fidel, Fidel, what does Fidel have, that the imperialists cannot overcome?”<span class="Apple-converted-space"> </span>I toyed with the idea of taking more liberties and trying to make.a rhyming chant like the original Spanish, something like “Fidel, Fidel, what Fidel has, makes the imperialist look like an ass”, similar to some of the liberties Hofstadter describes in his poetry translation efforts. <span class="Apple-converted-space"> </span>But while that might make it a more effective chant, I was hesitant about making the translation’s tone a bit too lighthearted.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">My favorite translation challenges were the ones that made me laugh out loud when first reading the Google translated version. <span class="Apple-converted-space"> </span>For example, when describing what he liked to do in the evenings as a child, Nelson wrote a phrase that Google translated as “watching the American dolls”. <span class="Apple-converted-space"> </span>Was life in Cuba really so boring that people enjoyed sitting and starting at dolls for hours? <span class="Apple-converted-space"> </span>This also conjured up images of those Chucky horror movies— maybe he was trying to make sure the Yankee imperialist dolls didn’t come to life and start chasing poor Communists with a knife? <span class="Apple-converted-space"> </span>Things in Cuba could be worse than I imagined. <span class="Apple-converted-space"> </span>As you would expect, with some help from the author, I eventually figured out what he was trying to say— he was talking about watching cartoons on TV. <span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">So, in the end, was my translation any good? <span class="Apple-converted-space"> </span>Nelson did a basic check by Google Translating each chapter back to Spanish after I delivered the draft— of course not very good Spanish, but at least sufficient to confirm that I got the general idea. <span class="Apple-converted-space"> </span>While I’m sure a professional Spanish-fluent translator would have done better, the most important goal was to make Nelson’s story available in English, and I’m confident we at least achieved that. <span class="Apple-converted-space"> </span>We were able to get the translation published, with some aid from a small foundation called the Liberty Sentinels Fund, and it is now available at Amazon and other online booksellers.<span class="Apple-converted-space"> </span>You can order it using the link in the show notes at <a href="http://mathmutation.com"><span class="s1">mathmutation.com</span></a> or just search for it on Amazon, and judge my translation for yourself.<span class="Apple-converted-space"> </span>The book is called “The Revolution of Promises”, by Nelson Rodriguez Chartrand. <span class="Apple-converted-space"> </span>If you like it, a good review on Amazon would also be helpful.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Douglas_Hofstadter">https://en.wikipedia.org/wiki/Douglas_Hofstadter</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.amazon.com/Revolution-Promises-Reflections-Cuban-Exile-ebook/dp/B0CG11SYWK/ref=sr_1_1">https://www.amazon.com/Revolution-Promises-Reflections-Cuban-Exile-ebook/dp/B0CG11SYWK/ref=sr_1_1</a></span></li></ul><p></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com1tag:blogger.com,1999:blog-672497650672656575.post-3788001848822548022023-06-29T17:27:00.001-07:002023-06-29T17:27:38.909-07:00285: Believe My Proofs Or Else<p><a href="http://www.erikseligman.com/mm/mm285.mp3"> Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Before I start, I’d like to thank listener ‘katenmkate’ for posting an updated review of Math Mutation on Apple Podcasts. <span class="Apple-converted-space"> </span>You’ve probably observed that the pace of new episodes has been slowing a bit, but seeing a nice review always helps to get me motivated!</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Recently I saw an amusing XKCD cartoon online that reminded me of a term I hadn’t heard in a while, “proof by intimidation”. <span class="Apple-converted-space"> </span>In the cartoon, a teacher explains, “The Axiom of Choice allows you to select one element from each set in a collection…. And have it executed as an example to the others.”<span class="Apple-converted-space"> </span>The caption underneath reads “My math teacher was a big believer in proof by intimidation.”<span class="Apple-converted-space"> </span>A nicely absurd joke— wouldn’t it be great if we could just intimidate the numbers and variables into making our proofs work out?<span class="Apple-converted-space"> </span>“Proof by intimidation” is a real term though. <span class="Apple-converted-space"> </span>To me, it conjures up an image of Tony Soprano knocking at your door, baseball bat in hand, telling you to deliver your proofs or face an unfortunate accident. <span class="Apple-converted-space"> </span>But according to the Wikipedia page, the term arose after some lectures in the 1930s by a mathematician named William Feller:<span class="Apple-converted-space"> </span>“He took umbrage when someone interrupted his lecturing by pointing out some glaring mistake. He became red in the face and raised his voice, often to full shouting range. It was reported that on occasion he had asked the objector to leave the classroom.”</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">These certainly match with the image that the term itself tends to conjure initially in our brains, but the real meaning is slightly less extreme. <span class="Apple-converted-space"> </span>As Wikipedia explains it, proof by intimidation “is a <a href="https://en.wikipedia.org/wiki/Jocular"><span class="s1" style="color: #274ec0;">jocular</span></a> phrase used mainly in <a href="https://en.wikipedia.org/wiki/Mathematics"><span class="s1" style="color: #274ec0;">mathematics</span></a> to refer to a specific form of <a href="https://en.wikipedia.org/wiki/Hand-waving"><span class="s1" style="color: #274ec0;">hand-waving</span></a>, whereby one attempts to advance an argument by marking it as obvious or <a href="https://en.wikipedia.org/wiki/Triviality_(mathematics)"><span class="s1" style="color: #274ec0;">trivial</span></a>, or by giving an argument loaded with jargon and obscure results.”<span class="Apple-converted-space"> </span>For example, while skipping a difficult step in proving some complex theorem, you might say, “You know the Zorac Theorem of Hyperbolic Manifold Theory, right?” <span class="Apple-converted-space"> </span>Chances are that the listener doesn’t know that theorem, and is too embarrassed about it to try to bring up & expose your missing argument.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">As you would expect, that type of argument certainly makes a proof invalid in many cases. <span class="Apple-converted-space"> </span>But there is one interesting wrinkle that most of the online explanations skip over.<span class="Apple-converted-space"> </span>Often, stating that something is obvious, or trivially follows from well-known results, is a very important part of a valid mathematical argument or paper <span class="Apple-converted-space"> </span>It simply wouldn’t be feasible to publish modern math, science, or engineering results (or at least, to publish them in an article of reasonable length) if you could not use the building blocks provided by past researchers without re-creating them.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">I remember when I started my first advanced math class as a freshman in college, the TA gave us some guidelines for what he expected in our homework, and one of them was that “This is obvious” or “This trivially follows” ARE perfectly acceptable— as long as they are accurate. <span class="Apple-converted-space"> </span>Apparently his burdens of grading homework were significantly complicated in the past by students who felt they had to justify the foundations of algebra in every answer. <span class="Apple-converted-space"> </span>For example, if for some step in a proof you need to say, “Let y be a prime number greater than x”, you don’t need to embed a copy of Euler’s proof of the existence of arbitrarily large prime numbers. <span class="Apple-converted-space"> </span>However, encouraging students to use the word “obvious” in our homework did leave him open to bluff attempts through proof by intimidation— he had to keep on his toes to make sure we really were only using it for trivial steps, rather than actually cutting larger corners in our proofs. <span class="Apple-converted-space"> </span>In one assignment that I was a bit rushed for, I actually did say “And obviously, …” while skipping over the hardest part of a problem. <span class="Apple-converted-space"> </span>The TA, giving me 0 points, commented, “If this was really obvious, we wouldn’t have bothered to assign the problem in your homework!”</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Proof by intimidation is just one representative of a family of related logical fallacies. <span class="Apple-converted-space"> </span>A humorous list by someone named Dana Angluin has been circulating the net for a while, with methods including “proof by vigorous handwaving”, “proof by cumbersome notation”, “proof by exhaustion” (meaning making the reader tired, not the mathematically valid proof by exhaustion method), “proof by obfuscation”, “proof by eminent authority”, “proof by importance”, “proof by appeal to intuition”, “proof by reduction to the wrong problem”, and of course the classic “proof by reference to inaccessible literature”. <span class="Apple-converted-space"> </span>As I read these, I think a lot of them seem to be common these days outside the domain of mathematics…. Though since this podcast isn’t focused on theology, media or politics, I’ll stay out of that quagmire.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="text-decoration-line: underline;"><a href="https://mitadmissions.org/blogs/entry/mathematics-for-computer-science-top-10-proof-techniques-not-allowed/">https://mitadmissions.org/blogs/entry/mathematics-for-computer-science-top-10-proof-techniques-not-allowed/</a></span></li><li><span class="s2" style="text-decoration-line: underline;"><a href="https://proftomcrick.com/2011/11/25/proof-by-intimidation/">https://proftomcrick.com/2011/11/25/proof-by-intimidation/</a></span></li><li><span class="s2" style="text-decoration-line: underline;"><a href="https://users.cs.northwestern.edu/~riesbeck/proofs.html">https://users.cs.northwestern.edu/~riesbeck/proofs.html</a></span></li><li><span class="s2" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Proof_by_intimidation">https://en.wikipedia.org/wiki/Proof_by_intimidation</a></span></li></ul><p></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com1tag:blogger.com,1999:blog-672497650672656575.post-48573092240864971422023-04-30T18:41:00.000-07:002023-04-30T18:41:02.697-07:00284: Don't Square Numbers, Triangle Them<p><a href="http://www.erikseligman.com/mm/mm284.mp3"> Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">As I mentioned in the last episode, recently I read an unusual book, “A Fuller Explanation”, that attempts to explain the mathematical philosophy and ideas of Buckminster Fuller.<span class="Apple-converted-space"> </span>You probably know Fuller’s name due to his popularization of geodesic domes, those sphere-like structures built of a large number of connected triangles, which evenly distribute forces to make the dome self-supporting. <span class="Apple-converted-space"> </span>His unique approach to mathematics, “synergetics”, probably didn’t make enough true mathematical contributions to justify all the new terminology he created, but does look at many conventional ideas in different ways and through unusual philosophical lenses, which likely helped him to derive new and original ideas in architecture. <span class="Apple-converted-space"> </span>Today we’ll discuss another of his key points, his central focus on triangles as a key foundation of mathematics.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">While most people who are conventionally educated tend to primarily think of geometry in terms of right angles and square coordinate systems, Fuller considered triangles to be a more fundamental element. <span class="Apple-converted-space"> </span>This follows directly from their essential ability to distribute forces without deforming. <span class="Apple-converted-space"> </span>Think about it for a minute:<span class="Apple-converted-space"> </span>if you apply pressure to the corner of a 4-sided figure, such as a square, its angles can be distorted to something different without changing the lengths of the sides: <span class="Apple-converted-space"> </span>for example, you can squish it into a parallelogram. <span class="Apple-converted-space"> </span>Triangles don’t have this flaw:<span class="Apple-converted-space"> </span>assuming the sides are rigid and well-fastened together, you can’t distort it at all. <span class="Apple-converted-space"> </span>This stems from the elementary geometric fact that the lengths of the three sides fully determine the triangle:<span class="Apple-converted-space"> </span>for any 3 side lengths, there is only one possible set of angles to go with them and make a triangle.<span class="Apple-converted-space"> </span>Thus, if you are building something in the real world, triangles and related 3-dimensional shapes naturally result in strong self-supporting structures.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">According to one of the many legends of Fuller’s life, he first discovered the strength of this type of construction when he was a child in kindergarten. <span class="Apple-converted-space"> </span>His teacher had given the class an exercise where they were to build a small model house out of toothpicks and semi-dried peas.<span class="Apple-converted-space"> </span>At the time, Fuller’s parents had not yet discovered his need for glasses, so he was nearly blind and could not see what his classmates were doing at all. <span class="Apple-converted-space"> </span>As he recalled it, “All the other kids, the minute they were told to make structures, immediately tried to imitate houses. <span class="Apple-converted-space"> </span>I couldn’t see, so I felt. <span class="Apple-converted-space"> </span>And a triangle felt great!”<span class="Apple-converted-space"> </span>Basing his decisions on how the strength of the building felt as he was constructing it, he ended up developing a triangle-based complex of tetrahedra and octahedra. <span class="Apple-converted-space"> </span>The unusual look accompanied by the unexpected solidity and strength of his little house surprised his teachers and classmates, and many years later was patented by Fuller as the “Octet Truss”.<span class="Apple-converted-space"> </span>Fuller wasn’t the first architect to discover the strength of triangles, of course, but he put a lot more thought into extending them to useful 3-D structures than many had in the past. <span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">With the strong memory of such early discoveries, it’s probably not surprising that Fuller decided he wanted to build an entire mathematical system based on triangles. <span class="Apple-converted-space"> </span>He took this philosophy into all areas of mathematics that he looked at, sometimes to a bit of an extreme. <span class="Apple-converted-space"> </span>For example, a fundamental building block of algebra is looking at higher powers of numbers, starting with squaring. <span class="Apple-converted-space"> </span>But Fuller considered the operation of “triangling” much more important.<span class="Apple-converted-space"> </span>Of course this is a useful aspect of general algebra:<span class="Apple-converted-space"> </span>you might recall that the triangular numbers represent the number of units it takes to form an equilateral triangle:<span class="Apple-converted-space"> </span>start with 1 ball, add a row of 2 under it, add a row of 3 under that, etc. <span class="Apple-converted-space"> </span>So the series of triangular numbers start with<span class="Apple-converted-space"> </span>1, 3, 6, 10, and so on, representing the value ((n squared - n) / 2) for each n. <span class="Apple-converted-space"> </span>Fuller considered these numbers especially significant because they also happen to represent the set of pairwise relationships between n items. <span class="Apple-converted-space"> </span>For example, if you have 4 people and want to connect phone lines so any 2 can always talk to each other, you need 6 phone lines, the 3rd triangular number. <span class="Apple-converted-space"> </span>For 5 people you need 10, the 4th triangular number, etc. <span class="Apple-converted-space"> </span>Fuller was pleased that this abstract concept of the number of mutual relationships had such a simple geometric counterpart. <span class="Apple-converted-space"> </span>This observation helped to guide him in many of his further explorations related to constructing strong 3-dimensional patterns that could be used in construction.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">So, are you convinced yet that triangles are important? <span class="Apple-converted-space"> </span>Of course these have been a fundamental element of mathematics for thousands of years, but it’s been relatively rare for people to view them and think about them exactly the way Fuller did. <span class="Apple-converted-space"> </span>That’s probably why, despite exploring areas of mathematics that had been relatively well-covered in the past, he still was able to make original and useful observations that led to significant practical results. <span class="Apple-converted-space"> </span>It’s a nice reminder that just because other people have already examined some area of mathematics, it doesn’t mean that there aren’t more fascinating discoveries waiting just under the surface to be uncovered.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Buckminster_Fuller">https://en.wikipedia.org/wiki/Buckminster_Fuller</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.amazon.com/Fuller-Explanation-Buckminster-Back-Action-ebook/dp/B002YQ2X5S">https://www.amazon.com/Fuller-Explanation-Buckminster-Back-Action-ebook/dp/B002YQ2X5S</a></span></li></ul><p></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 11px; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-23459056909709770732023-02-26T14:29:00.000-08:002023-02-26T14:29:18.776-08:00283: Escape From Infinity<p> <a href="http://www.erikseligman.com/mm/mm283.mp3">Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">Recently I read an interesting book, “A Fuller Explanation”, that attempts to explain the mathematical philosophy and ideas of Buckminster Fuller, written by a former student and colleague of his named Amy Edmondson.<span class="Apple-converted-space"> </span>Fuller is probably best known as the architect who popularized geodesic domes, those sphere-like structures built of a large number of connected triangles, which evenly distribute forces to make the dome self-supporting. <span class="Apple-converted-space"> </span>He had a unique approach to mathematics, defining a system of math, physics, and philosophy called “synergetics”, which coined a lot of new terms for what was essentially the geometry of 3-dimensional solids. <span class="Apple-converted-space"> </span>It’s a very difficult read, which is what led to the need for Edmonson’s book.</span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">One of the basic insights guiding Fuller was the distrust of the casual use of infinities and infinitesimals throughout standard mathematics. <span class="Apple-converted-space"> </span>He would not accept the definition of infinitely small points or infinitely thin lines, as these cannot exist in the physical world.<span class="Apple-converted-space"> </span>He was also bothered by the infinite digits of pi, which are needed to understand a perfect sphere. <span class="Apple-converted-space"> </span>Pi isn’t equal to 3.14, or 3.142, or 3.14159, but keeps needing more digits forever. <span class="Apple-converted-space"> </span>If a soap bubble is spherical, how does nature know when to stop expanding the digits?<span class="Apple-converted-space"> </span>Of course, in real life, we know a soap bubble isn’t truly spherical, as it’s made of a finite number of connected atoms.<span class="Apple-converted-space"> </span>Or as Fuller would term it, it’s a “mesh of energy events interrelated by a network of tiny vectors.” <span class="Apple-converted-space"> </span>But can you really do mathematics without accepting the idea of infinity?</span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">With a bit of googling, I was surprised to find that Fuller was not alone in his distrust of infinity: <span class="Apple-converted-space"> </span>there’s a school of mathematics, known as “finitism”, that does not accept any reasoning that involves the existence of infinite quantities.<span class="Apple-converted-space"> </span>At first, you might think this rules out many types of mathematics you learned in school. <span class="Apple-converted-space"> </span>Don’t we know that the infinite series 1/2 + 1/4 + 1/8 … sums up to 1? <span class="Apple-converted-space"> </span>And don’t we measure the areas of curves in calculus by adding up infinite numbers of infinitesimals?<span class="Apple-converted-space"> </span></span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">Actually, we don’t depend on infinity for these basic concepts as much as you might think. <span class="Apple-converted-space"> </span>If you look at the precise definitions used in many areas of modern mathematics, we are talking about limits— the convenient description of these things as infinites is really a shortcut. <span class="Apple-converted-space"> </span>For example, what are we really saying when we claim the infinite series 1/2+1/4+1/8+… adds up to 1?