214: In Search Of The Ultimate Math Game

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With the holiday season upon us, many of you out there are probably giving or getting new tech gadgets as gifts.    Once you unwrap your fancy new iPad, iPhone, or Android tablet, you’re probably asking yourself, “Now what do I do with this?”    While you will probably be downloading lots of fun games and apps, you will somehow need to justify all the hours you spend in front of the screen to your family.    One of the best ways to do this is to install a few math-related games, and provide some educational value for your children.   But there is a bewildering array of supposedly educational games available for these systems.   How do you know which ones to get?     Today I will share some suggestions based on my experiences.

The first thing I should point out is that there are hundreds of games out there that are basically glamorized flashcards, presenting math problems directly and giving some kind of in-game reward for correct solutions.    For example, they will put up a math problem, like “What is 5 x 5?”, and if the answer is correct, the player gets a few points.   These points can then be traded in for virtual stickers, virtual ammunition against alien robots, or similar rewards.   While there is nothing wrong with this type of game, and they have the advantage of being able to easily draw on large libraries of problems for different skill levels, I don’t find them very exciting.    My daughter will play them if I tell her she needs to practice her math, but doesn’t usually come to me asking to play them.   What I really want are original games that both teach math and can stand on their own as fun games.   Fortunately, I have found two games that fit these requirements.

The first game I want to highlight is called “DragonBox Elements”.    This game is designed to teach the basics of geometric proofs, a seemingly advanced topic, but they present it in a very accessible and intuitive way.    Each basic shape, triangles and quadrilaterals, can summon a basic type of monster related to that shape.    So if you can identify a quadrilateral among the shapes on the screen, you trace it out and summon a quadrilateral-monster.   Line segments and angles are also marked with colors, such that any two objects with the same color have equal length, and you can upgrade the monsters to “special” ones using these.   So, for example, if you notice a triangle-monster has two equal sides, you can click on them to upgrade to the slightly more powerful isoceles-monster.   The monsters also have powers, which essentially invert this process:  so if you have been given an isosceles-monster and its two equal sides are not yet colored, you can click on the monster and the two sides to mark them as equal.   They also introduce other powers related to ideas like opposite angles, radii of circles, and parallel lines.   So the basic Euclidean concepts of definitions, axioms, and theorems have been transformed into monsters and powers.    I’m not totally sure how this will translate to actual proof skills when my daughter reaches that level of math class, but laying the foundations at such a young age can’t hurt.   And more importantly, she loves this game, even asking to replay all the levels at “hard” difficulty after beating it once.

The second truly engaging iPad math game I have discovered is called “Calculords”.   This is a card game, where each turn you have a bunch of cards in hand that you can use to summon creatures for battle.   There are two types of cards, number cards and creature cards.   It’s not a simple energy system like in most popular collectible card games though:   in order to summon a creature for battle, you need to add, subtract, and multiply number cards to reach the creature’s number.   The creatures are then placed on a lane-based battlefield, where they fight the evil monsters summoned by an alien enemy.    For example, suppose you have a Hungry Blob card, a monster with a summoning cost of 15, and your number cards are 3, 3, 4, and 1.   You can form a 15 using 3 x 4 + 3, so you can play those cards to summon your blob.   But an additional wrinkle, adding to the mathematical challenge, is that you also gain extra bonuses if you precisely use up all your number or creature cards.   So a better move would be to play 3 x 4 x 1 + 3, which still reaches your 15, but uses up your numbers.    Since you have 9 creature cards and 9 number cards on each turn, the number of potential choices and calculations is quite large, and the strategy to summon the best set of monsters while trying to use up cards to get the bonus can get very involved.    But the game offers many enemies at a variety of difficulty levels;  my daughter has been playing at the easier levels since she was in 2nd grade.   This is another game that she and I have found quite addictive, and an amazing way to get her to eagerly practice her basic arithmetic.   And at the top difficulty levels, even I find it challenging, when I sneak in a chance to play on my own.

So, in short, these are the two truly original smartphone/tablet math games I currently recommend for elementary-age students:   DragonBox Elements and Calculords.   Naturally, these are heavily influenced by my 4th-grade daughter’s tastes, and their effectiveness probably varies a lot at older and younger ages.   DragonBox elements provides the amusing and engaging transformation of Euclidean definitions, axioms, and theorems into monsters and powers.   And Calculords provides a strategic challenge involving arithmetic calculations that is accessible to young children at lower levels, and fun even for adult math geeks at the hardest settings.   If you have kids at the upper elementary level who could use some extra math practice, be sure to take a look at these excellent games.   Also be sure to post reviews on iTunes or similar sites if you like them, as this will increase the chance of further games appearing from these talented authors..   And as always, I’ll be interested to hear from you on this topic:  with such an overwhelming number of smartphone and tablet games out there, I’m sure there are a few great ones that I haven’t discovered yet.   

