On the other hand… Gödel (but not Escher or Bach)

In my previous blog post, I wrote about — and included a cartoon about — one aspect of math teaching.

The cartoon asserted that “no one can impart perfect universal truths to their students,” with the punch line “…except math teachers.” The text of the post alluded to the comfort of having definite answers to problems in these times, with the implication that math does that. On the one hand, both of these claims are true — depending on what the meaning of “true” is. On the other hand, both claims are ultimately false.

Let’s talk about each side of this paradox:

  • First comes the argument that math teachers do impart universal truths and that the problems they assign do have definite answers. This is pretty clearly the view of the vast majority of the public (although they don’t articulate it that way). If you are like most people, you expect a math problem to have a definite answer. Perhaps you can’t find it — but surely someone is smart enough to find it! And a lot of people take comfort in that view. Most people expect that math consists of universal truths: 2 + 2 = 4 no matter what your beliefs, even if you’re a Republican. A lot of people take comfort in that view too. If you don’t know a lot of math, these views appear to be obviously true: every problem has a definite answer, and math consists of universal truths.
  • But if you know a tiny bit more math, you start playing games like “2 + 2 = 11 if you’re in base three.” But that’s not a serious objection. The clear but unwritten context is that we’re using base ten. We’re talking about numbers, not the numerals that name those numbers.
  • It’s only if you know a lot more math that you start to see the rest of the story, which we will briefly discuss here.

Two of my former students — who do know a lot more math — raised objections to my previous post, or at least expressed reservations about it. Their objections will mystify tenth-graders — and the general public, who don’t remember any math beyond tenth grade anyway (if that) — but we have to consider the reservations out of respect for the many people who take AP or college-level statistics, not to mention the somewhat smaller number of people who learn about Kurt Gödel. So here goes a pair of reservations:

Reservation One

A Weston precalculus student of mine — who shall go unnamed here — complained to me once about his AP statistics course:

I asked my teacher to prove what he was claiming, and he just said that the claim is true because “it seems to work.” That isn’t math!

Indeed. Real math requires axioms and theorems — proofs! Statistics isn’t “real math.”

But it does seem to work.

When I was an undergraduate, I wondered why some colleges (including Harvard, at least at the time) had completely separate departments of Mathematics and Applied Mathematics. One of my math professors openly looked down on the Applied Math department, as it wasn’t “real math.” No theorems, no proofs — you just wave your hands and say that “it seems to work.” Statistics is (mostly) part of applied math, and it does seem to work. In fact, it’s enormously useful, and our country would be better off if more people understood it. From electoral polls to COVID-19, it’s important to have a basic understanding of statistics. But it’s not really math, as it doesn’t rest on an axiomatic system. That’s one point of view, anyway; your mileage may vary.

Reservation Two

What’s perhaps more surprising is that even if you limit yourself to pure math, based on an axiomatic system, it turns out that you can’t get universal truths! Almost a century ago, Austrian mathematician Kurt Gödel demolished the massive attempt by Bertrand Russell and Alfred North Whitehead to build a mathematical system that was both consistent and complete. Gödel’s Incompleteness Theorems showed that you simply can’t have both: a consistent system isn’t complete (i.e., some true statements cannot be proved), and a complete system isn’t consistent (a statement can be both true and false — yikes!), so you pays your money and you takes your choice.

There are hundreds of sources available that explain Gödel’s proofs and their significance, but we have neither time nor space to go into them here. All I will do is recommend one webpage and one book:

  • The webpage is http://www.godel-universe.com/godels-theorems/, and it’s reasonably understandable, but only if you’re paying attention. You can’t skim.
  • The book is my #1 favorite: Douglas Hofstadter’s Gödel, Escher, Bach. As the title suggests, Hofstadter draws connections between music and math and between art and math — but not in the obvious ways that we might usually mention in a math course, such as the connections that exponential and trigonometric functions show when studying frequencies of musical notes. No, Hofstadter explores what he calls “strange loops,” a mix of recursion and transformations that appear visually in many of Escher’s works and aurally in many of Bach’s works. To see the connection with Gödel’s theorems, go read the book! You’ll find it a lot of fun as well as informative: it’s not a dry math textbook by any means.

So what’s the conclusion? My take is that it’s important to tear down some of the edifice built by the “definite answer” and “universal truths” beliefs, but only after students have come to adopt those beliefs and have faith in them. Otherwise it’s just destructive. When students are old enough — say, 11th or 12th grade — they are ready to poke holes in their previous beliefs and see where they’re true and where they aren’t.

Categories: Math, Teaching & Learning, Weston