<span class="Apple-converted-space"> </span>What we are saying is that if you want to get a value close to 1 within any given margin, I can tell you a number of terms to add that will get you there. <span class="Apple-converted-space"> </span>If you want to get within 25% of 1, add the first two terms, 1/2+1/4. <span class="Apple-converted-space"> </span>If you want to get within 15%, you need to add the three terms 1/2+1/4+1/8.<span class="Apple-converted-space"> </span>And so on. <span class="Apple-converted-space"> </span>(This is essentially the famous “epsilon-delta” proof method you may remember from high school.). <span class="Apple-converted-space"> </span>Thus we are never really adding infinite sums, the ‘1’ is just a marker indicating a value we can get arbitrarily close to by adding enough terms.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">You might object that this doesn’t cover certain other critical uses of infinity:<span class="Apple-converted-space"> </span>for example, how would we handle Zeno’s paradoxes?<span class="Apple-converted-space"> </span>You may recall that one of Zeno’s classic paradoxes says that to cross a street, we must first go 1/2 way across, then 1/4 of the distance, and so on, adding to an infinite number of tasks to do. <span class="Apple-converted-space"> </span>Since we can’t do an infinite number of tasks in real life, motion is impossible.<span class="Apple-converted-space"> </span>The traditional way to resolve it is by saying that this infinite series 1/2+1/4+… adds up to 1.<span class="Apple-converted-space"> </span>But if a finitist says we can’t do an infinite number of things, are we stuck? <span class="Apple-converted-space"> </span>Actually no— since the finitist also denies infinitesimal quantities, there is some lower limit to how much we can subdivide the distance. <span class="Apple-converted-space"> </span>After a certain amount of dividing by 2, we reach some minimum allowable distance. <span class="Apple-converted-space"> </span>This is not as outlandish as it seems, since physics does give us the Planck Length, about 10^-35 meters, which some interpret as the pixel size of the universe. <span class="Apple-converted-space"> </span>If we have to stop dividing by 2 at some point, Zeno’s issue goes away, as we are now adding a finite (but very large) number of tiny steps which take<span class="Apple-converted-space"> </span>a correspondingly tiny time each to complete.<span class="Apple-converted-space"> </span>Calculating distances using the sum of an infinite series again becomes<span class="Apple-converted-space"> </span>just a limit-based approximation, not a true use of infinity.</span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">Thus, you can perform most practical real-world mathematics without a true dependence on infinity.<span class="Apple-converted-space"> </span>One place where finitists do diverge from the rest of mathematics is in reasoning about infinity itself, or the various types of infinities.<span class="Apple-converted-space"> </span>You may recall, for example, Cantor’s ‘diagonal’ argument which proves that the infinity of real numbers is greater than the infinity of integers, leading to the idea of a hierarchy of infinities. <span class="Apple-converted-space"> </span>A finitist would consider this argument pointless, having no applicability to real life, even if it does logically follow from Cantor’s premises.</span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">In Fuller’s case, this refusal to accept infinities had some positive results.<span class="Apple-converted-space"> </span>By focusing his attention on viewing spheres as a mesh of finite elements and balanced force vectors, this probably set him on the path of understanding geodesic domes, which became his major architectural accomplishment. <span class="Apple-converted-space"> </span>As Edmondson describes it, Fuller’s alternate ways of looking at standard areas of mathematics enabled him and his followers to circumvent previous rigid assumptions and open “rusty mental gates that block discovery”.<span class="Apple-converted-space"> </span>Fuller’s views were also full of other odd mathematical quirks, such as the idea that we should be “triangling” instead of “squaring” numbers when wanting to look at higher dimensions; maybe we will discuss those in a future episode.</span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span style="font-size: medium;"><br /></span></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-size: medium;">And this has been your math mutation for today.</span></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Finitism">https://en.wikipedia.org/wiki/Finitism</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Buckminster_Fuller">https://en.wikipedia.org/wiki/Buckminster_Fuller</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.amazon.com/Fuller-Explanation-Buckminster-Back-Action-ebook/dp/B002YQ2X5S">https://www.amazon.com/Fuller-Explanation-Buckminster-Back-Action-ebook/dp/B002YQ2X5S</a></span></li></ul><p></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com1tag:blogger.com,1999:blog-672497650672656575.post-61529464590563811442022-12-20T14:52:00.003-08:002022-12-20T14:52:47.894-08:00282: The Man With The Map<p><a href="http://www.erikseligman.com/mm/mm282.mp3"> Audio Link </a></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">I was surprised to read about the recent passing of Maurice Karnaugh, a pioneering mathematician and engineer from the early days of computing. <span class="Apple-converted-space"> </span>Karnaugh originally earned his PhD from Yale, and went on to a distinguished career at Bell Labs and IBM, also becoming an IEEE Fellow in 1976. <span class="Apple-converted-space"> </span>My surprise came from the fact that he was still alive so recently:<span class="Apple-converted-space"> </span>he was born in 1924, and his key contributions were in the 1950s and 1960s, so I had assumed he died years ago. <span class="Apple-converted-space"> </span>In any case, to honor his memory, I thought it might be fun to look at one of his key contributions:<span class="Apple-converted-space"> </span>the Karnaugh Map, known to generations of engineering students as the K-map for short.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">So, what is a K-map? <span class="Apple-converted-space"> </span>Basically, it’s a way of depicting the definition of a Boolean function, that is a function that takes a bunch of inputs and generates an output, with all inputs and the output being a Boolean value, that is either 0 or 1.<span class="Apple-converted-space"> </span>As you probably know, such functions are fundamental to the design of computer chips and related devices. <span class="Apple-converted-space"> </span>When trying to design an electronic circuit schematic that implements such a function, you usually want to try to find a minimum set of basic logic gates, primarily AND, OR, and NOT gates, that defines it.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">For example, suppose you have a function that takes 4 inputs, A, B, C, and D, and outputs a 1 only if both A and B are true, or both C and D are true. <span class="Apple-converted-space"> </span>You can basically implement this with 3 gates: <span class="Apple-converted-space"> </span>(A AND B) , (C AND D), and an OR gate to look at those two results, outputting a 1 if either succeeded. <span class="Apple-converted-space"> </span>But often when defining such a function, you’re initially given a truth table, a table that lists every possible combination of inputs and the resulting output. <span class="Apple-converted-space"> </span>With 4 variables, the truth table would have 2^4, or 16, rows, 7 of which show an output of 1. <span class="Apple-converted-space"> </span>A naive translation of such a truth table directly to a circuit would result in one or more gates for every row of the table, so by default you would generate a much larger circuit than necessary. <span class="Apple-converted-space"> </span>The cool thing about a K-map is that even though it’s mathematically trivial— it actually just rewrites the 2^n lines of the truth table in a 2^(n/2) x 2^(n^2) square format— it makes a major difference in enabling humans to draw efficient schematics.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">So how did Karnaugh help here? <span class="Apple-converted-space"> </span>The key insight of the K-map is to define a different shape for the truth table, one that conveys the same information, but in a way that the human eye can easily find a near-minimal set of gates that would implement the desired circuit. <span class="Apple-converted-space"> </span>First, we make the table two-dimensional, by grouping half the variables for the rows, and half for the columns. <span class="Apple-converted-space"> </span>So there would be one row for AB = 00,<span class="Apple-converted-space"> </span>one row for AB = 01, etc, and a column for CD=00, another for CD=01, etc. <span class="Apple-converted-space"> </span>This doesn’t actually change the amount of information:<span class="Apple-converted-space"> </span>for each row in the original truth table, there is now a (row, column) pair, leading to a corresponding entry in the two-dimensional K-map. <span class="Apple-converted-space"> </span>Instead of 16 rows, we now have 4 rows and 4 columns, specifying the outputs for the same 16 input combinations.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">The second clever trick is to order each set of rows and columns according to a Gray code— that is, an ordering such that each pair of inputs only differs from the previous pair in one bit. <span class="Apple-converted-space"> </span>So rather than the conventional numerical ordering of 00, 01, 10, 11, corresponding to our ordinary base-10 of 0, 1, 2, 3 in order, we sort the rows as 00, 01, 11, 10. <span class="Apple-converted-space"> </span>These are out of order, but the fact that only one bit is changing at a time makes the combinations more convenient to visually analyze.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">One you have created this two-dimensional truth table with the gray code ordering, it has the very nice property that if you can spot rectangular patterns, they correspond to boolean expressions, or minterms, that enable an efficient representation of the function in terms of logic gates.<span class="Apple-converted-space"> </span>In our example, we would see that the row for AB=11 contains a 4x1 rectangle of 1s, and the column of CD=11 contains a 1x4 rectangle of 1s, leading us directly to the (A AND B) OR (C AND D) solution.<span class="Apple-converted-space"> </span>Of course, the details are a bit messy to convey in an audio podcast, but you can see more involved illustrations in online sources like the Wikipedia page in the show notes. <span class="Apple-converted-space"> </span>But the most important point is that in this two-dimensional truth table, you can now generate a minimal-gate representation by spotting rectangles of 1s, greatly enhancing the efficiency of your circuit designs.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">Over the years, as with many things in computer science, K-maps have faded in significance.<span class="Apple-converted-space"> </span>This is because the power of our electronic design software has grown exponentially:<span class="Apple-converted-space"> </span>these days, virtually nobody hand-draws a K-map to minimize a circuit.<span class="Apple-converted-space"> </span>Circuit synthesis software directly looks at high-level definitions like truth tables, and does a much better job at coming up with minimal gate implementations than any person could do by hand. <span class="Apple-converted-space"> </span>Some of the techniques used by this software relate to K-maps, but of course many more complex algorithms, most of which could not be effectively executed without a computer, have been developed in the intervening decades. <span class="Apple-converted-space"> </span>Despite this, Karnaugh’s contribution was a critical enabler in the early days of computer chip design, and the K-map is still remembered by generations of mathematics and computer science students.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Karnaugh_map">https://en.wikipedia.org/wiki/Karnaugh_map</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Maurice_Karnaugh">https://en.wikipedia.org/wiki/Maurice_Karnaugh</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.quora.com/What-is-the-practical-use-of-a-Karnaugh-map">https://www.quora.com/What-is-the-practical-use-of-a-Karnaugh-map</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.eetimes.com/karnaugh-maps-101/">https://www.eetimes.com/karnaugh-maps-101/</a></span></li></ul><p></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-30035443502173209162022-10-30T16:40:00.001-07:002022-10-30T16:40:08.179-07:00281: Pascal Vs Mathematics<p><a href="http://www.erikseligman.com/mm/mm281.mp3"> Audio Link</a></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">If you have enough interest in math to listen to this podcast, I’m pretty sure you’ll recognize the name of Blaise Pascal, the 17th-century French mathematician and physicist. <span class="Apple-converted-space"> </span>Among other achievements, he created Pascal’s Triangle, helped found probability theory, invented and manufactured the first major mechanical calculator, and made essential contributions to the development of fluid mechanics. <span class="Apple-converted-space"> </span>His name was eventually immortalized in the form of a computer language, a unit of pressure, a university in France, and an otter in the Animal Crossing videogame, among other things.. <span class="Apple-converted-space"> </span>But did you know that in the final decade of his life, he essentially renounced the study of mathematics to concentrate on philosophy and theology?<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">According to notes found after his death in 1662, Pascal had some kind of sudden religious experience in 1654, when he went into a trance for two hours, during which time he claims that God gave him some new insights on life. <span class="Apple-converted-space"> </span>At that point he dropped most of his friends and sold most of his possessions to give money to the poor, leaving himself barely able to afford food. <span class="Apple-converted-space"> </span>He also decided that comfort and happiness were immoral distractions, so started wearing an iron belt with interior spikes. <span class="Apple-converted-space"> </span>And, most disturbingly, he decided that math and science were no longer worthy of study, so he would devote all his time to religious philosophy. <span class="Apple-converted-space"> </span>He was still a very original and productive thinker, however, as in this period he wrote his great philosophical work known as the Pensees.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">There were a couple of reasons why he may have decided to give up on math and physics at this point.<span class="Apple-converted-space"> </span>Part of it was certainly just a change in emphasis:<span class="Apple-converted-space"> </span>he was concentrating on something else now, which he considered more important. <span class="Apple-converted-space"> </span>He also made comments about worldly studies being used to feed human egos, which means nothing in the eyes of God. <span class="Apple-converted-space"> </span>At one point he stated that he could barely remember what geometry was.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">He never completely suppressed his earlier love of mathematics though. <span class="Apple-converted-space"> </span>Ironically, at several points in the Pensees he uses clearly mathematical ideas.<span class="Apple-converted-space"> </span>Most famously, you may have heard of “Pascal’s Wager”, where he discusses the expected returns of believing in God vs not believing, based on his ideas of probability theory. <span class="Apple-converted-space"> </span>You have two choices, to believe or not believe. <span class="Apple-converted-space"> </span>If you choose to believe, you may suffer a finite net loss from time spent on religion, if God doesn’t exist— but if he does, you have an infinite payoff. <span class="Apple-converted-space"> </span>Choosing not to believe offers at best the savings from that lifetime loss. <span class="Apple-converted-space"> </span>Thus, the rational choice to maximize your expected gain is to believe in and worship God.</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">As many have pointed out since then, there is at least one huge hole in Pascal’s argument: <span class="Apple-converted-space"> </span>what if you choose to believe in the wrong God, or worship him in the wrong way? <span class="Apple-converted-space"> </span>Many world religions consider heresy significantly worse than nonbelief. <span class="Apple-converted-space"> </span>He has an implicit assumption of Christianity, in the form he knows, being the only option other than agnosticism or atheism. <span class="Apple-converted-space"> </span>I think Homer Simpson once refuted Pascal’s Wager effectively, when trying to get out of going to church with his wife:<span class="Apple-converted-space"> </span>“But Marge, what if we chose the wrong religion? Each week we just make God madder and madder.”</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">Another surprising use of math in the Pensees is Pascal’s comments on why the study of math and science may be pointless in general. <span class="Apple-converted-space"> </span>He compares the finite knowledge that man may gain by these studies against the infinite knowledge of God: <span class="Apple-converted-space"> </span>“… what matters it that man should have a little more knowledge of the universe?<span class="Apple-converted-space"> </span>If he has it, he but gets a little higher.<span class="Apple-converted-space"> </span>Is he not always infinitely removed from the end…? <span class="Apple-converted-space"> </span>In comparison with these Infinites all finites are equal, and I see no reason for fixing our imagination on one more than on another.” <span class="Apple-converted-space"> </span>While he doesn’t write any equations here, the ideas clearly have a basis in his previous studies related to finite and infinite values. <span class="Apple-converted-space"> </span>We could even consider this a self-contradiction:<span class="Apple-converted-space"> </span>wouldn’t the fact that his math just gave him some theological insight mean that it was, in fact, worthy of study to get closer to God?</p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">Pascal did also still engage a few times during this period in direct mathematical studies. <span class="Apple-converted-space"> </span>Most notably, in 1658, he started speculating on some properties of the cycloid, the curve traced by a point on a moving circle, to distract himself while suffering form a toothache. <span class="Apple-converted-space"> </span>When his ache got better, he took this as a sign from God to continue his work on this topic. <span class="Apple-converted-space"> </span>That excuse seems a bit thin to me:<span class="Apple-converted-space"> </span>clearly he never lost his inbuilt love for mathematics, even when he felt his theological speculations were pulling him in another direction.<span class="Apple-converted-space"> </span></p><p class="p2" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">And this has been your math mutation for today.</p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p><p class="p1" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">References: <span class="Apple-converted-space"> </span></p><p class="p1" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s1" style="text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Blaise_Pascal">https://en.wikipedia.org/wiki/Blaise_Pascal</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.amazon.com/Drunkards-Walk-Randomness-Rules-Lives/dp/0307275175">https://www.amazon.com/Drunkards-Walk-Randomness-Rules-Lives/dp/0307275175</a></span></li><li><span class="s1" style="text-decoration-line: underline;"><a href="https://www.gutenberg.org/ebooks/18269">https://www.gutenberg.org/ebooks/18269</a></span></li></ul><p></p><p class="p2" style="font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-28085648184549431512022-08-26T16:13:00.000-07:002022-08-26T16:13:08.440-07:00280: Rubik's Resurgence<p><a href="http://www.erikseligman.com/mm/mm280.mp3">Audio Link </a></p><p><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Do you remember the Rubik’s Cube, that 3x3x3 cube puzzle of multicolored squares, with each side able to rotate independently, that was a fad in the 1980s? <span class="Apple-converted-space"> </span>The goal was to take a cube that has been scrambled by someone else with a few rotations, and get it back to a configuration where all squares on each side match in color. <span class="Apple-converted-space"> </span>Originally developed by Hungarian architecture professor Erno Rubik in 1975, he had initially intended it to model how a structure can be designed with parts that move independently yet still hold together. <span class="Apple-converted-space"> </span>The legend is that after he demonstrated the independent motion of a few sides and had trouble rearranging it back to the original configuration, he realized he had an interesting puzzle. <span class="Apple-converted-space"> </span>In the early 1980s, it started winning various awards for best toy or puzzle, and quickly became the best selling toy of all time. <span class="Apple-converted-space"> </span>(A title which it apparently still holds.) <span class="Apple-converted-space"> </span>It was insanely difficult for the average person to solve, though typically with some trial and error you could get one or two sides done, the key to holding people’s interest.