And this has been your math mutation for today.

References:  

• http://calculords.com
• http://dragonbox.com/elements

213: Proof of the Fourth Dimension

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Rudolf Steiner was a prolific Austrian author and philosopher of the late 19th and early 20th centuries.    He felt a strong connection to mysticism and spiritualism, ever since he supposedly communicated with the ghost of a recently deceased aunt at the age of 9.   Steiner is well-known for having led a group that split off from the popular circle of European mystics known as the Theosophical Society, which seemed heavily inclined to regard the religions of East Asia as somehow providing the keys to understanding spirituality.   Steiner called his new group the Anthroposophical Society, and this competing group believed that Western science and culture were just as strongly connected to the spiritual-- it was just a matter of intepreting them properly.    One particular Western idea that Steiner was fond of was the concept of a fourth physical dimension, another mathematically-defined direction that we cannot percieve but is just as real as length, width, and height.     Steiner believed that our consciousness extended into this fourth dimension, and that phenomena like ghosts and ESP resulted from activity in this hidden dimension.      And most interestingly, he believed he had a simple philosophical proof that this fourth dimension really does exist, and our human minds really do extend into this additional dimension.

Here's how Steiner's proof goes.   We all know that a creature of a particular dimension, if it looks out at its world, really only sees a view that is one dimension smaller.   For example, a one-dimensional creature living in Lineland, a universe that exists entirely on a single straight line, can only perceive a single point on either side of himself:   a zero-dimensional view.    Similarly, a two-dimensional Flatlander, living on a plane, really only sees a line;  it is only us three-dimensional creatures, looking down on the plane from above, who can truly comprehend its full two-dimensional world.    And in real life, when we look out with our eyes, we are only seeing a plane.   Yet somehow we do believe we fully understand and perceive the three dimensions of our world.   Steiner draws what he believes is a natural conclusion from this:   "The fact that we can delineate external beings in three dimensions and manipulate three-dimensional spaces means that we ourselves must be four-dimensional...  We float in a sea of the fourth dimension just like ice cubes on water."   In other words, our ability to fully perceive our three-dimensional space shows that our minds must extend beyond those three dimensions.  

It's a fun thought, but you can see something fishy there right away, if you think about the world of modern computing.    I can think of all sorts of situations in which an object in three dimensions is represented by a model in fewer dimensions.     For example, most computer memories and circuits that power modern three-dimensional computer games are essentially stored in flat two-dimensional circuit boards.   While these are technically 3-D like all physical objects, the memory storage can be thought of as truly two-dimensional in some sense, as each (x,y) coordinate on the circuit board only stores one encoded value at any given time.    More basically, you may recall the concept of a Turing Machine discussed in some earlier podcasts:  this is a theoretical model of computing, based on writing and reading values from a long, essentially one-dimensional, tape.   It has been shown that any modern computer can be modelled by a very slow, but 100% accurate, Turing machine equivalent.    So even the 3-D models in a modern computer game could, with enough work, be represented in one dimension.

I think the main flaw in Steiner's argument is his fundamental premise, that a creature of n dimensions can only perceive n-1 dimensions.   It is true that through the sense of sight, a creature can only see one dimension lower, but our senses are not limited to sight.   Think about a blind man, who perceives the world mainly by walking around and tapping items with his cane to understand their form:  he can walk forward, back, right, or left, and even climb ladders up and down.    He is truly perceiving the full three dimensions of his world, travelling within all three of those dimensions and building a mental model based on his real experiences.    This applies to the lower-dimensional examples as well:   the flatlander can move around and perceive his full plane, and even the poor Linelander can move back and forth on his line.    Thus, the idea that perceiving your full dimensionality requires capabilities from a greater dimensionality does not really seem to ring true.    You need to think of perception much more generally than simple line-of-sight.

Naturally, this does not fundamentally prove that Steiner was wrong about our minds extending into the fourth dimension; it just means that the proof of such an idea is not so simple.   So it's still entirely possible that the concept of our mystical four-dimensional minds is correct but unproven, and the rest of Steiner's Anthroposophical Society ideas might still be valid.    This philosophy of the fourth dimension was just a launching point for a variety of mystic concepts, related to traveling along this fourth dimension to the astral plane where you could encounter ghosts, life after death, etc.   Some of Steiner's lectures get amusingly specific on details of the astral plane-- apparently he believed that his meditation and similar activities had actually taken him to this place, so he could talk about how astral dimensions mirrored our own, and writing there would appear backwards.   Personally, I'm a bit of a skeptic on this topic, but these kinds of ideas do seem to have a lasting appeal, as shown by the New Age sections you can find in many modern bookstores.   If you're into that stuff, try meditating hard enough, and maybe you too can follow Steiner's path into the astral plane through the fourth dimension.   While you're there. see if you can track down Steiner's spirit, to discuss the flaws in his philosophical proofs.

