Godel's Theorem, Diophantine Equations, and "Truth"

In this video I elaborate on a fact that I brought up in a previous video on Godel's theorem which can be found here • Godel's Theorem: An In...
My goal in this video is to point to some tension which exists between what it means to say that a mathematical statement is true. This tension exists between the the Platonist position which says that math is discovered and the formalist position which claims that math is invented. For more on how Godel's theorem relates to these questions one should check out Torkel Franzen's book Godel's Theorem: An Incomplete Guide to its Use and Abuse.

[@AlainGenestier-tj8fn]

As far as I know, the Robinson-Matijasevich theorem used in your video provides an effective procedure to translate a closed statement written in the langage of arithmetic (Peano arithmetic, or even primitive recursive arithmetic) into a diophantine equation. So, the Gödel undecidable statement attached to a theory TH (containing PRA) could in principle be made explicit. In our age of computers, I would be surprised if it hasn't been done for TH=PA or ZFC. Of course, the resulting diophantine equation is certainly ugly and unappetizing. At least, it has been done for the parametric diophantine equation expressing "p is prime", i.e. an explicit polynomial P(X_0,X_1,...,X_n) has been produced, such that p is prime iff the diophantine equation P(p,X_1,...,X_n)=0 has no solution. Of course, this amazing result has no practical use (it is the worst practical method to solve the question "is p prime" ?).

[@hungrymathprof]

That is interesting to think that one of the undecidable but true statements about Diophantine equations would be the translation of the Godel senentce!

[@AlainGenestier-tj8fn]

Also, I have the feeling that the "platonistic" view of an objective world of integers, pre-existing a formal system, is a mere reformulation of the categoricity of second order arithmetic. I see no reason to support this objectivity if you are not also committed to the existence of a "real world" of sets (at least, finite sets).

[@hungrymathprof]

I suspect that the categoricity result comes after the platonist intuition but I can see it going both ways. For instance, I think the intuition is that the twin prime conjecture (or replace with some other unknown conjecture about the natural numbers) is meaningful with or without set theory, and even if it cannot be proven in ZFC, the statement has an answer which in principle could be determined by checking every prime in some theoretical universe where we live forever. I don't take a side either way personally but I have become more sympathetic to the platonist position over time. I really enjoyed reading Godel's own philosophical writing where he tries to defend platonism for math objects. See Kurt Godel: Unpublished Philosophical Essays.

[@AlainGenestier-tj8fn]

@@hungrymathprof To refine the discussion, let's consider an arithmetic statement which is true for the second order arithmetic in ZF, but undecidable in PA (like the termination of all Goodstein sequences). Is it true for the "platonic integers" ? If you think it is, what compels you to think it is true for the "platonic integers" ?

[@AlainGenestier-tj8fn]

To clarify what I have just said, a perfectly respectable position is to also assume the existence of sets obeying ZF. But my question was more precisely "if you don't assume ZF, what compels you to think it is true for the platonic integers ?".

[@hungrymathprof]

Well, this question would be better asked of a philosopher who has taken the firm position of a platonist about mathematical objects. I am no such person. That being said, undecidable implies that there isn't a proof of finite length, but the fact that I could think of a proof of infinite length (a boring, brute force, and purely theoretical proof where one just checks every natural number) does seem to give one the intuition that the natural numbers must exist. Of course this is not a solid argument and is simply the expression of some feelings mathematicians have when doing mathematics. I am of the position that when it comes to metaphysical questions, one should not expect a completely compelling argument. Instead, one can just choose to take a leap of faith and believe in platonism about mathematical objects or not. The arguments about them perhaps just help us form intuition or perhaps just tickle our curiosity. My position is to remain agnostic about metaphysical questions and simply enjoy thinking about possible answers. Where do you fall on platonism about mathematical objects?

[@AlainGenestier-tj8fn]

@@hungrymathprof I am also a mathematician who just likes to think (from time to time) to such questions. To make precise the notion of a "proof of finite length" I remark that you already need a pre-existing notion of integer. A (dis)-proof of the vanishing of the Goodstein sequences would be a non-standard integer such that the Goodstein recursive sequence with this initial term doesn't terminate. From one perspective it is of infinite length (because the integer is non-standard) but for the (speculative) inhabitants of a "non-standard world" it would be finite...