<span class="Apple-converted-space"> </span>To get a sense of how popular it was back then, there were cube-solving guidebooks that sold millions of copies, and even a Saturday morning cartoon series about a cube-shaped superhero. <span class="Apple-converted-space"> </span>But by 1983 or so sales were dropping off, and the fad was considered over.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Like everyone else who was alive in the early 80s, I spent some time messing around with cubes, but found it too frustrating after a while, and eventually solved it with the aid of a guide book. <span class="Apple-converted-space"> </span>I remember being impressed by a classmate who swore he hadn’t read any guidebooks, but could take a scrambled cube from me, go work on it in a corner of the room, and come back with it fully solved. <span class="Apple-converted-space"> </span>His secret was to remove the colored stickers from the squares and put them back in the right configuration, without rotating the sides at all. <span class="Apple-converted-space"> </span>But the non-cheating solutions in the guidebooks typically revolve around identifying sequences of moves that can move around known sets of squares while keeping others in their current configuration, then getting the desired cubes in place layer by layer. <span class="Apple-converted-space"> </span>These sequences are tricky in that they appear to be completely scrambling the cube before restoring various parts, which is a key reason why average cubers would fail to discover them— you need to mess up your cube on the way to completing the solution.<span class="Apple-converted-space"> </span>It’s actually been mathematically proven that any reachable configuration can be solved in 20 moves or fewer.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">One aspect of the cube that is always fun to mock is its marketing campaign. <span class="Apple-converted-space"> </span>Typically the cube was sold and advertised with the phrase, “Over 3 Billion Combinations!” <span class="Apple-converted-space"> </span>But if you think about it for a minute, there are a lot more. <span class="Apple-converted-space"> </span>A cube has 8 corner pieces, so you could combine those in 8 factorial (8!) ways, which is 8 * 7 * 6 …<span class="Apple-converted-space"> </span>down to 1. <span class="Apple-converted-space"> </span>And since each of these combinations can have each corner piece in 3 different rotations, you need to multiply by 3 to the 8th power. <span class="Apple-converted-space"> </span>Similarly, the 12 side pieces can be arranged in 12 factorial (12!) ways, then you need to multiply by 2 to the 12th power.<span class="Apple-converted-space"> </span>So to find the total possibilities, we need to multiply 8! * 3^8 * 12! * 2^12.<span class="Apple-converted-space"> </span>It turns out that only 1/12 of these positions are actually reachable from a starting solved state, so we need to divide the result by 12. <span class="Apple-converted-space"> </span>But we still get a total a bit larger than 3 billion:<span class="Apple-converted-space"> </span>4.3 * 10^19. <span class="Apple-converted-space"> </span>So the marketing campaign was underestimating the possibilities by a ridiculous amount— in fact, if you square the 3 billion that they gave, you still don’t quite reach the true number of cube configurations. <span class="Apple-converted-space"> </span>Perhaps they were afraid that the terms for larger numbers, such as the quintillions needed for the true number, would confuse the average customer.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">I was surprised to read recently that Rubik’s Cube actually has actually gained in popularity again in the modern era, fueled by daring YouTubers who speed-solve cubes, solve them with their feet, and perform similar feats. <span class="Apple-converted-space"> </span>Lots of enthusiasts take their cubing very seriously, and there are Rubik’s Cube speed-solving championship events held regularly. <span class="Apple-converted-space"> </span>If you’re a professional-class cuber, you can buy specially made Rubik’s lubricants to enable you to rotate your sides faster.<span class="Apple-converted-space"> </span>The latest championship, this year in Toronto, included standard cube solving plus events on varying size cubes (up to 7x7x7), blindfolded cube solving, and one-handed cube solving. <span class="Apple-converted-space"> </span>(Apparently they eliminated the foot-only solving, though, concerned that it was unsanitary.) Champion Matty Inaba fully solved a 3x3x3 cube in 4.27 seconds, which sounds like about the time it usually takes me to rotate a side or two. <span class="Apple-converted-space"> </span>Author A.J. Jacobs, in “The Puzzler”, also points out that if you’re too intimidated to compete yourself, there are Fantasy Cubing leagues, similar to Fantasy Football, where you can bet on your favorite combination of winners.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">So, what does the future hold for the sport of Rubik’s Cubing? <span class="Apple-converted-space"> </span>Well, even though the big leagues are only competing up to the 7x7x7 level this year, Jacobs tracked down a French inventor who has put together a 33x33x33 one. <span class="Apple-converted-space"> </span>As you would guess, the larger cubes are pretty challenging to build— this one involved over 6000 moving parts and ended up the size of a medicine ball. <span class="Apple-converted-space"> </span>Experienced cubers do say, however, that the basic algorithms for solving are fundamentally the same for all cube sizes. <span class="Apple-converted-space"> </span>Due to the home manufacturing capabilities enabled by modern 3-D printing, one of Jacobs’ interviewees points out that we are in a “golden age of weird twisty puzzles”.<span class="Apple-converted-space"> </span>Hobbyists have invented many Rubik’s like variants that are not perfect cubes, and have numerous asymmetric parts, to create some extra challenge. <span class="Apple-converted-space"> </span>Personally I’m not sure I would ever have the patience to deal with anything beyond a basic 3x3x3 cube, though maybe I’ll look into joining a fantasy league sometime. <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Rubik%27s_Cube">https://en.wikipedia.org/wiki/Rubik%27s_Cube</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.worldcubeassociation.org/competitions/NAC2022">https://www.worldcubeassociation.org/competitions/NAC2022</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://ajjacobs.com/books/the-puzzler/">https://ajjacobs.com/books/the-puzzler/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://web.mit.edu/sp.268/www/rubik.pdf">https://web.mit.edu/sp.268/www/rubik.pdf</a></span></li></ul><p></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-39059586728019901492022-07-02T05:43:00.000-07:002022-07-02T05:43:19.548-07:00279: Improbable Envelopes<p><a href="http://www.erikseligman.com/mm/mm279.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Today we’re going to talk about a well-known paradox that a co-worker recently reminded me about, the Two Envelopes Paradox. <span class="Apple-converted-space"> </span>It’s similar to some others we have discussed in past episodes, such as the Monty Hall paradox, in that a slightly incorrect use of the laws of probability gives an apparent result that isn’t quite correct.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Here’s how the basic paradox goes. <span class="Apple-converted-space"> </span>You are shown two envelopes, and told that one contains twice as much money as the other one, with the choice of envelopes for each amount having been pre-determined by a secret coin flip.<span class="Apple-converted-space"> </span>No information is given as to the exact amount of money at stake. <span class="Apple-converted-space"> </span>You need to choose which envelope you want. <span class="Apple-converted-space"> </span>After you initially point at one, you are told the amount of money in it, and asked, “Would you like to switch to the other one?” <span class="Apple-converted-space"> </span>Since you have been presented no new information about which envelope has more money, it should be obvious that switching makes no difference at this point, as either way you have a 50% chance of having guessed the right one.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">But let’s calculate the expected value of switching envelopes. <span class="Apple-converted-space"> </span>Say the envelope you chose contains D dollars. <span class="Apple-converted-space"> </span>If you stick with your current envelope, the expected amount you will get is simply D.<span class="Apple-converted-space"> </span>The other one contains either 2D or D/2 dollars, with 1/2 probability for each. <span class="Apple-converted-space"> </span>The expected value if you switch is then 1/2*2D + 1/2*D/2, which adds up to (5/4)D.<span class="Apple-converted-space"> </span>Thus your expected winnings if you switch are greater than the D you would gain from your first choice, and you should always switch! <span class="Apple-converted-space"> </span>But this doesn’t make much sense, if you had no additional information. <span class="Apple-converted-space"> </span>Can you spot the flaw in this reasoning?</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The key is to recognize that you’re combining two different dollar amounts in your expected value calculation:<span class="Apple-converted-space"> </span>the D that exists in the case where you initially chose the smaller envelope is different from the D if you chose the larger one. <span class="Apple-converted-space"> </span>The easiest way to see this is if you define another variable, T, the total money in the combination of two envelopes. <span class="Apple-converted-space"> </span>In this case, the larger one contains (2/3)T, and the smaller has (1/3)T. <span class="Apple-converted-space"> </span>Now your expected winnings become the same as the expected value from switching, (1/2)*(2/3)T + (1/2)*(1/3)T, or simply T/2.<span class="Apple-converted-space"> </span>Alternatively, you could have come to the same conclusion by modifying our original reasoning using Bayes’ Theorem, replacing our reuse of D with correct calculations for the conditional values in each envelope.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">But weirdly enough, <a href="http://brilliant.org"><span class="s2" style="font-kerning: none;">brilliant.org</span></a> seems to point out an odd strategy that will let you choose whether to switch with a greater than 50% chance of getting the higher envelope.<span class="Apple-converted-space"> </span>Here’s how it works. <span class="Apple-converted-space"> </span>After you choose your first envelope, choose a random value N using a normal probability distribution. <span class="Apple-converted-space"> </span>This is the common “bell curve”, with the important property that any number on the number line has a probability of being chosen, as the ‘tail’ of the bell asymptotically approaches and never reaches 0. <span class="Apple-converted-space"> </span>So if the center of the normal distribution is at 100, you have the highest probability of choosing a number near 100, but still a small chance of choosing 1000, and a really tiny chance of choosing 10 billion. <span class="Apple-converted-space"> </span>Therefore, if you’ve chosen a number N using this distribution, there is some probability, P, that N is between the dollar values in the two envelopes. <span class="Apple-converted-space"> </span>Now assume the envelope you didn’t choose contains that number N, and choose to switch on that basis:<span class="Apple-converted-space"> </span>if N is greater than the amount in the envelope you chose, you switch, and otherwise keep your original envelope.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Why does using this random number help? <span class="Apple-converted-space"> </span>Well, if your random number N was smaller than both envelopes, then you will always keep your first<span class="Apple-converted-space"> </span>choice, and there is no change to the overall 50-50 chance of winning. <span class="Apple-converted-space"> </span>If it was larger than both, you’ll always switch, and again no change. <span class="Apple-converted-space"> </span>But what if you got lucky, and N was between the two values, which we know can happen with nonzero probability P? <span class="Apple-converted-space"> </span>Then you will make the right choice:<span class="Apple-converted-space"> </span>you will switch if and only if your initial envelope was the smaller one!<span class="Apple-converted-space"> </span>Thus, with probability P, the chance of N being in this lucky interval, you will be guaranteed a win, while the rest of the time you have the original odds. <span class="Apple-converted-space"> </span>So the overall chance of winning is (1/2)*(1-P) + 1*P, which is slightly greater than 1/2.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Something seems fishy about this reasoning, but I can’t spot an obvious error. <span class="Apple-converted-space"> </span>I remember also hearing about this solution from a professor back in grad school, and periodically tried searching the web for a refutation, but so far haven’t found one. <span class="Apple-converted-space"> </span>Of course, with no information about the actual value of P, this edge can be unpredictably small, so is probably of no real value in practical cases.<span class="Apple-converted-space"> </span>There also seems to be a philosophical challenge here:<span class="Apple-converted-space"> </span>How meaningful is an unpredictable bonus to your odds of an unknowable amount? <span class="Apple-converted-space"> </span>I’ll be interested to hear if any of you out there have some more insight into the problem, or pointers to further discussions of this bizarre probability trick.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://brilliant.org/wiki/two-envelope-paradox/">https://brilliant.org/wiki/two-envelope-paradox/</a></span><span class="s1" style="font-kerning: none;"> </span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-1582354613548214412022-05-22T17:24:00.000-07:002022-05-22T17:24:38.479-07:00278: Bicycle Repair Man<p><a href="http://www.erikseligman.com/mm/mm278.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">With the arrival of summer weather in many parts of the U.S., it’s time to think again about various outdoor activities.<span class="Apple-converted-space"> </span>Watching a few bicycles pass by my house the other day brought to mind a famous anecdote about pioneering mathematician and computer scientist Alan Turing. <span class="Apple-converted-space"> </span>As you may recall, Turing was the famous British thinker who not only founded theoretical computer science, but also was the primarily visionary in the project to crack the German Enigma code, a key contribution to the Allied victory in World War II. <span class="Apple-converted-space"> </span>As often happens with such geniuses, his personal life was very odd, though he usually had reasons for whatever he did. <span class="Apple-converted-space"> </span>For example, he was mocked as ridiculously paranoid for chaining his favorite coffee mug to a radiator.<span class="Apple-converted-space"> </span>But there are rumors that years later, a pile of smashed coffee mugs was found near his old office, apparently thrown away by a disgruntled co-worker.<span class="Apple-converted-space"> </span>Another crazy story involves Turing’s efforts to repair his bicycle.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">As the story goes, Turing noticed that every once in a while, the chain on his bike would come loose, and he would have to stop and kick it a certain way to get it back on its track again. <span class="Apple-converted-space"> </span>After a while, he noticed that the intervals at which this was happening seemed kind of regular, so decided to check that theory rigorously. <span class="Apple-converted-space"> </span>He attached a mechanical counter, and started measuring the exact interval at which this problem was occurring.<span class="Apple-converted-space"> </span>It turned out he was right— the intervals were regular. <span class="Apple-converted-space"> </span>The number of clicks between failures was proportional to the product of the number of spokes in his wheel, the number of links in the chain, and the cogs in the bicycle gears. <span class="Apple-converted-space"> </span>Once he discovered that, he took a close look at the components, and soon discovered the root cause: <span class="Apple-converted-space"> </span>there was a particular spoke that was slightly bent, and when this got too close to a particular link in the chain that was also damaged, the chain would be pulled off. <span class="Apple-converted-space"> </span>Armed with this knowledge, he was able to correctly fix the bent spoke and resolve the problem.<span class="Apple-converted-space"> </span>It’s been said that any competent bike repairman would have spotted the issue in a few minutes without bothering with counters and intervals, but the mathematics of that would be pretty boring.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Naturally, as with any anecdote about someone famous, there are some alternate versions of this story. <span class="Apple-converted-space"> </span>My favorite is the one that changes the ending slightly:<span class="Apple-converted-space"> </span>once Turing figured out the formula for when the chain would jump off, he started carefully calculating the intervals as he rode the bike, and stopping to kick it at the exact right times he calculated. <span class="Apple-converted-space"> </span>That’s a fun one, and certainly fits into the stereotypes of Turing’s eccentricity. <span class="Apple-converted-space"> </span>But I do find it a bit hard to believe.<span class="Apple-converted-space"> </span>When riding a bike outdoors, there are lots of variables involved to interrupt your concentration:<span class="Apple-converted-space"> </span>road obstacles, changing inclines, approaching cars, etc. <span class="Apple-converted-space"> </span>Could someone safely riding a bicycle successfully keep a running count of the wheel and chain rotations, over a continuous ride of several miles? <span class="Apple-converted-space"> </span>And in Turing’s case, it was further complicated by the fact that he always wore a gas mask as he rode, to prevent triggering his allergies. <span class="Apple-converted-space"> </span>But the alarm clock he was known to wear around his waist might have helped.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">In honor of this story, there was actually a proposal back in 2015 to name a bicycle bridge in Cambridge, England after Turing. <span class="Apple-converted-space"> </span>I didn’t see any further references to this online, so it looks like it didn’t pass. <span class="Apple-converted-space"> </span>But there’s plenty of non-bicycle-related stuff named after him in the computer science world.<span class="Apple-converted-space"> </span>If you want someone to propose a bicycle bridge in your name, next time your bike breaks down, think about the clever mathematical tricks you might use to diagnose the issue. <span class="Apple-converted-space"> </span>Also, remember to found a branch of mathematics and win a world war.<span class="Apple-converted-space"> </span>Or forget about the bridge, and just take your bike to a competent repair shop.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.washingtonpost.com/archive/1999/06/09/alan-turing/3640bb61-b23d-41df-9f39-d00a6b2e30cc/">https://www.washingtonpost.com/archive/1999/06/09/alan-turing/3640bb61-b23d-41df-9f39-d00a6b2e30cc/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="http://cozybeehive.blogspot.com/2009/09/that-strange-bicyclist-alan-turing.html">http://cozybeehive.blogspot.com/2009/09/that-strange-bicyclist-alan-turing.html</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.mub.eps.manchester.ac.uk/science-engineering/2020/02/20/alan-turing-did-you-know/">https://www.mub.eps.manchester.ac.uk/science-engineering/2020/02/20/alan-turing-did-you-know/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://road.cc/content/news/170689-cambridge-cycle-bridge-should-be-named-after-alan-turing-and-nimbys-should-back">https://road.cc/content/news/170689-cambridge-cycle-bridge-should-be-named-after-alan-turing-and-nimbys-should-back</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com2tag:blogger.com,1999:blog-672497650672656575.post-49646381929278171652022-03-30T15:28:00.002-07:002022-03-30T15:28:37.160-07:00277: Bad Career Advice<p><a href="http://www.erikseligman.com/mm/mm277.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">As I mentioned in the last episode, I’ve recently reread Nassim Nicholas Taleb’s<span class="Apple-converted-space"> </span>classic book “The Black Swan” , about the disproportionate role of unlikely extreme events, the Black Swans, in shaping our lives and our history.<span class="Apple-converted-space"> </span>Today I’d like to discuss another of the intriguing ideas he discusses in the book:<span class="Apple-converted-space"> </span>scalable and unscalable jobs, and which you should choose if starting out your career.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Back when he was in college, Taleb received some advice from a business student:<span class="Apple-converted-space"> </span>choose a scalable, rather than a nonscalable, profession in order to become rich.