And this has been your math mutation for today.



References:

• http://www.goodreads.com/book/show/471326.The_Fourth_Dimension
• http://wn.rsarchive.org/Lectures/19220624p01.html
• https://en.wikipedia.org/wiki/Rudolf_Steiner

211: Saving A Few Million Years

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Those of you who follow the Math Mutation facebook feed may have noticed that a book I co-authored was just released: "Formal Verification: An Essential Toolkit for Modern VLSI Design". Now, I need to caution you that this is not a Math Mutation book-- it's a technical book, aimed at electrical and computer engineers invovled in chip design. However, I do think Math Mutation listeners might have some interest in the core concepts on which the book is based. So, today I'll try to give you a brief summary of what Formal Verification is, and why it's worth writing a book about.

You're probably aware that modern computer chips are pretty complicated, by many measures the most complex devices ever created by man. For new ones coming out this year, the number of transistors is measured in the billions. So it makes sense to ask the question: how do we know these things will work? It would be prohibitively expensive to just build them first and then test them, so we need to discover and fix as many of the bugs as possible during the design stage. The process of finding these bugs is known as design validation. For many years, the most popular technology used in design validation has been simulation: testing how a software model of the design will behave for various sets of inputs.

Unfortunately, even a simple chip design has so many possible behaviors that there is no way to simulate them all. For example, suppose you are designing a simple integer adder: it will take 2 numbers as inputs, each expressed with 32 bits, or binary digits, which can each be 1 or 0. It will then output their sum. How many possible input sets are there for this design? Since each input has 32 bits, it has 2^32 possible values, thus resulting in 2^64 possible values for the pair. If we assume we have a fast simulator that can check 2^20 values per second, that means that we will need 2^44 seconds to check all possible values-- over half a million years. Most managers are not very happy when given a time estimate on this order to finish a project. And needless to say, most chips sold today are many orders of magnitude more complex than a simple adder.

So, what can we do to make sure our chip designs really will work? A variety of technologies have been developed over the past few decades to try to find a good set of example values to simulate. And they do seem to be doing a decent job: most electronic devices you buy today seem to more-or-less do what you want them to. But it still seems like there should be a better way to validate them: no matter how good you make it, simulation and related methods can never cover more than a tiny portion of your design's possible behaviors.

That's where formal verification comes in. The idea of formal verification is to take a totally different approach: instead of trying specific values for your design, why not just mathematically prove that it will always be correct? That way you don't have to worry about trying every possible test case. If there is a single set of values that would generate an incorrect result, your proof will fail, and you will know your design has a bug. If you do succeed in mathematically proving your design correct, then you know that there is no bug, and do not need to waste time simulating lots of testcases. In effect, formally verifying a design is equivalent to simulating all possible values. Many would argue that philosophically, this is really the "right" way to validate chip designs. You may have heard the famous Guindon quote, "Writing is nature’s way of letting you know how sloppy your thinking is." Formal Verification pioneer Leslie Lamport expanded on this with "Math is nature's way of letting you know how sloppy your writing is.", and later added "Formal math is nature's way of letting you know how sloppy your math is."

You've probably guessed by now that there has to be a catch. Formal verification is easier defined than done: when billions of transistors are involved, how do we even get our heads around the problem of creating mathematical proofs? It's far beyond what anyone could manually do, so to make this method a possibility, humans need to be aided by intelligent software that helps to automate proofs. To further complicate matters, it's also been shown that any formal verification system needs to internally solve what are known as NP-complete problems. If you remember our discussion way back in episode 13, an NP-complete problem is "provably hard" in some sense, meaning that no piece of computer software can ever solve it efficiently in 100% of cases. However, researchers have worked for many years to try to develop practical software that could utilize clever tricks to enable real proofs on a wide variety of actual industrial product designs.

The good news is that, in the past decade, formal technology has advanced to the point where it really is practical for an average design engineer to use in many cases. While formal verification software can't handle full multi-billion-transistor chip designs, it can often enable an engineer to create solid proofs on major sub-blocks that go into a chip design, massively reducing overall risk of bugs. Using formal verification software remains a bit of an art though. Due to the NP-completeness issue, the software may get stuck or progress very slowly: the user must often give subtle hints and suggest shortcuts to enable the proofs to complete. In addition, formal verification is a problem that is impossible to fully automate: no matter how good your software gets at proving stuff, a human still has to somehow be able to tell it what stuff to prove-- what is the overall intent of the design in the first place? Ultimately, someone has to carefully transfer the design intent from their human brain into a machine-readable form, and understand the possible limitations and pitfalls in this process. Sadly, computer software that directly plugs into your brain is probably still many years away, and even then I have the feeling that many of us think too sloppily to enable this level of verification directly.

