Q: How many theorems are there?

Physicist: Every sub-field in math and physics has at least hundreds, and there are hundreds or thousands of sub-fields.  So overall we’ve proven… At least millions?

This is one of those things that can’t really have an exact answer, or even a ball-park answer.  One person’s theorem is another person’s corollary (“corollary” = “by the way”, in mathspeak).  More than that, not all theorems are winners.  For example, you’d be hard-pressed to find a professional nerd who thinks that the Pythagorean theorem isn’t a theorem, but something like “If N is an integer, then N(N+1) is even” generally isn’t considered profound enough for theorem status.  Not all theorems can be the BEST theorem.

So, counting theorems is kinda like counting art.  If you include finger-paint portraits and urinals, then the number of art gets a lot higher.  There have been lists of theorems made, but they’re always woefully incomplete.  Get a list, crack open any fairly specific/advanced math/physics book, pick a theorem at random, and it probably won’t be on the list.  At the very least, you can say that there are so many recognized theorems out there that no one could possibly live long enough to learn them all, or even any more than a small fraction.  You’d have better luck collecting all of the art.

Is it a theorem just because you hang it on a wall?

It’s been known for thousands of years that there are buckets of theorems, but it was generally assumed that at some point we’d find all of them and be done with math.  However, in 1931 Gödel added a pair of new theorems to the list, that was essentially about the list itself.  “Gödel’s incompleteness theorems” say, among other things, that there are an infinite number of axioms (“true, but unprovable statements”).  Extending that idea, we can expect that there should be an infinite number of theorems that relate these axioms to each other and talk about their consequences.  Math, and every strongly math-dependent discipline (which is all the good ones) are infinite, incompletable sciences.

In one wrenching insight, Gödel managed to simultaneously secure jobs for mathematicians in perpetuity and also to make their ultimate goal of proving/disproving everything utterly unattainable.

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4 Responses to Q: How many theorems are there?

  1. Greg says:

    The last paragraph is just brilliant!

  2. Joe says:

    I asked this question last week. This is a much fuller answer than I received. Thanks!


    Ô lôco meu!

  4. Amir says:

    The last paragraph is just brilliant! [2]

    And the ones before are awesome as usual.

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