<span class="Apple-converted-space"> </span>It’s a pretty simple concept:<span class="Apple-converted-space"> </span>a scalable job is one where you are paid for ideas, not hourly labor, and thus can affect many people with a small amount of work. <span class="Apple-converted-space"> </span>This contrasts with nonscalable jobs, where you are directly paid for the labor you perform.<span class="Apple-converted-space"> </span>An example of a scalable job is a corporate CEO, a derivatives trader, or an author: <span class="Apple-converted-space"> </span>in any of these professions, a small amount of work can impact a massive number of people.<span class="Apple-converted-space"> </span>On the opposite end of the spectrum, you can think of cases like a dentist or a chef: <span class="Apple-converted-space"> </span>your services are inherently delivered one-on-one, and your output is essentially dependent on the time you spend. <span class="Apple-converted-space"> </span>Some like to refer to scalable professions as “idea” professions, and nonscalable ones as “labor” professions.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">You can see why scalable work has the potential to earn much more money, since it’s a matter of simple math. <span class="Apple-converted-space"> </span>If your actions affect one or a small number of people, there is fundamentally less money to go around, since whatever you earn must ultimately come from the people you serve.<span class="Apple-converted-space"> </span>If your actions affect millions of people, then there is the capacity to draw in money from many directions; as Taleb phrases it, this can “add zeroes to your output… at little or no extra effort.” <span class="Apple-converted-space"> </span>An idea person does the same amount of work no matter how many people it affects. <span class="Apple-converted-space"> </span>The CEO gives his orders once, the derivatives trader presses the same button regardless of the quantity traded, and the author writes the book once. <span class="Apple-converted-space"> </span>Taleb also glosses over the fact that some professions are kind of in-between:<span class="Apple-converted-space"> </span>for example, a chip design engineer at Intel impacts many millions of customers, though that impact is shared with about 100,000 other co-workers, creating a pretty good income potential, though unlikely to make someone super-rich unless they progress high on the even-more-scalable management ladder. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">So, was the advice correct, to choose a scalable rather than a nonscalable profession? <span class="Apple-converted-space"> </span>Well, it is true that if you go around looking at super-rich people, almost all did it through being in the scalable world.<span class="Apple-converted-space"> </span>But be careful: <span class="Apple-converted-space"> </span>if you make this kind of observation, you are making a fundamental logical fallacy, confusing “A implies B” for “B implies A”.<span class="Apple-converted-space"> </span>If someone is rich, they probably got that way through a scalable profession— but does that mean that if an arbitrary person chooses a scalable profession, they are likely to become rich? <span class="Apple-converted-space"> </span>The answer to that is a definite NO. <span class="Apple-converted-space"> </span>For every J.K Rowling or Nassim Nicholas Taleb, there are thousands of aspiring authors who barely sell a copy of their book. <span class="Apple-converted-space"> </span>(I won’t comment on how the Math Mutation book fits into this discussion.) <span class="Apple-converted-space"> </span>The scalable professions are massively profitable for the small number of people who are successful, but what’s a lot less visible are the corresponding masses of unsuccessful aspirants to these careers who failed miserably.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Thus, Taleb points out that the advice he got was not very good, even though he happened to follow it and succeed himself.<span class="Apple-converted-space"> </span>In a nonscalable profession, the average worker makes a decent living— in some, like skilled tradesmen, engineers, or doctors, they can be pretty sure of heading towards the upper middle class if they do a good job, even though they are not likely to become rich. <span class="Apple-converted-space"> </span>Choosing a scalable profession is like entering a lottery, while nonscalable or mid-range jobs give you much better odds of earning a solid living. <span class="Apple-converted-space"> </span>And of course, there is usually the possibility of later moving into management of whatever career path you’re on, if you decide later that you want to gamble on the scalable route, while having a solid profession to fall back on if you don’t win that lottery. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Another interesting point Taleb makes is that the nature of some professions changes over time. <span class="Apple-converted-space"> </span>If you look to the 19th century or earlier, being a singer was a nonscalable, labor-type position:<span class="Apple-converted-space"> </span>you had to be physically present before a relatively small audience, and repeat that activity every time you wanted to earn money for your music. <span class="Apple-converted-space"> </span>Thus many singers across the world could make a living producing music for eager audiences. <span class="Apple-converted-space"> </span>Then came the 20th-century revolution in recorded sound. <span class="Apple-converted-space"> </span>Once that happened, you could see superstars like Elvis, Pavarotti, or the Beatles become household names— and the majority of the money that in previous years would have been distributed among thousands of local performers ended up in their pockets. <span class="Apple-converted-space"> </span>For those who think this situation was unfair to the local singers, think about the effect of the printing press on monks, or of the invention of the alphabet on traveling storytellers. <span class="Apple-converted-space"> </span>The truth is, making a job more scalable almost always benefits huge numbers of people who consume the goods or services being produced, while creating an inconvenience for those who had previously profited off the nonscalability of their jobs.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Another interesting point Taleb makes is to look at this concept at a national level.<span class="Apple-converted-space"> </span>In the case of the United States, it seems to have been much more economically successful in the past century than the intellectual European nations of, as Taleb puts it, “museumgoers and equation solvers”.<span class="Apple-converted-space"> </span>He attributes this to the US’s much greater tolerance for creative tinkering and trial-and-error, which result in the development of new concepts and ideas. <span class="Apple-converted-space"> </span>Because the economic benefits of concepts and ideas are scalable, this has resulted in a large multiplier on the money that can be made by US companies in general.<span class="Apple-converted-space"> </span>The much-lamented loss of US manufacturing jobs is just a reflection of this shift of focus. <span class="Apple-converted-space"> </span>Nike can design a shoe, or Boeing can design an airplane, with a relatively small number of ideas, and subcontract the grunt work to foreign companies.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">I wish I could say that I pondered all these insights when initially setting out on my career, but like most of us, I just blundered my way around until I settled into something that seemed good. <span class="Apple-converted-space"> </span>It seems to have worked out pretty well for me.<span class="Apple-converted-space"> </span>But if you’re at an earlier stage of your life, Taleb’s ideas are worth strong consideration.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/Black-Swan-Improbable-Robustness-Fragility/dp/081297381X">https://www.amazon.com/Black-Swan-Improbable-Robustness-Fragility/dp/081297381X</a></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-26372658962868858092022-02-20T06:24:00.003-08:002022-02-20T06:24:55.306-08:00276: Don't Believe the Math<p><a href="http://www.erikseligman.com/mm/mm276.mp3">Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The last episode’s discussion of randomness brought to mind the classic book “The Black Swan” by economist-philosopher Nassim Nicholas Taleb.<span class="Apple-converted-space"> </span>His books discuss the disproportionate role of unlikely extreme events, the Black Swans, in shaping our lives and our history.<span class="Apple-converted-space"> </span>Noticing online that there is a 2nd edition now, I decided to reread Taleb’s book, and got many intriguing new ideas for podcast episodes. <span class="Apple-converted-space"> </span>Today we will talk about the “Ludic Fallacy”, the incorrect use of mathematical models and games to predict real-life events. <span class="Apple-converted-space"> </span>To understand this better, let’s look at one of his key examples.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Suppose we do an experiment with two observers in a room, a professor and a gambler. <span class="Apple-converted-space"> </span>We present them the following mathematical puzzle: <span class="Apple-converted-space"> </span>I have a fair coin that I plan to flip 100 times, with everyone watching. <span class="Apple-converted-space"> </span>The first 99 flips are all heads. <span class="Apple-converted-space"> </span>The two observers are asked to estimate the probability that the next flip will turn up heads.<span class="Apple-converted-space"> </span>The professor confidently answers, “Since you said it’s a fair coin, previous flips have no influence on future flips. <span class="Apple-converted-space"> </span>So the chance is the same as always, exactly 50%.” <span class="Apple-converted-space"> </span>On the other hand, the gambler answers, with equal confidence, “If you got 99 heads, I’m almost certain that the coin is biased in some way, regardless of whether you said it’s fair. <span class="Apple-converted-space"> </span>So I’ll estimate a 99% chance that the next flip is heads.”<span class="Apple-converted-space"> </span>Naturally, in a purely mathematical sense, the professor was right, according to the information we provided. <span class="Apple-converted-space"> </span>But if this were a real-life situation, and you had to bet money on the outcome of the next flip, which answer would you go with? <span class="Apple-converted-space"> </span>The gambler probably has a point.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this is Taleb’s key insight that forms the Ludic Fallacy: <span class="Apple-converted-space"> </span>while abstract mathematical models may provide some insight into possibilities, you cannot consider them reliable models of real life. <span class="Apple-converted-space"> </span>Issues or events that are outside your simple model may have a huge effect. <span class="Apple-converted-space"> </span>Taleb criticizes a lot of professionals who spend their lives creating complex mathematical models, and claim that they deserve large salaries or become media darlings for using them to make intricate predictions about the future, which then turn out to have little more accuracy than random chance. <span class="Apple-converted-space"> </span>Economists are some of the most notorious in this regard.<span class="Apple-converted-space"> </span>You may recall that back in the 1990s, a large hedge fund called Long Term Capital Management (or LTCM) was built around some insights from supposedly genius economists who had Nobel Prizes. <span class="Apple-converted-space"> </span>But when its “mathematically proven” strategy led to buying massive numbers of Russian bonds with borrowed money, which then defaulted, LTCM failed so badly that it needed a multi-billion dollar bailout to avoid crashing the world economy. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">There are plenty of other examples like this, and it’s not just experts who fall for this kind of fallacy. <span class="Apple-converted-space"> </span>Taleb is a bit critical of the modern software, such as features in Microsoft’s Excel spreadsheets, that make it easy for even ordinary workers to mathematically extend existing data into future extrapolations, which are very rarely accurate in the face of unpredictable real-life events. <span class="Apple-converted-space"> </span>In effect, computers allow anyone to transform themselves into an incompetent economist with high self-confidence.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">I think my favorite example that Taleb cites is the story of a casino he consulted with in Las Vegas.<span class="Apple-converted-space"> </span>They had very meticulously modeled all the ways that a gambler could cheat, or that low-probability events in the games might threaten their cash flow, and had invested massive amounts of money in gambling theory, security, high-tech surveillance, and insurance to guard against these events. <span class="Apple-converted-space"> </span>So what did the four largest money-losing incidents in their casino turn out to be? <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;"><span class="Apple-tab-span" style="white-space: pre;"> </span>1. The irreplaceable loss of their headline performer when he was maimed by one of this trained tigers.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;"><span class="Apple-tab-span" style="white-space: pre;"> </span>2. A disgruntled worker, who had been injured on the job, attempted to blow up the casino.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;"><span class="Apple-tab-span" style="white-space: pre;"> </span>3. An incompetent employee had been putting some required IRS forms in a drawer and failing to send them in, resulting in massive fines.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;"><span class="Apple-tab-span" style="white-space: pre;"> </span>4. The owner’s daughter was kidnapped, and he illegally took money from the casino in order to ransom<span class="Apple-converted-space"> </span>her.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now of course, it would have been very hard for any of these to be predicted by the models the casino was using. <span class="Apple-converted-space"> </span>That’s Taleb’s point:<span class="Apple-converted-space"> </span>no mathematical modeling could cover every conceivable low-probability event.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">This is also an important reason why Taleb opposes centrally-planned economies. <span class="Apple-converted-space"> </span>One of the few Nobel-winning economists who Taleb respects is F.A.Hayek, whose 1974 Nobel speech offered a harsh critique of his fellow economists who fall back on math due to their physics envy,<span class="Apple-converted-space"> </span>and try to claim that their equations model the world just like the hard sciences.<span class="Apple-converted-space"> </span>No matter how many measurable elements they factor into their equations, the real world is much too complicated to model accurately and make exact predictions. <span class="Apple-converted-space"> </span>Modern free economies are largely successful because millions of individuals make small-scale decisions based on local information, and are free to take educated risks with occasional huge payoffs for society in general. <span class="Apple-converted-space"> </span>In his conclusion Hayek wrote, “The recognition of the insuperable limits to his knowledge ought indeed to teach the student of society a lesson of humility which should guard him against becoming an accomplice in men’s fatal striving to control society – a striving which makes him not only a tyrant over his fellows, but which may well make him the destroyer of a civilization which no brain has designed but which has grown from the free efforts of millions of individuals.”</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">We should mention, however, that Taleb and Hayek are not showing that mathematical models are totally useless— we just need to recognize their limitations. <span class="Apple-converted-space"> </span>They can be powerful in finding possibilities of what might happen, and opening our eyes to potential consequences of our basic assumptions. <span class="Apple-converted-space"> </span>For example, let’s look again at the coin flipping case.<span class="Apple-converted-space"> </span>Suppose instead of 99 heads, our example had shown a variety of results, including a run of 5 heads in a row somewhere in the middle. <span class="Apple-converted-space"> </span>The gambler might spot that and initially have a gut feeling that this is an indication of bias. <span class="Apple-converted-space"> </span>But the professor could then walk him through a calculation, based on the ideal fair coin, that if you flip a coin 100 times, there is over an 80% chance of seeing a run of length 5 at some point. <span class="Apple-converted-space"> </span>So using the insight from his modeling, the gambler can determine that this run is not evidence of bias, and make a more educated guess, considering that the initial promise of a fair coin has not yet been proven false.<span class="Apple-converted-space"> </span>Remember, however, that the gambler still needs to consider the possibilities of external factors that are not covered by the modeling— maybe as he is making his final bet, that disgruntled employee will return to the casino with an angry tiger.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">So, in short, you can continue to use mathematical models to gain limited insight, but they are not confident sources for practical predictions. <span class="Apple-converted-space"> </span>Don’t get overconfident and fool yourself into making big bets that your model will guarantee discovery of all real-life risks.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.gregegan.net/QUARANTINE/Runs/Runs.html">https://www.gregegan.net/QUARANTINE/Runs/Runs.html</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/Black-Swan-Improbable-Robustness-Fragility/dp/081297381X">https://www.amazon.com/Black-Swan-Improbable-Robustness-Fragility/dp/081297381X</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.investopedia.com/terms/l/longtermcapital.asp">https://www.investopedia.com/terms/l/longtermcapital.asp</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.nobelprize.org/prizes/economic-sciences/1974/hayek/lecture/">https://www.nobelprize.org/prizes/economic-sciences/1974/hayek/lecture/</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-33680296257756833982022-01-22T07:24:00.001-08:002022-01-22T07:24:20.194-08:00275: Demanding Fair Dice<p><a href="http://www.erikseligman.com/mm/mm275.mp3"> Audio Link</a></p><p><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Recently I was amused to hear about a strange machine called the “dice-o-matic”, a monstrous 7-foot tall machine that can physically roll thousands of dice per day. <span class="Apple-converted-space"> </span>It uses a camera to automatically capture the results and serve them up on the internet, for players of online board and role-playing games who need trustworthy rolls of virtual dice. <span class="Apple-converted-space"> </span>This was created by the hosts of the website <a href="http://gamesbyemail.com"><span class="s2" style="font-kerning: none;">gamesbyemail.com</span></a>, as a result of demands by players for real dice rolls to apply to their games.<span class="Apple-converted-space"> </span>At first I thought this was rather silly— after all, nearly every programming language these days offers an option to generate a random number, so why would someone need physical dice rolls? <span class="Apple-converted-space"> </span>But as I read more about it, I could sort of understand the motivation.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">To start with, let’s review how most random numbers are generated on a computer. <span class="Apple-converted-space"> </span>Typically these are not truly random numbers, but pseudo-random: <span class="Apple-converted-space"> </span>each number is derived from the previous one, but in such a way that they would effectively appear random to an external observer not aware of the algorithm used.<span class="Apple-converted-space"> </span>For a given “seed”, or starting number, they will always generate the same sequence. <span class="Apple-converted-space"> </span>The most common algorithm for such numbers is the Linear Congruential number generator. <span class="Apple-converted-space"> </span>For example, suppose you are trying to generate a random number between 0 and 7. <span class="Apple-converted-space"> </span>A simple Linear Congruential generator might take the current number, multiply it by 5, add 1, and then calculate the remainder modulo 8. <span class="Apple-converted-space"> </span>If we start with the number 0, we get the sequence 0,1,6,7,4,5,2,3, a pretty random-looking sequence of 8 numbers.<span class="Apple-converted-space"> </span>That sequence would then repeat forever— this type of generator naturally has that property. <span class="Apple-converted-space"> </span>But in real life, the numbers used for the multiplication, addition, and modulo operators would be much higher, so the cyclic nature would almost never be exposed. <span class="Apple-converted-space"> </span>By the way, the deterministic nature of these pseudo-random numbers can actually be beneficial in many applications:<span class="Apple-converted-space"> </span>for example, if testing a microprocessor design with random simulations, knowing the seed that led to discovering a malfunction enables you to easily repeat the exact same random test.