Thus, the need for a Formal Verification book. While there have been many books on Formal Verification published over the past few decades, most have focused on internal algorithms that would be needed to develop the software involved. Our book is one of the first real practical manuals designed to help deesign and validation engineers use formal verification software on real-life design targets.
Anyway, that quick summary should give you an idea of what our new book is about. If you're in a field where you do chip design or something related, please visit our book's website at http://formalverificationbook.com, order a copy, and tell all your friends about it!   If you're not in this field, the book probably won't make much sense to you, but hopefully you've still enjoyed this episode of the podcast.


And this has been your Math Mutation for today.


References:

• http://formalverificationbook.com
• https://en.wikipedia.org/wiki/Logic_simulation
• https://en.wikipedia.org/wiki/Formal_verification
• http://joeduffyblog.com/2009/01/09/tla-math-is-natures-way-of-letting-you-know-how-sloppy-your-writing-is/
• http://mathmutation.blogspot.com/2011/12/13-another-problem-for-millenium.html

204: What Happened To Grigori Perelman?

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Before we start, I'd like to thank listeners katenmkate and EdB, who recently posted nice reviews on iTunes. I'd also like to welcome our many new listeners-- from the hits on the Facebook page, I'm guessing a bunch of you out there just got new smartphones for Xmas and started listening to podcasts. Remember, posting good reviews on iTunes helps spread the word about Math Mutation, as well as motivating me to get to work on the next episode.

Anyway, on to today's topic. We often think of mathematical history as something that happened far in the past, rather than something that is still going on. This is understandable to some degree, as until you get to the most advanced level of college math classes, you generally are learning about discoveries and theorems proven centuries ago. But even since this podcast began in 2007, the mathematical world has not stood still. In particular, way back in episode 12, we discussed the strange case of Grigori Perelman, the Russian genius who had refused the Fields Medal, widely viewed as math's equivalent of the Nobel Prize. Perelman is still alive, and his saga has just continued to get more bizarre.

As you may recall, Grigori Perelman was the first person to solve one of the Clay Institute's celebrated "Millennium Problems", a set of major problems identified by leading mathematicians in the year 2000 as key challenges for the 21st century. Just two years later, Perelman posted a series of internet articles containing a proof of the Poincare Conjecture, a millennium problem involving the shapes of certain multidimensional spaces. But because he had posted it on the internet instead of in a refereed journal, there was some confusion about when or how he would qualify for the prize. And amid this controversy, a group of Chinese mathematicians published a journal article claiming they had completed the proof, apparently claiming credit for themselves for solving this problem. The confusion was compounded by the fact that so few mathematicians in the world could fully understand the proof to begin with. Apparently all this bickering left a bitter taste in Perelman's mouth, and even though he was selected to receive the Fields Medal, he refused it, quit professional mathematics altogether, and moved back to Russia to quietly live with his mother.

That was pretty much where things stood at the time we discussed Perelman in podcast 12. My curiosity about his fate was revived a few months ago when I read Masha Gessen's excellent biography of Perlman, "Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century". It gives a great overview of Perelman's early life, where he became a superstar in Russian math competitions but still had to contend with Soviet anti-semitism when moving on to university level. It also continues a little beyond the events of 2006, describing a somewhat happy postscript: eventually the competing group of Chinese mathematicians retitled their paper " Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture", explicitly removing any attempt to claim credit for the proof, and recasting their contribution as merely providing a more readable explanation of Perelman's proof. Sadly, this did not cause Perelman to rejoin the mathematical community: he has continued to live in poverty and seclusion with his mother, remaining retired from mathematics and refusing any kind of interviews with the media.

As you would expect, this reclusiveness just served to pique the curiosity of the world media, and there were many attempts to get him to give interviews or return to public life. Even when researching her biography, Masha Gessen was unable to get an interview. In 2010, the Clay institute finally decided to officially award him the million dollar prize for solving the Poincare Conjecture There had been some concern that his refusal to publish in a traditional journal would disqualify him for the prize, but the Institute seemed willing to modify the rules in this case. Still, Perelman refused to accept the prize or rejoin the mathematical community. He claimed that this was partially because he thought Richard Hamilton, another mathematician whose work he had built upon for the proof, was just as deserving as he was. He also said that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust." Responding to a persistent reporter through the closed door of his apartment, he later clarified that he didn't want "to be on display like an animal in a zoo." Even more paradoxically, he added "I'm not a hero of mathematics. I'm not even that successful." Perhaps he just holds himself and everyone else to impossibly high standards.

Meanwhile, Perelman's elusiveness to the media has continued. In 2011 a Russian studio filmed a documentary about him, again without cooperation or participation from Perelman himself. A Russian journalist named Alexander Zabrovsky claimed later that year to have successfully interviewed Perelman and published a report, but experienced analysts, including biographer Masha Gessen, poked that report full of holes, pointing out various unlikely statements and contradictions. One critic provided the amusing summary "All those thoughts about nanotechnologies and the ideas of filling hollowness look like rabbi's thoughts about pork flavor properties." A more believable 2012 article by journalist Brett Forrest describes a brief, and rather unenlightening, conversation he was able to have with Perelman after staking out his apartment for several days and finally catching him while the mathematician and his mother were out for a walk.

