Math Mutation: 2023

Sunday, August 27, 2023

286: Adventures In Translation

Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host. And now, on to the math.

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. 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”. I found his discussions of translation challenges fascinating: 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. 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. 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.

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, 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. Curious to read this memoir, and with no English translation having been planned by the author, I volunteered to take this on myself. 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. 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. 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. Thus, I went ahead and began translating.

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. 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. It is useful, however, as a first step: as I approached each paragraph, I used that tool to create a “gloss”, an initial awkward translation to use as a starting point. From there, I would try to understand the core concepts being expressed, and try to rewrite the sentences in more natural sounding English. 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. The most entertaining challenges were the cases where the initial version simply didn’t make sense at all.

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”. That doesn’t make much sense initially. 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. It could also be a question, asking “Fidel, Fidel, what does Fidel have, that the imperialists cannot overcome?” 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. But while that might make it a more effective chant, I was hesitant about making the translation’s tone a bit too lighthearted.

My favorite translation challenges were the ones that made me laugh out loud when first reading the Google translated version. 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”. Was life in Cuba really so boring that people enjoyed sitting and starting at dolls for hours? 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? Things in Cuba could be worse than I imagined. 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.

So, in the end, was my translation any good? 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. 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. 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. You can order it using the link in the show notes at mathmutation.com or just search for it on Amazon, and judge my translation for yourself. The book is called “The Revolution of Promises”, by Nelson Rodriguez Chartrand. If you like it, a good review on Amazon would also be helpful.

And this has been your math mutation for today.

References:

Thursday, June 29, 2023

285: Believe My Proofs Or Else

Audio Link

Before I start, I’d like to thank listener ‘katenmkate’ for posting an updated review of Math Mutation on Apple Podcasts. 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!

Recently I saw an amusing XKCD cartoon online that reminded me of a term I hadn’t heard in a while, “proof by intimidation”. 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.” The caption underneath reads “My math teacher was a big believer in proof by intimidation.” A nicely absurd joke— wouldn’t it be great if we could just intimidate the numbers and variables into making our proofs work out? “Proof by intimidation” is a real term though. 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. But according to the Wikipedia page, the term arose after some lectures in the 1930s by a mathematician named William Feller: “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.”

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. As Wikipedia explains it, proof by intimidation “is a jocular phrase used mainly in mathematics to refer to a specific form of hand-waving, whereby one attempts to advance an argument by marking it as obvious or trivial, or by giving an argument loaded with jargon and obscure results.” 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?” 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.

As you would expect, that type of argument certainly makes a proof invalid in many cases. But there is one interesting wrinkle that most of the online explanations skip over. 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 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.

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. 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. 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. 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. 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. The TA, giving me 0 points, commented, “If this was really obvious, we wouldn’t have bothered to assign the problem in your homework!”

Proof by intimidation is just one representative of a family of related logical fallacies. 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”. 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.

And this has been your math mutation for today.

References:

Sunday, April 30, 2023

284: Don't Square Numbers, Triangle Them

Audio Link

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. 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. 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. Today we’ll discuss another of his key points, his central focus on triangles as a key foundation of mathematics.

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. This follows directly from their essential ability to distribute forces without deforming. Think about it for a minute: 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: for example, you can squish it into a parallelogram. Triangles don’t have this flaw: assuming the sides are rigid and well-fastened together, you can’t distort it at all. This stems from the elementary geometric fact that the lengths of the three sides fully determine the triangle: for any 3 side lengths, there is only one possible set of angles to go with them and make a triangle. Thus, if you are building something in the real world, triangles and related 3-dimensional shapes naturally result in strong self-supporting structures.

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. His teacher had given the class an exercise where they were to build a small model house out of toothpicks and semi-dried peas. 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. As he recalled it, “All the other kids, the minute they were told to make structures, immediately tried to imitate houses. I couldn’t see, so I felt. And a triangle felt great!” 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. 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”. 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.