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Usually you would try to get some number from a random physical source to set the seed, so you would not have the same sequence of numbers every time.<span class="Apple-converted-space"> </span>On unix systems, the /dev/random file looks for environmental noise, such as device driver delays, for storage in an “entropy pool” of random starting seeds. <span class="Apple-converted-space"> </span>There are other even better methods:<span class="Apple-converted-space"> </span>quantum physics supplies true randomness, through the timing of radioactive decays, for example. <span class="Apple-converted-space"> </span>And there are silly-sounding methods online that can be quite effective, such as the program at <a href="http://lavarand.org"><span class="s2" style="font-kerning: none;">lavarand.org</span></a> for generating random values using patterns observed in lava lamps.<span class="Apple-converted-space"> </span>Various online services provide true random numbers generated through other methods.<span class="Apple-converted-space"> </span>Of course, if you can generate these seeds with true randomness, you might argue that you should generate all your random numbers this way instead of using a pseudo-random algorithm.<span class="Apple-converted-space"> </span>But remember that current microprocessors run blazingly fast, with billions of operations per second. <span class="Apple-converted-space"> </span>Anything requiring input from the external world is going to create extra delays that are huge compared to internal operations.<span class="Apple-converted-space"> </span>So it’s much more common to use these true random sources just for the seed.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Which brings us back to the motivation for the dice-o-matic. <span class="Apple-converted-space"> </span>According to the description on their website, users were dissatisfied with the randomness of their earlier pseudo-random method. <span class="Apple-converted-space"> </span>They write:<span class="Apple-converted-space"> </span>“Some players have put more effort into statistical analysis of the rolls than they put into their doctoral dissertation.” <span class="Apple-converted-space"> </span>Personally, I’m a bit skeptical— <span class="Apple-converted-space"> </span>in any set of random numbers, you are going to see streaks of values that appear non-random. <span class="Apple-converted-space"> </span>There may be a run of 20 low numbers in a row, or an ascending 1-2-3-4-5-6 in a few places, etc.<span class="Apple-converted-space"> </span>I bet most of the complaining users had lost a game or had their D&D characters killed due to bad rolls that appeared in such a sequence, and were looking for external causes to blame. <span class="Apple-converted-space"> </span>On the other hand, if they put in enough effort, it should often be possible to demonstrate the non-randomness of pseudo-random numbers. <span class="Apple-converted-space"> </span>The website <a href="http://random.org"><span class="s2" style="font-kerning: none;">random.org</span></a> points out some applications, such as scientific simulation of virus infections, where this lack of randomness has had detectable effects.<span class="Apple-converted-space"> </span>In any case, the <a href="http://gamesbyemail.com"><span class="s2" style="font-kerning: none;">gamesbyemail.com</span></a> people took the issue seriously, and the dice-o-matic was their solution, freeing them from the dangers of pseudo-randomness.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">We should also keep in mind, however, that dice actually aren’t the greatest sources of true random numbers either: <span class="Apple-converted-space"> </span>wear and tear will cause them to regularly depart from true randomness over time, as well as other possible issues depending on the manufacturing process, like the different amounts of material cut out to make the pips on each side.<span class="Apple-converted-space"> </span>Personally, I think the gamesbyemail people might have been better off installing a few number-generating lava lamps.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="http://gamesbyemail.com/news/diceomatic">http://gamesbyemail.com/news/diceomatic</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="http://www.columbia.edu/~ks20/4106-18-Fall/Simulation-LCG.pdf">http://www.columbia.edu/~ks20/4106-18-Fall/Simulation-LCG.pdf</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.random.org/randomness/">https://www.random.org/randomness/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.lavarand.org/news/lavadiff.html">https://www.lavarand.org/news/lavadiff.html</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.networkworld.com/article/3208389/unix-how-random-is-random.html">https://www.networkworld.com/article/3208389/unix-how-random-is-random.html</a></span></li></ul><p></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-11750739263048322152021-11-30T16:17:00.000-08:002021-11-30T16:17:16.701-08:00274: The Gomboc<p><a href="http://www.erikseligman.com/mm/mm274.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">If you heard the title of this podcast, you’re probably asking yourself, “What did he say?” <span class="Apple-converted-space"> </span>Today we are discussing a 3-dimensional shape that was first discovered only in the 21st century, by a pair of Hungarians named<span class="Apple-converted-space"> </span>Gábor Domokos and Péter Várkonyi,.<span class="Apple-converted-space"> </span>It’s called the Gomboc, spelled g-o-m-b-o-c,<span class="Apple-converted-space"> </span>with two dots over each of the o’s due to its Hungarian origin. <span class="Apple-converted-space"> </span>It looks kind of like a spherical stress ball whose top has been partially squeezed in; we can’t really do it justice in a verbal description, but you can find links to articles with pictures in the show notes at <a href="http://mathmutation.com"><span class="s2" style="font-kerning: none;">mathmutation.com</span></a>. <span class="Apple-converted-space"> </span>But most importantly, this new shape has the amazing property that it has only one stable equilibrium position:<span class="Apple-converted-space"> </span>no matter what position you put it down in, it will roll around and right itself.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now at first, you might not think this is very interesting. <span class="Apple-converted-space"> </span>When you were a child, you very likely played with a toy like that, a small egg-shaped doll, which would always pull itself upright no matter how you placed it. <span class="Apple-converted-space"> </span>These are called roly-poly toys in some places, but in the US during my childhood, they were marketed as “Weebles”, with the advertising slogan that “Weebles wobble but they don’t fall down”. <span class="Apple-converted-space"> </span>They worked because of a weighted bottom:<span class="Apple-converted-space"> </span>the bottom half of the toy weighed significantly more than the top.<span class="Apple-converted-space"> </span>The Gomboc, on the other hand, is made of a single material with uniform density, and miraculously still has this self-righting property.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Another interesting fact about the Gomboc is that, until a few decades ago, it was not obvious that such a shape should exist at all. <span class="Apple-converted-space"> </span>If you look at the 2-dimensional analog of the problem, trying to find a two-dimensional shape of uniform density that will always right itself when placed on a line that has a gravitational pull, there is no solution.<span class="Apple-converted-space"> </span>Any convex shape you create will have a least two stable equilibrium positions. <span class="Apple-converted-space"> </span>Domokos had originally set out to try to prove the 3-dimensional analog of this theorem, which would have demonstrated that nothing like the Gomboc could actually exist. <span class="Apple-converted-space"> </span>But after a conversation with a Russian mathematician named Vladimir Arnold, he realized that things were a bit different in three dimensions, and the theorem might not hold.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Domokos and Varkonyi then started working with computer modeling and trying to figure out what such a self-righting shape would look like. <span class="Apple-converted-space"> </span>At one point, Domokos came up with the idea that maybe nature had solved the problem already, and started experimenting with pebbles found at a beach, to see if any had naturally assumed a self-righting shape. <span class="Apple-converted-space"> </span>After checking over 2000 pebbles, he was disappointed. <span class="Apple-converted-space"> </span>Their breakthrough came when they started defining some key parameters of the shapes they were creating, called “flatness” and “thinness”, and realized they would have to minimize both together to come up with their desired shape.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Continuing their computer modeling, but with these parameters in mind, they were finally able to start describing 3-D self-righting shapes. <span class="Apple-converted-space"> </span>The first one they came up with was very close to a sphere— disappointingly, so close that they could not manufacture one in practice.<span class="Apple-converted-space"> </span>Because it only differed from a true sphere by a factor of 10^-5, and even microscopic variations from their intended design would kill its self-righting property, attempts to manufacture it would just lead to the creation of ordinary spheres. <span class="Apple-converted-space"> </span>But by realizing that they could sacrifice smoothness, and allow some sharp edges between sphere-like segments, they were then able to come up with a more practical shape.<span class="Apple-converted-space"> </span>It’s still incredibly challenging to manufacture: <span class="Apple-converted-space"> </span>to get one you can hold in your hand, it has to be made with a precision down to the width of a human hair.<span class="Apple-converted-space"> </span>Apparently interest is now wide enough that they are being mass-produced, and can be ordered at the “gomboc shop” website.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Another interesting thing about this shape— Domokos realized afterwards that his insight about nature developing in this direction wasn’t totally off base. <span class="Apple-converted-space"> </span>He couldn’t find it in pebbles, because the tight margin for error meant that any Gomboc pebble would quickly wear away at one of its edges into a non-self-righting shape. <span class="Apple-converted-space"> </span>But evolution painstakingly comes up with precise designs over millions of years— and after carefully searching species of tortoises, he did find two with shell designs very close to his Gomboc. <span class="Apple-converted-space"> </span>A tortoise stuck on its back is very vulnerable, so it really does make sense that they would evolve an improved ability to right themselves as needed.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References:</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://gomboc.eu/en/mathematics/">https://gomboc.eu/en/mathematics/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.wired.com/2008/02/the-gomboc-the/">https://www.wired.com/2008/02/the-gomboc-the/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.youtube.com/watch?v=rvVF5QWSYF4">https://www.youtube.com/watch?v=rvVF5QWSYF4</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.maplesoft.com/company/casestudies/Professional/Gomboc.aspx">https://www.maplesoft.com/company/casestudies/Professional/Gomboc.aspx</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://gomboc-shop.com/">https://gomboc-shop.com/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://plus.maths.org/content/story-goumlmboumlc">https://plus.maths.org/content/story-goumlmboumlc</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://findwords.info/term/weeble">https://findwords.info/term/weeble</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-79723885044897370662021-10-31T08:11:00.005-07:002021-10-31T08:12:55.855-07:00273: A Maze Of Labyrinths<p><a href="http://www.erikseligman.com/mm/mm273.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Mazes, or labyrinths, are probably the one type of somewhat mathematical game that nearly everyone is familiar with. <span class="Apple-converted-space"> </span>Books of activities for young children, at least in the US, often contain simple mazes, drawings with many different twisty passages leading from one side of the page to the other, with a challenge to trace the right path. <span class="Apple-converted-space"> </span>And when I was first learning to program computers while growing up, one of my favorite types of simple programs to write and experiment with were ones that generate and draw mazes and labyrinths. <span class="Apple-converted-space"> </span>As you are likely aware, these have been around in both literary and visual form since ancient times.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Recently I was reading a book on this topic by a scholar named Penelope Doob, and she pointed out an intriguing mystery related to these constructs. <span class="Apple-converted-space"> </span>There are two major types of labyrinths:<span class="Apple-converted-space"> </span>unicursal ones, where there is ultimately only one path (though a very twisty one) that will eventually lead anyone entering to the goal; and multicursal ones, where you have to make choices along the way among many paths, most of which lead to dead ends. <span class="Apple-converted-space"> </span>The word “labyrinth” can refer to both, though these days “maze” usually refers to the multicursal type. <span class="Apple-converted-space"> </span>But here’s the strange thing: <span class="Apple-converted-space"> </span>in ancient and medieval sources, nearly every visual or physically built maze is unicursal or single-path, while nearly every literary description of labyrinths describes a multicursal or multiple-path one. <span class="Apple-converted-space"> </span>Why would we have this strange division? <span class="Apple-converted-space"> </span>Why weren’t people drawing and building the same labyrinths they described in their stories?</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Let’s start by looking in more detail at unicursal labyrinths, the ones with only one path from beginning to end. <span class="Apple-converted-space"> </span>This one path is guaranteed to get anyone entering to their goal eventually, but it will take a lot longer than one might guess from the size of the labyrinth.<span class="Apple-converted-space"> </span>This is often seen as a spiritual metaphor: <span class="Apple-converted-space"> </span>the path to true faith or Enlightenment may be confusing and long, but as long as you remain steady and stick to the path, following the prescribed moral choices that move you forward, you will reach your goal. <span class="Apple-converted-space"> </span>Similarly, when laid into the ground or in a garden, it can be seen as a meditation aid, providing a simple path to slowly follow which doesn’t require any choices or decisions, and helping to free your mind of conscious thought. <span class="Apple-converted-space"> </span>Actually, that’s not fully true:<span class="Apple-converted-space"> </span>you do have the initial choice to enter the labyrinth or not, as well as the choice to turn back at any time.<span class="Apple-converted-space"> </span>But this also can work well with the metaphor of following the steady path to God: <span class="Apple-converted-space"> </span>you make the critical choice of whether to embark on the path, and at any moment along the way, you either maintain your faith or turn away. <span class="Apple-converted-space"> </span>As mentioned above, nearly all visually depicted labyrinths in ancient and medieval times are unicursal, with only a handful of exceptions. <span class="Apple-converted-space"> </span>Doob’s book shows some interesting examples dating back to stone carvings between 1800-1400 BC.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The multicursal type of labyrinth, where you have many choices to make along the way and can end up in a dead end or trap, is the kind most suited to games and puzzles.<span class="Apple-converted-space"> </span>If you’re a modern gamer, I’m sure you've wandered around an endless set of 3-D mazes in various videogames or in tabletop games like Dungeons & Dragons. <span class="Apple-converted-space"> </span>The literary precedents for this type of labyrinth go back to ancient times as well— who can forget the Greek legend of the Minotaur, who lived at the center of a confusing and dangerous labyrinth. <span class="Apple-converted-space"> </span>Theseus only managed to make it out alive because the princess Ariadne gave him some thread which he could slowly unwind to mark his path. <span class="Apple-converted-space"> </span>These labyrinths can also provide a slightly different metaphor for the path to religious enlightenment:<span class="Apple-converted-space"> </span>the constant presence of temptations to sin, which will lead you away from your goal, possibly forever. <span class="Apple-converted-space"> </span>Whether you succeed and make it to the goal or spend the rest of your life wandering in confusion, it’s due to your own choices.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">But we should not forget that there are a number of elements that both types of labyrinths have in common. <span class="Apple-converted-space"> </span>When you look at either a unicursal or multicursal maze from above, they are a visually busy picture that’s hard to fully comprehend at a glance; in most cases, you can’t even be sure a labyrinth is of one type or the other without starting to carefully trace the path.<span class="Apple-converted-space"> </span>Both seem to trigger the same set of concepts in our brain, at least at first glance.<span class="Apple-converted-space"> </span>Also in both cases, because the maze twists back and forth many times within a small confined area, it’s very hard to know how far along the path you are:<span class="Apple-converted-space"> </span>while the unicursal maze might not provide false paths, it still conceals the total distance to the goal. <span class="Apple-converted-space"> </span>So they both have the same effects in attacking your self-confidence and triggering confusion, <span class="Apple-converted-space"> </span>These commonalities are strong enough that the ancients generally used the same language to describe both, variants of the words that led to our modern “labyrinth”.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Ultimately, these commonalities provide a key to understanding the strange issue of visually depicted unicursal mazes and literary multicursal ones. <span class="Apple-converted-space"> </span>As Doob describes it, “the best solution that can be found to the mystery is that classical and medieval eyes saw insufficient difference in the implications of the two models to warrant a new design.” <span class="Apple-converted-space"> </span>Unicursal mazes are somewhat simpler to draw or illustrate, since you don’t have to worry about accidentally failing to create a non-dead-end path to the goal <span class="Apple-converted-space"> </span>And it was very common for most art to be based somewhat on earlier art, so once drawings of unicursal labyrinths became common, those who needed to draw something similar followed the patterns set by their predecessors.<span class="Apple-converted-space"> </span>When artists were creating illustrations of labyrinths, they didn’t consult the corresponding literary works to check for an exact match. <span class="Apple-converted-space"> </span>On the other hand, one cannot deny that when telling stories, multicursal mazes are much more terrifying, providing the possibility of getting lost or confused forever without reaching your goal, so it makes a lot of sense that these would feature prominently in myths and legends.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.cornellpress.cornell.edu/book/9781501738456/the-idea-of-the-labyrinth-from-classical-antiquity-through-the-middle-ages">https://www.cornellpress.cornell.edu/book/9781501738456/the-idea-of-the-labyrinth-from-classical-antiquity-through-the-middle-ages</a></span><span class="s1" style="font-kerning: none;"> </span></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-73173873323003652842021-09-20T16:33:00.000-07:002021-09-20T16:33:06.691-07:00272: The Mathematics of Jackie Mason<p><a href="http://www.erikseligman.com/mm/mm272.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.<span class="Apple-converted-space"> </span>And now, on to the math.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">To celebrate this month’s Jewish holidays, I thought it might be fun to talk about the legendary Jewish comedian Jackie Mason, who passed away recently at the age of 93. <span class="Apple-converted-space"> </span>Now you might be wondering what this topic has to do with math. <span class="Apple-converted-space"> </span>This episode was inspired by an intriguing quote that’s shown up in some of his online obituaries:<span class="Apple-converted-space"> </span>“The Talmud is the study of logic… Every time I see a contradiction or hypocrisy in somebody’s behavior, I think of the Talmud and build the joke from there.” <span class="Apple-converted-space"> </span>You may recall that before becoming a comedian, Mason studied to become a rabbi. <span class="Apple-converted-space"> </span>The Talmud, in case you’re not familiar with it, is the ancient book of Jewish law.<span class="Apple-converted-space"> </span>Could this book really be viewed as a study of logic?