Probably the most intriguing possibility here is that Perelman has not actually abandoned mathematics, but has merely abandoned the organized research community, and is using his seclusion to quietly work on the problems that truly interest him. Fellow mathematician Yakov Eliashberg claimed in 2007 that Perelman had privately confided that he was working on some new problems, but did not yet have any results worth reporting. Meanwhile, Perelman continues to ignore the world around him, as he and his mother quietly live in their small apartment in St Petersburg, Russia. Something tells me that this not quite the end of the Perelman story, or of his contributions to mathematics.

And this has been your math mutation for today.
　

References:

• http://en.wikipedia.org/wiki/Grigori_Perelman
• http://news.bbc.co.uk/2/hi/europe/8585407.stm
• http://phys.org/news197209671.html
• http://english.pravda.ru/society/stories/06-05-2011/117816-grigori_perelman-0/
• http://www.brettforrest.com/shattered-genius/
• http://www.lepoint.fr/monde/le-genie-qui-s-est-retire-du-monde-30-09-2010-1246189_24.php
• http://www.amazon.com/Perfect-Rigor-Mathematical-Breakthrough-Century/dp/015101406X/ref=pd_sim_sbs_b_1?ie=UTF8&refRID=1GHDAFJMBTA8HSWC3H31

      

200: Answering Common Listener Questions

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Wow, I can't believe we've made it to 200 episodes.  Thanks everyone for sticking with me all this time, or at least for discovering this podcast and not immediately deleting it.   Actually, if we're being technical, this is the 201st episode, since I started at number 0.   But we all suffer from the common human fascination with big round numbers, so I think reaching number 200 is still something to celebrate.  

Finding a sufficiently momentous topic for this episode has been a challenge.  Wimping out somewhat, I think a good use of it is to answer a number of listener questions I have received by email over the past 7 years in which I've been podcasting.   Of course I have tried to send individual answers to each of you who has emailed me-- please continue emailing me at erik (e-r-i-k) at mathmutation.com-- but on the theory that each emailer represents a large number of listeners who are too busy or lazy to email, they are probably worth answering here.

1.  Who listens to this podcast?   According to my ISP, I've been getting about a thousand downloads per week on average.   Oddly, a slight majority seem to be from China, a country from which I've never received a listener email, as far as I can tell.   Chinese listeners, please email me to say hi!   Or perhaps the Communist spies there have determined that my podcast is of strategic importance to the United States and needs to be monitored.  If that's the case, I'll look forward to an elevated status under our new overlords after the invasion.  Assuming, that is, that they don't connect me to any of my non-podcast political writings, and toss me into the laogai instead.

2.  How is this podcast funded?    Well, you can probably guess from the average level of audio quality that I'm not doing this from a professional studio; just a decent laptop microphone, plus some cheap/shareware utilities including the Podcast RSS Buddy and the Audacity sound editor, along with a cheap server account at 1 and 1 Internet.  So I actually don't spend a noticeable amount on the podcast.  That's why rather than asking for donations, I ask that if you like the podcast enough to motivate you, you donate to your favorite charity in honor of Math Mutation and email me.
On a side note, I have fantasized about trying to amp up the quality and frequency and make this podcast a profitable venture.   Many of us small podcasters were inspired a few years ago when Brian Dunning of the Skeptoid podcast quit his day job and announced he was podcasting full time.   However, that dream died somewhat when it was revealed earlier this year that that Brian's lifestyle was partially funded by some kind of internet fraud, and he was sentenced to a jail term.

3.  Why don't you release episodes more often, and/or record longer episodes?  First of all, thanks for the vote of confidence, and I'm glad you're enjoying the podcast enough to want more!  During my first year or two of Math Mutation, I had lots of great ideas in the back of my mind, so coming up with topics & preparing episodes was pretty easy.   But now I'm at a point where I've cleared the backlog in my brain, and now I have to think pretty hard to come up with cool topics, and spend a nontrivial amount of time researching each one before I can talk about it.  This is also combined with many non-podcast responsibilities in my daily life, including a wife and daughter who somehow like to hang out with me, and an elected position on the local school board, at the 4th largest district in Oregon.   So I'm afraid I won't be able to increase the pace anytime soon.  Perhaps in a few years, after I've been tarred, feathered, and removed from public office, and my daughter becomes a teenager and hates me, I'll have a bit more podcasting time though.