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. He took this philosophy into all areas of mathematics that he looked at, sometimes to a bit of an extreme. For example, a fundamental building block of algebra is looking at higher powers of numbers, starting with squaring. But Fuller considered the operation of “triangling” much more important. Of course this is a useful aspect of general algebra: you might recall that the triangular numbers represent the number of units it takes to form an equilateral triangle: start with 1 ball, add a row of 2 under it, add a row of 3 under that, etc. So the series of triangular numbers start with 1, 3, 6, 10, and so on, representing the value ((n squared - n) / 2) for each n. Fuller considered these numbers especially significant because they also happen to represent the set of pairwise relationships between n items. 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. For 5 people you need 10, the 4th triangular number, etc. Fuller was pleased that this abstract concept of the number of mutual relationships had such a simple geometric counterpart. 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.

So, are you convinced yet that triangles are important? 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. 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. 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.

And this has been your math mutation for today.

References:

Sunday, February 26, 2023

283: Escape From Infinity

Audio Link

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. 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. 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. It’s a very difficult read, which is what led to the need for Edmonson’s book.

One of the basic insights guiding Fuller was the distrust of the casual use of infinities and infinitesimals throughout standard mathematics. He would not accept the definition of infinitely small points or infinitely thin lines, as these cannot exist in the physical world. He was also bothered by the infinite digits of pi, which are needed to understand a perfect sphere. Pi isn’t equal to 3.14, or 3.142, or 3.14159, but keeps needing more digits forever. If a soap bubble is spherical, how does nature know when to stop expanding the digits? 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. Or as Fuller would term it, it’s a “mesh of energy events interrelated by a network of tiny vectors.” But can you really do mathematics without accepting the idea of infinity?

With a bit of googling, I was surprised to find that Fuller was not alone in his distrust of infinity: there’s a school of mathematics, known as “finitism”, that does not accept any reasoning that involves the existence of infinite quantities. At first, you might think this rules out many types of mathematics you learned in school. Don’t we know that the infinite series 1/2 + 1/4 + 1/8 … sums up to 1? And don’t we measure the areas of curves in calculus by adding up infinite numbers of infinitesimals?

Actually, we don’t depend on infinity for these basic concepts as much as you might think. 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. For example, what are we really saying when we claim the infinite series 1/2+1/4+1/8+… adds up to 1? 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. If you want to get within 25% of 1, add the first two terms, 1/2+1/4. If you want to get within 15%, you need to add the three terms 1/2+1/4+1/8. And so on. (This is essentially the famous “epsilon-delta” proof method you may remember from high school.). 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.

You might object that this doesn’t cover certain other critical uses of infinity: for example, how would we handle Zeno’s paradoxes? 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. Since we can’t do an infinite number of tasks in real life, motion is impossible. The traditional way to resolve it is by saying that this infinite series 1/2+1/4+… adds up to 1. But if a finitist says we can’t do an infinite number of things, are we stuck? Actually no— since the finitist also denies infinitesimal quantities, there is some lower limit to how much we can subdivide the distance. After a certain amount of dividing by 2, we reach some minimum allowable distance. 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. 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 a correspondingly tiny time each to complete. Calculating distances using the sum of an infinite series again becomes just a limit-based approximation, not a true use of infinity.

Thus, you can perform most practical real-world mathematics without a true dependence on infinity. One place where finitists do diverge from the rest of mathematics is in reasoning about infinity itself, or the various types of infinities. 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. A finitist would consider this argument pointless, having no applicability to real life, even if it does logically follow from Cantor’s premises.

In Fuller’s case, this refusal to accept infinities had some positive results. 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. 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”. 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.

And this has been your math mutation for today.

References:

What Is This Blog?

Welcome to Math Mutation, a short podcast for people of all ages, where we explore fun, interesting, or just plain weird corners of mathematics that you probably didn't hear in school.

This blog page contains the transcripts and show notes for the Math Mutation podcast, available on iTunes or at this RSS feed. If you like it, also be sure to join the Facebook fan page!

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

If you're interested in education issues, also check out Erik's education blog at http://seligman4schools.blogspot.com .