</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">To test this theory, I picked a joke I remembered from Mason’s show that I saw many years ago, and decided to see if I could relate it to formal logic in the Talmud. <span class="Apple-converted-space"> </span>This joke was referencing the battles in Israeli courts at the time, over the legitimacy of conversion to Judaism by various types of rabbis.<span class="Apple-converted-space"> </span>Here’s a paraphrase, as I remember it: <span class="Apple-converted-space"> </span>“The Israeli Supreme court finally made up its mind on the Jewish conversion laws. <span class="Apple-converted-space"> </span>If you were converted by an Orthodox rabbi, you’re a Jew. <span class="Apple-converted-space"> </span>If you were converted by a Conservative rabbi, you’re a half Jew. <span class="Apple-converted-space"> </span>If you were converted by a reform rabbi, you’re a Puerto Rican.”<span class="Apple-converted-space"> </span>In case you didn’t get the joke, it references the tensions among the many ethnic groups in New York City, which in the 20th century often put Jews and Puerto Ricans at odds with each other. <span class="Apple-converted-space"> </span>Trust me, it’s funny, though I can’t match Mason’s delivery!</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Searching the Talmud for relevant quotes on this topic, I found an interesting one, discussing how to handle a candidate for conversion. <span class="Apple-converted-space"> </span>“If he accepts, we circumcise him at once… Two learned men stand nearby, reminding him of some of the easy mitzvot and some of the hard ones. As soon as he emerges and dries himself, he is an Israelite in all respects. “<span class="Apple-converted-space"> </span>This would seem to indicate that being Jewish after conversion is a binary value, of 0 or 1— you are or you aren’t. <span class="Apple-converted-space"> </span>Thus many proposals that came up in the conversion debate might be considered absurd, attempting to apply different degrees of Jewishness:<span class="Apple-converted-space"> </span>as defined by this quote, you’re Jewish as soon as the ritual is done. <span class="Apple-converted-space"> </span>So that joke might be said to be mocking the contradiction of attempting to assign fractions to a 0 or 1 value, and thus fall out of the logic of the Talmud.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now, I’m not claiming to be a Talmudic scholar, so I could be way off here. <span class="Apple-converted-space"> </span>I did a bit more Google searching, and found that Mason was not alone:<span class="Apple-converted-space"> </span>there are many philosophers and logicians who do believe the Talmud makes extensive use of formal logic, though couched in confusing human-language phrases. <span class="Apple-converted-space"> </span>One example is a book by someone named Avi Sion, called “Logic in the Talmud”.<span class="Apple-converted-space"> </span>He points out that unlike many other religious books, rather than just pronouncing laws commanded from above, the Talmud often uses formal logical arguments to show why these laws must follow from earlier premises.<span class="Apple-converted-space"> </span>One common example is <i>a fortiori</i> arguments, which mainly work as follows: <span class="Apple-converted-space"> </span>Suppose we have two subjects, P and Q, which have some attribute R, and we want to find out if P also has a related attribute S. <span class="Apple-converted-space"> </span>If we agree that P has more of R than Q,<span class="Apple-converted-space"> </span>but Q has enough R to automatically possess S, then P must therefore also possess S.<span class="Apple-converted-space"> </span>As my cousin Ben David puts it, “ The humdrum example people are given of the principle is that if person P is stronger than person Q, and if person Q can lift a certain weight, then certainly person P can lift it.”</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now let’s look at one of the examples Sion points out in the Talmud, a debate over how much a woman should be shamed for offending God. <span class="Apple-converted-space"> </span>The Talmud says:<span class="Apple-converted-space"> </span>“If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again. ” <span class="Apple-converted-space"> </span>We should point out that this spitting in the face was meant to symbolize that the woman had offended her father somehow, though it probably would not be considered very appropriate on the father’s side these days. <span class="Apple-converted-space"> </span>Looking at the elements of the logic, in this case, the subjects P and Q are God and one’s father, and R is the amount of offensiveness. <span class="Apple-converted-space"> </span>If in a religious framework where you believe God is more important than any one human, certainly the idea of offending P contains more R, offensiveness, than offending Q. <span class="Apple-converted-space"> </span>Since offending Q produces punishment S, seven days’ worth of shame, certainly offending P should result in at least that much punishment as well.<span class="Apple-converted-space"> </span>So applying the formal logic does seem to produce the result here.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Anyway, as you can see, this does get a bit confusing.<span class="Apple-converted-space"> </span>It requires understanding both the cultural norms of the time— like the face-spitting— and a lot of previous context, like the established rules about shame for offending one’s family. <span class="Apple-converted-space"> </span>Thus we can see there may be some merit to observing that the Talmud actually does follow rules of logic to some degree, but with the need for a lot of human interpretation.<span class="Apple-converted-space"> </span>I’m not really into all this complex religious law stuff, but I am happy that Mason was able to successfully convert it into a bunch of great jokes.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/Logic-Talmud-Compilation-Avi-Sion-ebook/dp/B07BMYHLCQ">https://www.amazon.com/Logic-Talmud-Compilation-Avi-Sion-ebook/dp/B07BMYHLCQ</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.washingtonpost.com/local/obituaries/jackie-mason-dies/2021/07/24/62717c86-ecea-11eb-97a0-a09d10181e36_story.html">https://www.washingtonpost.com/local/obituaries/jackie-mason-dies/2021/07/24/62717c86-ecea-11eb-97a0-a09d10181e36_story.html</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.thejc.com/judaism/books/what-the-talmud-says-about-converts-1.9461">https://www.thejc.com/judaism/books/what-the-talmud-says-about-converts-1.9461</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-56296490170814247992021-07-30T08:48:00.002-07:002021-07-30T08:48:53.665-07:00271: Too Much Testing<p><a href="http://www.erikseligman.com/mm/mm271.mp3"> Audio Link</a></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our new headquarters in Wichita, Kansas, this is Erik Seligman, your host.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Recently the infamous Elizabeth Holmes of Theranos has been in the news again, apparently filing new motions to delay her trial. <span class="Apple-converted-space"> </span>As you may recall, Theranos was the company that claimed to have developed powerful blood testing kits, which could run hundreds of standard medical tests at home on a single drop of blood.<span class="Apple-converted-space"> </span>It turned out that the invention just didn’t work, and Holmes was eventually charged with fraud as the company collapsed.<span class="Apple-converted-space"> </span>But not enough people have noticed that the lies about the science-fiction technology weren’t the only problems in Theranos’s basic concept.<span class="Apple-converted-space"> </span>We need to think about the flaws in their fundamental premise to “democratize” your health information. <span class="Apple-converted-space"> </span>This is the idea that average consumers should be encouraged to run lots of tests, for rare diseases or issues, on their own blood.<span class="Apple-converted-space"> </span>We especially need to pay attention now that numerous non-fraudulent companies, like the well-intentioned Everlywell, have entered this space. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">At first glance, the core concept sounds like an unmitigated benefit.<span class="Apple-converted-space"> </span>Why not let everyone run their own blood tests, without worrying about expensive doctors?<span class="Apple-converted-space"> </span>And there are good philosophical arguments why this should be allowed, as a matter of individual freedom, regardless of the mathematical issues I’m about to discuss.<span class="Apple-converted-space"> </span>(I won’t be getting into those arguments, as that’s beyond the scope of this podcast!) <span class="Apple-converted-space"> </span>But there is a key element of the math behind these tests that too many consumers are likely to overlook or be unaware of: <span class="Apple-converted-space"> </span>the fact that if a highly accurate test shows positive for an extremely rare disease, you probably DON’T actually suffer from that disease.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"> </span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">To make this more concrete, let’s assume there is a blood test which can, with 99% accuracy, determine if you suffer from the deadly virus of Math Madness, or MM; and in the general population, only one person out of every million has this disease. <span class="Apple-converted-space"> </span>You run the test, and it shows up positive. <span class="Apple-converted-space"> </span>You might intuitively think you are 99% likely to have MM.<span class="Apple-converted-space"> </span>However, let’s think about the total numbers here. <span class="Apple-converted-space"> </span>Out of every million people tested, only one has MM, due to its frequency in the population. <span class="Apple-converted-space"> </span>Yet with a 99% accurate test, 1% of the approximately 1 million healthy people, or 10,000 people, are going to incorrectly test positive. <span class="Apple-converted-space"> </span>So a given person who tests positive only has about a 1 in 10000 chance of carrying the disease.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">How did our basic intuition fail us here?<span class="Apple-converted-space"> </span>The key problem is that we need to realize the conditional probability of A given B is quite different from the probability of B given A. <span class="Apple-converted-space"> </span>That 99% represents the probability of a positive test given that we have the disease, and the probability of a negative test given that we don’t have the disease. <span class="Apple-converted-space"> </span>But it doesn’t accurately measure the chance we have the disease given a positive test, the converse of what that 99% is about. <span class="Apple-converted-space"> </span>When we reverse the terms like that, we need to convert the probability using Bayes’ Theorem: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;"><span class="Apple-tab-span" style="white-space: pre;"> </span>P(A|B) = P(B|A)P(A)/P(B)</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">That P(A) term, or the prior probability of the condition being tested, is the key factor here that drastically cuts down the ultimate chance of having the disease. <span class="Apple-converted-space"> </span>For our MM example, that gives us .99 * (1/1000000)/(1/100), or approximately 1/10000.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now you might point out that a false positive test is OK, as this is just an initial check to see if we should consult the doctor for more accurate testing and followup. <span class="Apple-converted-space"> </span>But the problem is that once the “easy” tests are out of the way, often much more intrusive, stressful, and life-altering testing and treatment is required.<span class="Apple-converted-space"> </span>This was brought home to me by an interesting poster I saw at my doctor’s office, provided by the US Preventative Services Task Force, on whether people under 70 should get PSA tests for prostate cancer.<span class="Apple-converted-space"> </span>Prostate cancer is an interesting case because, while deadly in the worst cases like all cancer, mild versions of it often do little harm and can be ignored. <span class="Apple-converted-space"> </span>The poster points out that out of every 1000 men given the PSA test, 1 death from prostate cancer will be prevented. <span class="Apple-converted-space"> </span>But: <span class="Apple-converted-space"> </span>240 of those men will initially test positive, and have to go through a painful biopsy. <span class="Apple-converted-space"> </span>Then 80 of them will, after testing positive at biopsy, go through long, painful (and unnecessary) courses of surgery or radiation treatment, after which 50 will permanently suffer erectile dysfunction, and 15 will suffer from permanent urinary incontinence.<span class="Apple-converted-space"> </span>So we’re 65x more likely to have really painful lifetime consequences than we are to save our life by taking the test. <span class="Apple-converted-space"> </span>It might still be worth it, but you really have to think hard.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Thus, ultimately, it often makes the most sense to avoid medical testing for rare conditions unless there is some overt symptom that causes your doctor to suspect an issue.<span class="Apple-converted-space"> </span>Otherwise the followup resulting from the test can actually provide many types of very negative patient outcomes.<span class="Apple-converted-space"> </span>This just naturally falls out of the common fallacy where people fail to apply Bayes’ Theorem, which requires that you factor in the prior probability of an event before you can properly interpret a test’s results.<span class="Apple-converted-space"> </span>Can average consumers be expected to understand these issues, and the reasons why running every possible test on your blood might not be the wisest course of action? <span class="Apple-converted-space"> </span>At the very least, I think companies entering this space should be very clear about the issue, and put up posters like the one at my doctor’s office, so their customers will approach the topic with their eyes open.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.uspreventiveservicestaskforce.org/Home/GetFileByID/3795">https://www.uspreventiveservicestaskforce.org/Home/GetFileByID/3795</a></span><span class="s1" style="font-kerning: none;"> </span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Bayes%27_theorem">https://en.wikipedia.org/wiki/Bayes%27_theorem</a></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.inc.com/christine-lagorio-chafkin/everlywell-democratizing-health-information.html">https://www.inc.com/christine-lagorio-chafkin/everlywell-democratizing-health-information.html</a></span><span class="s1" style="font-kerning: none;"> </span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;"><br /></span></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-14590028426318106542021-06-15T17:52:00.001-07:002021-06-15T17:52:17.587-07:00270: Which Way To Turn<p><a href="http://www.erikseligman.com/mm/mm270.mp3"> Audio Link</a></p><p><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. <span class="Apple-converted-space"> </span>Recording from our new headquarters in Wichita, Kansas, this is Erik Seligman, your host.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Apologies for the long gap since the last episode— as you just heard, we have been relocating Math Mutation headquarters across the country. <span class="Apple-converted-space"> </span>You can probably guess that this involves lots of details to handle. <span class="Apple-converted-space"> </span>We should soon be back on our almost-regular monthly schedule.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Anyway, with all the moving stuff going on, I recalled a topic I had been considering a while back. <span class="Apple-converted-space"> </span>It’s a pretty simple geometry question, yet one with a major (if subtle) effect on all our lives:<span class="Apple-converted-space"> </span>why are screws generally right-handed?<span class="Apple-converted-space"> </span>If you’ve ever had to screw something together, you probably remember the saying “lefty loosey righty tighty’, which reflects the common design of these basic and universal tools. <span class="Apple-converted-space"> </span>Screws are typically considered one of the basic six “simple machines”, along with inclined planes, levers, pulleys, wedges, and wheels. <span class="Apple-converted-space"> </span>So why are they always designed to rotate in one direction?</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">With a quick web search, it seems almost unanimous that this convention simply derives from the typical handedness of humans:<span class="Apple-converted-space"> </span>if you’re right-handed, then a right-handed screw is easiest to screw in.<span class="Apple-converted-space"> </span>It’s just as easy to manufacture screws in either direction, but when they first became standardized (with the Whitworth design in 1841), a single design became ubiquitious. <span class="Apple-converted-space"> </span>I guess my poor left-handed daughter will forever be a victim of this society-wide conspiracy.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">But the more surprising fact I discovered when researching this topic is that left-handed screw threads do exist, and are used for a variety of specialized applications.<span class="Apple-converted-space"> </span>Perhaps the most obvious is for situations where the natural rotation of an object would tend to loosen a right-handed screw:<span class="Apple-converted-space"> </span>for example, the left-side pedals on a bicycle, certain lug nuts on the left side of cars, or connections that secure other machine parts that are rotating the wrong way.<span class="Apple-converted-space"> </span>There are also cases where it’s useful to couple a left-handed and a right-handed connection, in a pipe fitting for example, so rotation in a single direction helps connect at both ends.<span class="Apple-converted-space"> </span>A less obvious usage is for safety: <span class="Apple-converted-space"> </span>in many applications involving flammable gas lines, the connections for the gas line use left-handed rather than right-handed threads, so nobody connects the wrong pipe by accident.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">On the other hand, a few of the uses described just seemed kind of silly, though I suppose they were for valid reasons.<span class="Apple-converted-space"> </span>In the early 20th century, many lamps used in subways were specially designed to have bulbs that screw in with left-handed threads, so nobody could steal them and use them at home. <span class="Apple-converted-space"> </span>It’s probably a sign of our society’s growing prosperity over the last century that most people don’t steal public light bulbs anymore.<span class="Apple-converted-space"> </span>More bafflingly, I found a few references to the use of left-hand threads in early ballpoint pens, to provide a “secret method” of disassembly. <span class="Apple-converted-space"> </span>I guess business meetings were a lot less boring back then— given the amount of idle fiddling I’ve typically done with my pens on a normal day at work, I can’t imagine that secret lasting very long.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.accu.co.uk/en/p/119-why-are-screws-right-handed">https://www.accu.co.uk/en/p/119-why-are-screws-right-handed</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="http://www.todayifoundout.com/index.php/2014/09/decided-screws-turn-clockwise/">http://www.todayifoundout.com/index.php/2014/09/decided-screws-turn-clockwise/</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Screw_thread">https://en.wikipedia.org/wiki/Screw_thread</a></span></li></ul><p></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com1tag:blogger.com,1999:blog-672497650672656575.post-43495216618343473942021-04-25T08:31:00.002-07:002021-04-25T08:31:37.220-07:00269: A Good Use of Downtime<p><a href="http://www.erikseligman.com/mm/mm269.mp3"> Audio Link</a></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The recent observation of Holocaust Remembrance Day reminded me of one of the stranger stories to come out of World War II.<span class="Apple-converted-space"> </span>John Kerrich was a South African mathematician who made the ill-fated decision to visit some family members in Denmark in 1940 just before the Germans invaded, and soon found himself interned in a prison camp.<span class="Apple-converted-space"> </span>We should point out that he was one of the luckier ones, as the Germans allowed the Danes to run their prison camps locally, and thus lived in extremely humane conditions compared to the majority of prisoners in German-occupied territories, <span class="Apple-converted-space"> </span>But being imprisoned still left him with many hours of time to fill over the course of the war. <span class="Apple-converted-space"> </span>Kerrich decided to fill this time by doing some experiments to demonstrate the laws of probability.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Kerrich’s main experiment was a very simple one: <span class="Apple-converted-space"> </span>he and a fellow prisoner, Eric Chirstensen, flipped a coin ten thousand times and recorded the results.<span class="Apple-converted-space"> </span>Now you might scratch your head in confusion when first hearing this— why would someone bother with such an experiment, when it’s so easy for anyone to do at home?