4.  Can you help me solve this insanely difficult math problem:  (insert problem here)?   I've received a number of queries of this form.   I'm flattered that my podcasting persona has led you to believe I'm a matehmatical genius of some kind, but to clarify, I would put myself more in the category of an interested hobbyist, nowhere near the level of a professional mathematician.   I did earn a B.A. in math many years ago, but my M.S. is in computer science, and I work as an engineer, using and developing software that applies known mathematical techniques to practical issues in chip design at Intel.   If you're a math or science major or graduate student at a decent college, and have a problem that is challenging for you, it's probably way over my head!   So if you're one of the numerous people who sent me a question of this kind & didn't get a good answer, don't think that I'm withholding my brilliant insights, you've probably just left me totally baffled.   And you're probably way more likely to solve it than I am anyway.

5. What other podcasts do you listen to?   To start with, I don't listen to other math podcasts.  This is partially because I'm afraid I'll be intimidated at how much more professional they are.   But mostly I'm worried that I'll subconsciously remember them and accidentally repeat the same topic in my own podcast, as humans are prone to do.    I do avidly listen to podcasts in other genres though.   As you might suspect from some of my topics, I'm a big fan of the world of "science skepticism" podcasts, such as Skeptoid, QuackCast, Skeptic's Guide to the Universe, and Oh No Ross and Carrie.   Those are always fun, although occasionally a bit pretentious in their claims to teach other people how to think.   I'm also a bit of a history buff, really enjoying Robin Pierson's "History of Byzantium", Harris and Reily's "Life of Caesar", and the eclectic "History According to Bob".   Rounding out my playlist is the odd Australian comedy/culture podcast "Sunday Night Safran", where a Catholic priest and a Jewish atheist have a weekly debate on cultural issues.

Anyway, I think those are probably the most common questions I have received from listeners.   I always love to hear from you though, so don't hesitate to email me if you have more ideas, questions, or requests for the podcasts.   If I receive enough emails, I might not wait until episode 400 before doing another Q&A. 

And this has been your math mutation for today.

References:

• http://en.wikipedia.org/wiki/Brian_Dunning_(author)
• http://www.theskepticsguide.org/
• http://skeptoid.com
• http://moremark.squarespace.com/quackcast-home/
• http://ohnopodcast.com
• http://thehistoryofbyzantium.com/
• http://lifeofcaesar.com/
• http://historyaccordingtobob.com
• http://www.abc.net.au/triplej/safran/
• http://audacity.sourceforge.net/
• http://www.tolley.info/rssbuddy/
• http://www.1and1.com/

199: Precaution or Paranoia?

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Before we start, I'd like to thank listeners Dim Questor, who posted another nice review on iTunes, and Jordan Mahoney, who shared our podcast link on Facebook. Remember, you too can have your name,or bizarre iTunes nickname, immortalized in my podcast by following in Dim or Jordan's shoes!

Now, on to today's topic. You may have heard some online skeptics derisively referring to the "precautionary principle", the idea popular among certain activists that if a new technology has any risks at all, or some new product has a nonzero amount of toxic contamination, we need to take the safe path and ban it. While a certain level of caution is always wise, the fact is that anything you do contains some level of risk, and any product contains some level of contamination. For example, we all still ride in cars, even though an average of 0.72 people die for every 100 million passenger miles driven in cars. Should cars be banned because this number in .72 rather than 0? If we were to follow this principle in general, we would effectively prevent all forms of scientfic and technological advance. So I was surprised to see some buzz on the internet indicating that Nassim Nicholas Taleb, a famous mathematical philosopher who has written several books on risk and probablility, and who I respect a lot, has been arguing that the Precautionary Principle needs to be applied to the concept of genentically modified foods, or GMOs, and that they should therefore be banned. I downloaded his paper, linked in the show notes, to learn a little more about this somewhat surprising position.

Let's begin by reviewing Taleb's "Black Swan" concept, which you may remember me intruducing back in podcast 84, "How to Bankrupt Your Boss And Get Rich". The idea here is that when managing risk, we have to consider both the probablility of a negative event, and the magnitude of its effect. If a low-probablility event, known as a Black Swan, causes a huge penalty, this may outweigh all the cumulative benefits of taking a risk over time. For example, suppose we make a bet that we will flip a coin once per month, and I have to pay you $100 for each head, and you pay me $1000 for each tail. There are risks to each coin flip- I might gain or lose money- but not much overall risk to me by playing this game, since it's nearly impossible to lose a lot of money, and in the long term I will profit nicely on average.

Now suppose we add a new rule: if we see the "black swan" event of five heads in a row, I need to pay you a million dollars. With this rule, the game has changed a lot. If a certain low-probability but not impossible result happens, I will be totally ruined, losing more money than I have ever won in this game. The expected number of tosses to get 5 heads in a row is about 62, so even though it might seem like a profitable game for a year or two, I'm virtually guaranteed to be ruined in the long run. Of course this game is pretty silly, but as Taleb has pointed out, many traders of complex derivatives in the stock market are essentially playing it: they please their bosses by showing steady market-beating profits, and collect their annual bonuses, until one day some low-probability Black Swan event causes them to lose their company much more money than their accumulated total profits. Often by that point they have built up enough savings so they don't care if they are fired.

