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.
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.