<span class="Apple-converted-space"> </span>We need to keep in mind that back in 1940, the idea that everyone would have a computer at home (or, as we now do, in their pocket) that they could use for seemingly endless numbers of simulated coin flips, would have seemed like a crazy sci-fi fantasy.<span class="Apple-converted-space"> </span>Back then, most people had to manually engage in a physical coin flip or roll a die to generate a random number, a very tedious process.<span class="Apple-converted-space"> </span>Technically there were some advanced computers under development at the time that could do the simulation if programmed, but these were being run under highly classified conditions by major government entities. <span class="Apple-converted-space"> </span>So recording the value of ten thousand coin flips actually did seem like a useful contribution to math and science at the time.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">So, what did Kerrich accomplish with his coin flips?<span class="Apple-converted-space"> </span>The main purpose was to demonstrate the Law of Large Numbers.<span class="Apple-converted-space"> </span>This is the theorem that says that if you perform an experiment a large number of times, the average result will asymptotically approach the expected value.<span class="Apple-converted-space"> </span>In other words, if you have a coin that has 50-50 odds of coming up heads or tails, if you perform lots of trials, you will over time get closer and closer to 50% heads and 50% tails. <span class="Apple-converted-space"> </span>Kerrich’s coins got precisely 5,067 heads, and over the course of the experiment got closer and closer to the 50-50 ratio, thus providing reasonable evidence for the Law.<span class="Apple-converted-space"> </span>(In any 10000 flips, there is about an 18% chance that we will be off by at least this amount from the precise 50-50 ratio, so this result is reasonable for a single trial.)<span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Of course, it might make sense to request another trial of 10000 flips to confirm, for improved confidence in the result. <span class="Apple-converted-space"> </span>But apparently even in prison you don’t get bored enough for that— in his book, Kerrich wrote, “A way of answering the… question would be for the original experimenter to obtain a second sequence of 10000 spins, <span class="Apple-converted-space"> </span>Now it takes a long time to spin a coin 10000 times, and the man who did it objects strenuously to having to take the trouble of preparing further sets.”</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Kerrich and Christensen also did some other experiments along similar lines.<span class="Apple-converted-space"> </span>By constructing a fake coin with wood and lead, they created a biased coin to flip, and over the course of many flips demonstrated 70/30 odds for the two sides. <span class="Apple-converted-space"> </span>This experiment was probably less interesting because, unlike a standard coin, there likely wasn’t a good way to estimate its expected probabilities before the flips. <span class="Apple-converted-space"> </span>A more interesting experiment was the demonstration of Bayes’ Theorem using colored ping-pong balls in a box. <span class="Apple-converted-space"> </span>This theorem, as you may recall, helps us calculate the probability of an event when you have some knowledge prior conditions that affect the likelihood of each outcome.<span class="Apple-converted-space"> </span>The simple coin flip experiment seems to be the one that has resonated the most with reporters on the Internet though, perhaps because it’s the easiest to understand for anyone without much math background.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">In 1946, after the end of the war, Kerrich published his book, “An Experimental Introduction to the Theory of Probability.”.<span class="Apple-converted-space"> </span>Again, while it may seem silly these days to worry about publishing experimental confirmation of something so easy to simulate, and which has been theoretically proven on paper with very high confidence anyway, this really did seem like a useful contribution in the days before widespread computers.<span class="Apple-converted-space"> </span>The book seems intended for college math students seeking an introduction to probability, and in it Kerrich goes over many basics of the field as demonstrated by his simple experiments.<span class="Apple-converted-space"> </span>If you’re curious about the details and the graphs of Kerrich’s results, you can read the book online at <a href="http://openlibrary.org"><span class="s2" style="font-kerning: none;">openlibrary.org</span></a>, or click the link in our show notes at <a href="http://mathmutation.com"><span class="s2" style="font-kerning: none;">mathmutation.com</span></a> . <span class="Apple-converted-space"> </span>Overall, we have to give credit to Kerrich for managing to do something mathematically useful during his World War II imprisonment. <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/John_Edmund_Kerrich">https://en.wikipedia.org/wiki/John_Edmund_Kerrich</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Law_of_large_numbers">https://en.wikipedia.org/wiki/Law_of_large_numbers</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Bayes%27_theorem">https://en.wikipedia.org/wiki/Bayes%27_theorem</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://stats.stackexchange.com/questions/76663/john-kerrich-coin-flip-data">https://stats.stackexchange.com/questions/76663/john-kerrich-coin-flip-data</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://openlibrary.org/">https://openlibrary.org/</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.slader.com/discussion/question/while-he-was-a-prisoner-of-war-during-world-war-ii-john-kerrich-tossed-a-coin-10000-times-he-got-506/">https://www.slader.com/discussion/question/while-he-was-a-prisoner-of-war-during-world-war-ii-john-kerrich-tossed-a-coin-10000-times-he-got-506/</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/dp/B014RT1M1U/">https://www.amazon.com/dp/B014RT1M1U/</a></span></li></ul><p></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-52969434183894123412021-03-20T05:22:00.002-07:002021-03-20T05:22:22.905-07:00268: The Right Way to Gamble<p><a href="http://www.erikseligman.com/mm/mm268.mp3">Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">At some point, you’ve probably heard an urban legend like this: <span class="Apple-converted-space"> </span>someone walks into a Las Vegas casino with his life savings, converts it into chips, and bets it all on one spin of the roulette wheel.<span class="Apple-converted-space"> </span>In the version where he wins, he walks out very wealthy. <span class="Apple-converted-space"> </span>But, in the more likely scenario where he loses, he leaves totally ruined.<span class="Apple-converted-space"> </span>This isn’t just an urban legend, by the way, but it has actually happened numerous times— for example, in 2004, someone named Ashley Revell did this with $135,000, and ended up doubling his money in one spin at roulette.<span class="Apple-converted-space"> </span>I most recently read about this incident in a book called “Chancing it:<span class="Apple-converted-space"> </span>The Laws of Chance and How They Can Work for You”, by Robert Matthews. <span class="Apple-converted-space"> </span>And Matthews makes the intriguing point that if you are desperate to significantly increase your wealth ASAP, and want to maximize the chance of this happening, what Revell did might not be entirely irrational.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now, before we get into the details, I want to make it clear that I’m not recommending casinos or condoning gambling.<span class="Apple-converted-space"> </span>Occasionally while on vacation my wife & I will visit a casino, and here’s my foolproof winning strategy.<span class="Apple-converted-space"> </span>First you walk in, let yourself take in the dazzling atmosphere of the flashing lights, sounds, and excitement.<span class="Apple-converted-space"> </span>Then walk over to the bar, plop down ten bucks or so, and buy yourself a tasty drink. <span class="Apple-converted-space"> </span>Sit down at the counter, take out your smartphone, start up the Kindle app, and read a good book. <span class="Apple-converted-space"> </span>(The Matthews book might be a nice choice, linked in the show notes at mathmutation.com.)<span class="Apple-converted-space"> </span>Then relax in the comfortable knowledge that you’re ahead of the casino by one pina colada, which you probably value more than the ten dollars at that particular time. <span class="Apple-converted-space"> </span>Don’t waste time attempting any of the actual casino games, which always have odds that fundamentally are designed to make you you lose your money.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Anyway, getting back to the Revell story, let’s think for a minute about those rigged odds in a casino. <span class="Apple-converted-space"> </span>Basically, the expected value of your winnings, or the probability of winning each value times the amount of money, is always negative. <span class="Apple-converted-space"> </span>So, for example, let’s look at betting red or black in roulette.<span class="Apple-converted-space"> </span>This might seem like a low-risk bet, since there are two colors, and the payout is 1:1. <span class="Apple-converted-space"> </span>When you look closely at the wheel, though, you’ll see that in addition to the 36 red or black numbers, there are 2 others, a green 0 and 00. <span class="Apple-converted-space"> </span>Thus, if you bet on red, your chances of winning aren’t 18/36, but 18/38, or about 47.37%. <span class="Apple-converted-space"> </span>That’s the sneaky way the casinos get their edge in this case. <span class="Apple-converted-space"> </span>As a result, your expected winnings for each dollar you bet are around negative 5.26 cents.<span class="Apple-converted-space"> </span>This means that if you play for a large number of games, you will probably suffer a net loss of a little over 5% of your money.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">So let’s assume that Revell desperately needed to double his money in one day— perhaps his stockbroker had told him that if he didn’t have $200,000 by midnight, he would lose the chance to invest in the Math Mutation IPO, and he couldn’t bear the thought of missing out on such a cultural milestone. <span class="Apple-converted-space"> </span>Would it make more sense for him to divide his money into small bets, say $1000, and play roulette 135 times, or gamble it all at once? <span class="Apple-converted-space"> </span>Well, we know that betting it all at once gave about a 47% chance of doubling it— pretty good, almost 50-50 odds, even though the casino still has its slight edge. <span class="Apple-converted-space"> </span>But if he had bet it slowly over 100+ games, then the chances would be very high that his overall net winnings would be close to the expected value— so he would expect to lose about 5% of his money, even if he put his winnings in a separate pocket rather than gambling them away.<span class="Apple-converted-space"> </span>In other words, in order to quickly double his money in a casino, Revell’s single bold bet really was the most rational way to do it.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/dp/B014RT1M1U/">https://www.amazon.com/dp/B014RT1M1U/</a></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-43427542523665542352021-02-07T09:34:00.001-08:002021-02-07T09:34:07.862-08:00267: Free Will Isn't Free <p><a href="http://www.erikseligman.com/mm/mm267.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">If you’ve browsed the web sometime in the last three decades, especially if you visit any New Age or philosophy websites, you’ve probably come across the argument that quantum mechanics proves the existence of free will. <span class="Apple-converted-space"> </span>Superficially, this seems somewhat plausible, in that quantum mechanics blows away our traditional idea that we can fully understand the future behavior of the universe based on observable properties of its particles and forces. <span class="Apple-converted-space"> </span>Free will seems like a nice way to fill in the gap. <span class="Apple-converted-space"> </span>But when you think about it in slightly more detail, there seems to be a fatal flaw in this argument. <span class="Apple-converted-space"> </span>So let’s take a closer look.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">To start with, what is so special about quantum mechanics? <span class="Apple-converted-space"> </span>This is the area of physics, developed in the 20th century, that tries to explain the behaviors we observe in subatomic particles. <span class="Apple-converted-space"> </span>What’s bizarre about it is that according to its calculations, as Einstein stated, God is playing dice with the universe:<span class="Apple-converted-space"> </span>we can calculate probabilities of the position and momentum of particles, but not the exact values until we observe them. <span class="Apple-converted-space"> </span>For example, suppose I am throwing a toy mouse for my cat to catch.<span class="Apple-converted-space"> </span>Since this is a macroscopic object, classical physics works fine:<span class="Apple-converted-space"> </span>my cat can take out a calculator, and based on the force I throw with, the mass of the mouse, and the effects of gravity, he will be able to figure out exactly where to pounce. <span class="Apple-converted-space"> </span>But now suppose I am throwing a photon for him to catch. <span class="Apple-converted-space"> </span>Even if he knows everything that is theoretically knowable about my throw, he will not be able to calculate exactly where it will land— he can only calculate a set of probabilities, and then figure out where the photon went by observing it.<span class="Apple-converted-space"> </span>Of course, my cat uses the observation technique even with macroscopic mice, so no doubt he has read a few books on quantum physics and is trying to use the most generally applicable hunting method.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">But how does this lead to free will? <span class="Apple-converted-space"> </span>The most common argument is that since things happen in the universe that cannot be precisely calculated from all the known properties of its particles and energies, there must be another factor that determines what is happening. <span class="Apple-converted-space"> </span>Many physicists seem to believe that a hidden physical factor has been largely ruled out. <span class="Apple-converted-space"> </span>That means the missing piece, according to the free will argument, is likely to be human consciousness.<span class="Apple-converted-space"> </span>When the activities of the particles in our brain could determine multiple possible outcomes, it is our consciousness that chooses which one will actually happen.<span class="Apple-converted-space"> </span>The quantum events in my brain, for instance, could lead with equal probability to me recording a podcast this afternoon, or playing video games. <span class="Apple-converted-space"> </span>My free will is needed to make the final choice.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Now time for the fundamental flaw in this argument.<span class="Apple-converted-space"> </span>Suppose the quantum activities in my brain do ultimately give me a 50% chance of recording a podcast this afternoon, and a 50% chance of playing videogames. <span class="Apple-converted-space"> </span>The implicit assumption in the free will argument is that if the properties of my brain determined completely that I would make a podcast this afternoon, then I would not have free will. <span class="Apple-converted-space"> </span>Since there are two alternatives and I have to choose one, there is free will.<span class="Apple-converted-space"> </span>However, remember that the quantum calculations provided an exact probability for each event, not just a vague uncertainty, and these probabilities have been well-confirmed in the lab. <span class="Apple-converted-space"> </span>If I’m required to roll a 6-sided die, and record the podcast if numbers 1-3 come up, or play videogames if 4-6 comes up, aren’t I just as constrained as if I were only allowed one of those options? <span class="Apple-converted-space"> </span>I still don’t get any freedom here.<span class="Apple-converted-space"> </span>Either way, it’s not a question of my consciousness, it’s just another calculation, though one whose result cannot be predicted. <span class="Apple-converted-space"> </span>If I’m forced to do certain things with known probabilities, that’s the opposite of free will.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">There is one more wrinkle here though, which may rescue the free will argument. <span class="Apple-converted-space"> </span>Suppose we interpret the quantum calculation slightly differently. <span class="Apple-converted-space"> </span>Yes, there is a 50% chance of me recording the podcast, and 50% of playing a videogame. <span class="Apple-converted-space"> </span>But maybe this isn’t the universe rolling a die— maybe it’s a measure of the type of personality created by the sum total of quantum interactions in my brain. <span class="Apple-converted-space"> </span>So rather than being forced to roll a virtual die and decide, the universe has just built me into the kind of guy who, given a choice, would have a 50-50 chance of podcasting or gaming this afternoon.<span class="Apple-converted-space"> </span>All the quantum calculations about my brain activity are just figuring out the type of personality that has been created by its construction. <span class="Apple-converted-space"> </span>That argument rescues free will in the presence of quantum probabilities. <span class="Apple-converted-space"> </span>Though it does put us in the odd position of wanting to ascribe some kind of conscious will to subatomic particles observed in a physics lab, or to assume some hidden spirits in the room are making decisions for them.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">So what’s the answer here?<span class="Apple-converted-space"> </span>Well, I’m afraid that as often happens with these types of questions, we will not resolve it in a 5-minute podcast.<span class="Apple-converted-space"> </span>You can also find some more subtle arguments that incorporate other aspects of quantum physics, though those don’t look too convincing to me either. <span class="Apple-converted-space"> </span>If humanity truly doesn’t have free will, I’ll look forward to the day when someone writes a treatise on how a large Big Bang of hydrogen atoms fundamentally leads to Math Mutation podcasts a few billion years later.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; text-align: left;">And this has been your math mutation for today.</p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://bigthink.com/experts-corner/quantum-mechanics-supports-free-will">https://bigthink.com/experts-corner/quantum-mechanics-supports-free-will</a></span><span class="s1" style="font-kerning: none;"> </span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://conscienceandconsciousness.com/2020/12/11/does-quantum-mechanics-allow-for-free-will/">https://conscienceandconsciousness.com/2020/12/11/does-quantum-mechanics-allow-for-free-will/</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com1tag:blogger.com,1999:blog-672497650672656575.post-22664859049125746002020-12-25T18:02:00.000-08:002020-12-25T18:02:11.883-08:00266: A Number is a Number<p><a href="http://www.erikseligman.com/mm/mm266.mp3"> Audio Link</a></p><p><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">You may recall that in our last episode we discussed the results of 19th-century attempts to rigorously define the concept of whole numbers.<span class="Apple-converted-space"> </span>These attempts culminated in the Peano axioms, a set of simple properties that defined numbers on the basis of the primitive concepts of zero and succession, plus a few related rules. <span class="Apple-converted-space"> </span>While this definition has its merits, Bertrand Russell pointed out that it also has some major flaws:<span class="Apple-converted-space"> </span>sets that we don’t think of as whole numbers, such as all numbers above 100, all even numbers, or all inverted powers of two, could also satisfy these axioms. <span class="Apple-converted-space"> </span>So, how did Russell propose to define numbers?</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Here’s Russell’s definition: <span class="Apple-converted-space"> </span>“A number is anything which is the number of some class”.<span class="Apple-converted-space"> </span>Great, problem solved, we can end the podcast early today! <span class="Apple-converted-space"> </span>Or…<span class="Apple-converted-space"> </span>maybe not. <span class="Apple-converted-space"> </span>Let’s explore Russell’s concepts a bit, to figure out why this definition isn’t as circular as it seems.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The basic concept here is that we think of a whole number as a description of a class of sets, all sets which contain that number of elements.<span class="Apple-converted-space"> </span>Let’s take a look at the number 2. <span class="Apple-converted-space"> </span>The set of major US political parties, the set of Mars’s moons, and the set of my daughter’s cats are all described by the number two. <span class="Apple-converted-space"> </span>But how do we know this? <span class="Apple-converted-space"> </span>You might say we could just count the elements in each one to see they have the same number— but Russell points out that that would be cheating, since the concept of counting can only exist if we already know about whole numbers.