The Black Swan concept goes beyond finances, of course, which is where the GMO discussion comes in. Taleb points out that when taking any action which provides risk of global ruin, we need apply the Precautionary Principle, because when you multiply a small probability risk by the infinite costs of the global ruin scenario, you can see that the risk just isn't worth it. In other words, in cases where the cost is total ruin, the Precautionary Principle should apply, and we should refuse to take any risk, no matter how small it seems. We need to be careful here, though, in that Taleb is not advocating a universal embrace of the precautionary principle, but just advocating it for very specific scenarios. He still rides in cars and planes, for example. He also continues to support nuclear power, because even though a nuclear meltdown is not good, the risks are all local: a single nuclear accident, though bad for the immediate area, will not blow up the planet, and thus can be planned for within the domain of normal risk management.

If I'm reading Taleb's paper correctly, his main concern with GMOs is that large near-monopoly companies are selling newly existing forms of life to a global market, and in effect creating a monoculture, where a huge proportion of the world's crops in each category are a single, globally genetically identical species. Furthermore, these are new species, lacking the centuries of farming experience we have with naturally existing ones. This means that we have a high risk of problems with these new species, and many such problems carry risk of infinite-cost global ruin. If it has hidden long-term effects on human health, nearly the whole planet will be affected at once. If it is capable of virally spreading and displacing other species, this will all happen at once worldwide. And if some new disease can kill this species of crop, it will devastate the whole world's food produciton. Thus we are taking an infinite potential cost, which can affect agriculture or humanity worldwide, and offsetting it against a marginal benefit of improved crops due to the GMO. Incurring the risk of an infinite-cost Black Swan event, for a finite benefit due to GMO crops, does not balance out mathematically, and thus we shouldn't do it.

It seems to me that his most compelling point is really about the risk of monoculture- having a tiny set of large companies as global seed suppliers- rather than the risks of GMOs themselves. If Monsanto disavowed GMOs in favor of old-fashioned selective breeding, this monoculture issue would still be present. In general, Taleb has good points about monopolies and central control-- we do need to be careful about transforming local risks to global ones by making things the same everywhere. This is similar to his critique of the economic theory of Comparative Advantage, which we discussed back in episode 165. We need to watch for and prevent a true worldwide dependence on a tiny set of near-genetically-identical species, and make sure any new species is extensively tested in a small, local environment before being propagated further. And we do need to make sure we never reach a point where too large a proportion of the world is dependent on a single species of crop. But as long as we are taking action to avoid this worldwide monoculture, the concept behind GMO foods is basically the same as traditional selective breeding-- producing a new species based on existing forms of life-- so it's hard for me to see why it should be treated so differently. Many new plant varieties have been developed and deployed based on selective breeding over the past century as well.

Whether GMOs truly provide a risk of global ruin, or really just a finite risk of some bad crops in local markets, becomes even more important when we balance these risks against the potenital benefits. Improving overall crop yields and nutritional value, one of the potential GMO benefits, is a life-or-death question for many impoverished populations worldwide. Nobel Peace Prize winner Norman Bourlaug, known as "the man who saved a billion lives" due to his work improving the yields of Third World agriculture, stated before he passed away that we are approaching natural limits in making use of arable land, and that GMOs would be required in addition to fully end world hunger. Taleb dismisses such GMO benefits because we could theoretically solve the problems through other means-- but I don't think it's valid to argue on the basis of what can be theoretically done, if nobody is currently doing it. Our society, markets, and culture currently have the will and ability to solve these problems through the careful use of GMOs, and are not doing it through other methods suggested by Taleb.

So, should we follow Taleb's recommendations and ban GMOs, due to the imbalance of a finite benefit vs an infinite risk? Or embrace GMOs with open arms and dismiss Taleb as a kook on this issue? Neither is quite right. I think Taleb does have a good point about the risks of monoculture, and we need to make sure we create policies that ensure agricultural variety, and prevent reaching a point where nearly the whole world is depending on a fragile handful of plant species. But it looks to me like as long as we avoid this pitfall, the other risks of GMOs are comparable to those of naturally bred species, and the potential benefits are massive, potentially saving millions of starving and malnourished people. Thus we need to think carefully about all the benefits involved, and the true level of risk, before taking any hasty government action.

And this has been your math mutation for today.



References:

• https://docs.google.com/file/d/0B8nhAlfIk3QIbGFzOXF5UUN3N2c/edit
• https://www.facebook.com/notes/david-boxenhorn/comments-on-the-gmo-section-of-the-precautionary-principle-by-n-n-taleb-et-al/10154410752320551
• http://theweek.com/article/index/246552/the-odds-are-11-million-to-1-that-youll-die-in-a-plane-crash
• http://en.wikipedia.org/wiki/Norman_Borlaug#Future_of_global_farming_and_food_supply

198: We're All Pasteurized

Audio Link

Before we start, I'd like to thank listener Dan Unger, who posted another nice review on iTunes. Thanks Dan!