<span class="Apple-converted-space"> </span>So what do we do?</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The concept of 1-1 correspondence between sets comes to the rescue.<span class="Apple-converted-space"> </span>While we can’t count the elements of a set before we define whole numbers, we can describe<i> similar </i>sets:<span class="Apple-converted-space"> </span>a pair of sets are similar if their elements can be put into direct 1-1 correspondence, without any left out. <span class="Apple-converted-space"> </span>So despite lacking the intellectual power to count to 2, I can figure out that the number of my daughter’s kittens and the number of moons of Mars are the same:<span class="Apple-converted-space"> </span>I’ll map Mars’s moon Phobos to Harvey, and Mars’s moon Deimos to Freya, and see that there are no moons or cats left over.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Thus, we are now able to look at two sets and figure out if they belong in the same class of similar sets. <span class="Apple-converted-space"> </span>Russell defines the <i>number </i>of a class of sets as the class of all sets that are similar to it. <span class="Apple-converted-space"> </span>Personally, I think this would have been a bit clearer if Russell hadn’t chosen to overload the term ‘number’ here, using it twice with slightly different definitions.<span class="Apple-converted-space"> </span>So let’s call the class of similar sets a <i>numerical grouping</i> for clarity. <span class="Apple-converted-space"> </span>Then the definition we started with, “A number is anything which is the number of some class”, becomes “A number is anything which is the numerical grouping of some class”, which at least doesn’t sound quite as circular.<span class="Apple-converted-space"> </span>The wording gets a little tricky here, and I’m sure some Russell scholars might be offended at my attempt to clarify it, but I think the key concept is this: <span class="Apple-converted-space"> </span>A number is defined as a class of sets, all of which can be put into 1-1 correspondence with each other, and which, if we conventionally count them (not allowed in the definition, but used here for clarification), have that number of elements.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Is this more satisfying than the Peano axioms?<span class="Apple-converted-space"> </span>Well, if we identify zero with the empty set and the succession operation with adding an element to a set and finding its new number, we can see that those axioms are still satisfied. <span class="Apple-converted-space"> </span>Furthermore, this interpretation does seem to rule out the pathological examples Russell mentions:<span class="Apple-converted-space"> </span>the numbers greater than 100, even numbers, and inverted powers of two all fail to meet this set-based definition.<span class="Apple-converted-space"> </span>And Russell successfully used this definition as the basis for numerous further significant mathematical works. <span class="Apple-converted-space"> </span>On the other hand, Russell’s method was not the final word on the matter: <span class="Apple-converted-space"> </span>philosophers of mathematics continue to propose and debate alternate definitions of whole numbers to this day.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Personally, I know a whole number when I see it, and maybe that’s all the definition a non-philosopher needs on a normal day. <span class="Apple-converted-space"> </span>But it’s nice to know there are people out there somewhere thinking hard about why this is true.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Peano_axioms">https://en.wikipedia.org/wiki/Peano_axioms</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/Introduction-Mathematical-Philosophy-Bertrand-Russell/dp/1684221447">https://www.amazon.com/Introduction-Mathematical-Philosophy-Bertrand-Russell/dp/1684221447</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers">https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers</a></span></li></ul><p></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0tag:blogger.com,1999:blog-672497650672656575.post-17443311398976048842020-11-30T18:15:00.002-08:002020-11-30T18:15:45.754-08:00265: Defining Numbers, Sort Of<p><a href="http://www.erikseligman.com/mm/mm265.mp3"> Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">One of the amazing things about mathematics in general is the way we can continuously discover and prove new results on the basis of simple definitions and axioms.<span class="Apple-converted-space"> </span>But in a way, this concept is similar to those ancient myths that our planet is sitting on the back of a giant turtle. <span class="Apple-converted-space"> </span>It sits on another turtle, which sits on another, and it’s turtles all the way down. <span class="Apple-converted-space"> </span>Where’s the bottom?<span class="Apple-converted-space"> </span>In order to prove anything, we need to start from somewhere, right? <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">For millennia after the dawn of mathematics, people generally assumed that the starting point had to be the whole numbers and their basic operations: addition, multiplication, etc.<span class="Apple-converted-space"> </span>How much simpler could you get than that? <span class="Apple-converted-space"> </span>But in the 19th century, philosophers and mathematicians began serious efforts at trying to improve the overall rigor of their endeavor, by defining simpler notions from which you could derive whole numbers and prove their basic properties. <span class="Apple-converted-space"> </span>The average person may not need proofs that whole numbers exist, but mathematicians can be a bit picky sometimes. <span class="Apple-converted-space"> </span>One of the most successful of these efforts was by Giuseppe Peano from Italy, who published his set of axioms in 1889.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The Peano axioms can be stated in several equivalent forms, but for the moment we’ll use the version in Bertrand Russell’s nice “Introduction to Mathematical Philosophy”.<span class="Apple-converted-space"> </span>As Russell states it, they are based on three primitive notions and 5 axioms.<span class="Apple-converted-space"> </span>The notions are the concepts of zero, number, and successor. <span class="Apple-converted-space"> </span>Note that he’s not saying we start by knowing what all the numbers are, just that we are assuming some set exists which we are calling “numbers”.<span class="Apple-converted-space"> </span>Based on these, the 5 axioms are:</span></p><ol class="ol1"><li class="li1" style="-webkit-text-stroke: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"></span><span class="s1" style="font-kerning: none;">Zero is a number.</span></li><li class="li1" style="-webkit-text-stroke: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"></span><span class="s1" style="font-kerning: none;">The successor of any number is a number.</span></li><li class="li1" style="-webkit-text-stroke: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"></span><span class="s1" style="font-kerning: none;">No two numbers have the same successor.</span></li><li class="li1" style="-webkit-text-stroke: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"></span><span class="s1" style="font-kerning: none;">Zero is not the successor of any number. <span class="Apple-converted-space"> </span>(Remember that we’re just defining whole numbers here; negative numbers will potentially be a later extension to the system.)</span></li><li class="li1" style="-webkit-text-stroke: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s2" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"></span><span class="s1" style="font-kerning: none;">Induction works: <span class="Apple-converted-space"> </span>if some property P belongs to 0, and we can prove that if P is true for some number, it’s true for its successor, then P is true for all numbers.<span class="Apple-converted-space"> </span></span></li></ol><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">With these primitive notions, we can derive the existence of all the whole numbers, without having them known at the start.<span class="Apple-converted-space"> </span>For example, we know Zero has a successor which is a number, so let’s label that 1. <span class="Apple-converted-space"> </span>Then 1 has a successor number as well, so let’s call that 2, and so on. <span class="Apple-converted-space"> </span>We can also define the basic operations we’re used to, simply building on these axioms:<span class="Apple-converted-space"> </span>for example, let’s define addition.<span class="Apple-converted-space"> </span>We’ll create a plus operation, and define “a + 0” as equal to a for any number a.<span class="Apple-converted-space"> </span>Then we can define “a + the successor of b” as “the successor of (a+b)”.<span class="Apple-converted-space"> </span>So, for example, a + 1 equals a + “the successor of 0”, which becomes “the successor of a + 0”; by our original definition, this boils down to “the successor of a”. <span class="Apple-converted-space"> </span>Thus we have shown that the operation a+1 always leads to a’s successor, a good sign that we are using a reasonable definition of addition. <span class="Apple-converted-space"> </span>We can similarly define other operations such as multiplication and inequalities.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">The Peano axioms were quite successful and useful, and a great influence on the progression of the foundations of mathematics.<span class="Apple-converted-space"> </span>Yet Russell points out that they had a few key flaws. <span class="Apple-converted-space"> </span>Think again about the ideas we started with:<span class="Apple-converted-space"> </span>zero, successors, and induction. <span class="Apple-converted-space"> </span>They certainly apply to the natural numbers… but could they apply to other things, that are not what we would think of as the set of natural numbers? <span class="Apple-converted-space"> </span>The answer is yes— there are numerous other sets that can satisfy the axioms. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">- One example:<span class="Apple-converted-space"> </span>let’s just define our “zero” for these axioms as the conventional whole number 100. <span class="Apple-converted-space"> </span>Then what is being described is the set of whole numbers above 100. <span class="Apple-converted-space"> </span>If you think about it for a minute, this won’t violate any of Peano’s axioms—<span class="Apple-converted-space"> </span>we are still defining a set of unique numbers with successors, none of which is below our “zero”, and to which induction applies. <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">- As another example, let’s keep zero as our conventional zero, but define the “successor” operation as adding 2. <span class="Apple-converted-space"> </span>Now our axioms describe the set of even whole numbers.<span class="Apple-converted-space"> </span>A very useful set, indeed, but not the true set of whole numbers we were aiming to describe.</span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">- As an even more absurd case, let’s define our “zero” as the conventional number 1, and the successor operation as division by 2. <span class="Apple-converted-space"> </span>Then we are describing the infinite progression 1, 1/2, 1/4, 1/8, and so on.<span class="Apple-converted-space"> </span>Another very useful series, but not at all matching our intention of describing the whole numbers.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Choosing alternate interpretations of this set, naturally, can lead to very weird interpretations of our derived operations such as addition and multiplication. <span class="Apple-converted-space"> </span>But Russell’s point is that despite the power and utility of Peano’s axioms, there is clearly something lacking.<span class="Apple-converted-space"> </span>This situation actually reminds me a bit of some challenges we encounter in my day job, proving that computer chip designs work correctly: <span class="Apple-converted-space"> </span>it’s nice if you can prove that your axioms lead to desired conclusions for your design, but you also need evidence that there aren’t also BAD designs that would satisfy those axioms equally well. <span class="Apple-converted-space"> </span>If such bad designs do exist, your job isn’t quite done.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Russell’s answer to this issue was to seek improved approaches to defining whole numbers based on set theory, which would more precisely correspond to our notion of what these numbers really are.<span class="Apple-converted-space"> </span>We’ll discuss this topic in a future podcast. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s3" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Peano_axioms">https://en.wikipedia.org/wiki/Peano_axioms</a></span></li><li><span class="s3" style="font-kerning: none; text-decoration-line: underline;"><a href="https://www.amazon.com/Introduction-Mathematical-Philosophy-Bertrand-Russell/dp/1684221447">https://www.amazon.com/Introduction-Mathematical-Philosophy-Bertrand-Russell/dp/1684221447</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com10tag:blogger.com,1999:blog-672497650672656575.post-75728487845762485742020-10-11T18:07:00.003-07:002020-10-11T18:07:31.678-07:00264: Unbalanced Society<p><a href="http://www.erikseligman.com/mm/mm264.mp3">Audio Link</a></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">You may recall that in several past episodes I mentioned the odd pop philosopher Alfred Korzybski and his early 20th-century movement known as General Semantics. <span class="Apple-converted-space"> </span>Korzybski believed that the imprecision and misuse of language was responsible for many of society’s ills. <span class="Apple-converted-space"> </span>He came up with many supposedly practical ideas to fix this such as using “indexing” and “dating” to add numerical tags to objects you reference, and minimizing the use of the verb “to be” due to its many possible meanings.<span class="Apple-converted-space"> </span>A few weeks ago I discovered that one of his books, “Manhood of Humanity”, was downloadable for free at Project Gutenberg, and couldn’t resist taking a look to see if there were any more amusing ideas there. <span class="Apple-converted-space"> </span>And I did find one:<span class="Apple-converted-space"> </span>a supposed mathematical explanation for the many societal upheavals and conflicts of the past century.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Basically, Korzybski was looking at the pace of change in various fields of knowledge that we have been acquiring over time. <span class="Apple-converted-space"> </span>He made much of his observation that we are what he calls “time-binding” creatures: <span class="Apple-converted-space"> </span>unlike any other creature on the planet, we can learn things and pass them down to our descendants, so the process of learning and development happens at the level of human society, rather than just of individuals.<span class="Apple-converted-space"> </span>According to him, the simple, natural way most knowledge accumulates is through linear or arithmetic progressions: <span class="Apple-converted-space"> </span>a series like 2, 4, 6, 8, 10, where you steadily move forward by small jumps. <span class="Apple-converted-space"> </span>However, there are certain domains, in areas of science and technology, where in recent centuries every piece of new knowledge has drawn on massive amounts of previous ideas, creating a geometric, or exponential, progression, like 2, 4, 8, 16, 32, etc.<span class="Apple-converted-space"> </span>This disconnect is the source of many of our problems. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">In other words, early historical growth in social and political areas roughly tracked with growth of technology, but now technology has zoomed ahead of our social development due to this disconnect. <span class="Apple-converted-space"> </span>And this cannot be good for humanity as a whole. <span class="Apple-converted-space"> </span>As Korzybski stated, “It is plain as the noon-day sun that, if progress in one of the matters advances according to the law of a geometric progression and the other in accordance with a law of an arithmetical progression, progress in the former matter will very quickly and ever more rapidly outstrip progress in the latter, so that, if the two interests would be interdependent (as they always are), a strain is gradually produced in human affairs, social equilibrium is at length destroyed; there follows a period of readjustment by violence and force.” <span class="Apple-converted-space"> </span>He then goes on to state that this is a key cause of insurrections, revolutions, and wars.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">This idea seems like it might actually have something to it, to some degree.<span class="Apple-converted-space"> </span>But we do have to be careful here— while everyone is always focused on their own time, and our news media love to sensationalize wars and violence, there has been violence and war throughout the history of society. <span class="Apple-converted-space"> </span>Many argue that the real oddity of modern times is the proportion of humanity who can live out their lives secure from violence. <span class="Apple-converted-space"> </span>Korzybski did write this just after World War I, though, so we can understand why it might have looked to him like society was falling apart.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">Where he really went off the rails though is when he tried to prescribe a solution for this disconnect in societal vs technological growth: <span class="Apple-converted-space"> </span>everyone must use his system of more precise and scientifically defined language, correctly defining ideas like “good”, “bad”, and “truth”, and then we will enable exponential growth in all fields of knowledge. <span class="Apple-converted-space"> </span>In his words:<span class="Apple-converted-space"> </span>“If only these three words could be scientifically defined, philosophy, law, ethics, and psychology would cease to be private theories or verbalism and they would advance to the rank and dignity of sciences.” <span class="Apple-converted-space"> </span>He even claimed that such correct definitions would have lead to the kind of scientific societal reasoning that could have predicted and prevented World War I. <span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">But when he tried to actually apply his reasoning to a practical matter, we see some slightly more concerning comments.<span class="Apple-converted-space"> </span>He tried to use the need to come up with a common base to combine like terms to derive the need for a government to unite the people. <span class="Apple-converted-space"> </span>Just as you cannot combine algebraic terms like x^a + y^b without finding<span class="Apple-converted-space"> </span>common base, you must find the people a “common base” to unite them. <span class="Apple-converted-space"> </span>He wrote, “Germany united the powers of living men and women and children; it gave them a common base; it gave them one common social mood and aim; they all became consolidated in service of that which is called the State… they worked, lived, and died for the State.”<span class="Apple-converted-space"> </span>He seemed to like this idea, only complaining that the German leadership then chose the wrong aims for their “united terms”.<span class="Apple-converted-space"> </span></span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">This concept of making the social sciences more precise and mathematical seems to appear continuously in 20th-century writing, from many authors. <span class="Apple-converted-space"> </span>It always ultimately fails:<span class="Apple-converted-space"> </span>aside from the inherent imprecision of the concepts involved,<span class="Apple-converted-space"> </span>there is an obvious need for value judgements that cannot be sensibly derived from any mathematics.<span class="Apple-converted-space"> </span>In the end, Korzybski provided a few intriguing ideas, buried within loads and loads of sophistry and nonsense. <span class="Apple-converted-space"> </span>I always find it amusing to read this kind of stuff, as long as we all remember not to take it too seriously.<span class="Apple-converted-space"> </span>If we want to solve modern society’s problems, we can’t just lie back & let the math provide a magic formula.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">And this has been your math mutation for today.</span></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span class="s1" style="font-kerning: none;">References: <span class="Apple-converted-space"> </span></span></p><p class="p1" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"></p><ul style="text-align: left;"><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="https://en.wikipedia.org/wiki/Alfred_Korzybski">https://en.wikipedia.org/wiki/Alfred_Korzybski</a></span></li><li><span class="s2" style="font-kerning: none; text-decoration-line: underline;"><a href="http://www.gutenberg.org/files/25457/25457-pdf.pdf">http://www.gutenberg.org/files/25457/25457-pdf.pdf</a></span></li></ul><p></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p class="p2" style="-webkit-text-stroke-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 11px; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 13px;"><span class="s1" style="font-kerning: none;"></span><br /></p><p><br /></p>erik seligmanhttp://www.blogger.com/profile/13081739911203981726noreply@blogger.com0