Now, on to today's topic. Most of us know the name of Louis Pasteur, the 19th-century French biologist and chemist, from seeing his name embedded in the term 'pasteurized' on milk cartons. And he is rightly remembered for his discoveries in the area of the germ theory of disease, which led to the development of many vaccines as well as the famous process for heating milk to reduce the bacterial content, collectively saving human lives in the millions over the past two centuries. But did you know that before his famous medical contributions, he made an equally revolutionary discovery in chemistry: the fact that molecules could posses chirality, that is left-handedness or right-handedness, just like ordinary macroscopic 3-D objects?

We're all familiar with the concept of left/right asymmetry in the real world. Simple examples are a glove, which cannot be changed from left-handed to right-handed without turning it inside out, or an ordinary screw, where we have to remember the "lefty loosey righty tighty" rule to use it properly. It seems natural that molecules should be able to exhibit such asymmetry as well, but at the dawn of the 19th century, scientists hadn't really discussed that idea much. In 1848, a young Louis Pasteur was studying a chemical called tartaric acid, a byproduct of wine production. He was trying to understand a strange anomaly related to this substance and a very similar one called racemic acid. Scientists had determined the chemical compositions of both acids, and they were exactly the same. Yet they had some very different physical properties: in particular, tartaric acid in solution would rotate a beam of polarized light passing through it, while racemic acid had no such effect.

Pasteur decided that there must be some geometric difference at the molecular level, so generated crystals of both acids to examine under a microscope. Remember that crystals are essentially an endless repetition of a small molecular structure, so studying the shapes of pure crystals can grant some insight into the shapes of the molecules themselves. Other scientists thought he was wasting his time, since the experiment had already been done by others, and they found that tartaric and racemic acid crystals were exactly the same shape. But Pasteur noticed something that his colleagues had missed: while tartaric acid crystals were all the same shape, racemic acid was a mixture of two types: the tartaric acid crystals, plus another form that was a mirror image of the other. Since the form was somewhat asymmetric, this meant that the two forms were truly different, even though other scientists had dismissed these differences as insignificant: no rotation in our physical space could transform one into the other.

Pasteur then came up with a clever experiment. He crystallized some racemic acid, then with a microscope and needle, carefully separated the left-handed and right-handed crystals into two piles. Then he created solutions of the two types of crystals. As he suspected, one of the solutions was a solution of tartaric acid, and rotated polarized light in the same way that tartaric acid normally would. But the other was a new substance, which rotated light in the opposite direction. In other words, the reason racemic acid did not usually bend light was that it was a mix of right-handed and left-handed molecules, while tartartic acid consisted purely of the left-handed form. Pasteur had separated out the right-handed component of the racemic acid. The result was so shocking to the scientists of the day that the French Academy of Sciences made Pasteur repeat the experiment in front of witnesses before they would accept it.

Today awareness of the chirality, or handedness, of molecules plays a critical role in biochemical and medical research. This is mainly due to the fact that life is "homochiral", a fancy way of saying that our basic building blocks all have a single handedness. This differs from most naturally occurring nonliving materials, which tend to exhibit both kinds of handedness randomly in roughly even proportions. The biological origin of Pasteur's tartaric acid was responsible for its one-sided content. Almost all amino acids found in living creatures are left-handed, and we use them to interact with right-handed sugars to supply most of our energy. At first, scientists found this very surprising, since attempts to artificially synthesize most biological molecules result in roughly equal quantities of right- and left- handed forms. This can have tragic consequences: the infamous pregnancy drug thalidomide was a great treatment for morning sickness in its left-handed form, but the right-handed version caused serious birth defects.

The reason for life's left-handedness is one of science's great mysteries. You can see links to articles in the show notes with a few different theories. One is that it stems from a fundamental left-handedness in physics, since certain types of radioactive decay have a leftward electron spin. The effect is tiny, though, and you could argue that this is just begging the question, since you still have to explain the left-handedness of physics. Another theory is that life was seeded by left-handed molecules from space: meteors with both types of amino acids may have happened to pass through regions of space with polarized light that destroyed one kind more often than the other. Or it could just be luck-- maybe left-handed life randomly formed first in the primordial soup, and once it had a toehold it crowded out any other possibilities. Whatever the real answer is, this geometry is critically important to every cell in our bodies.

And this has been your math mutation for today.




References:

• http://www.astrobio.net/news-exclusive/how-life-shatters-chemistrys-mirror/
• http://www.foundersofscience.net/interest1.htm
• http://www.simsoup.info/Origin_Issues_Homochirality.html
• http://en.wikipedia.org/wiki/Louis_Pasteur