Good point, in fact it’s not clear (afaik) whether Mochizuki’s recent claimed proof of the abc conjecture satisfies ZFC – but suppose his argument turns out to be quite persuasive, but requiring an addition axiom(s) – how would the mathematics community respond?

I was fascinated by the Goodstein Theorem as a student, since it was a very concrete example of a sequence of numbers which converge to zero (incredibly after massive explosive growth) – but the convergence can not be proved in Peano arithmetic. There are no infinities involved, the sequence just grows very large but finite and always descends back to zero. However, you just need to add an axiom to allow transfinite ordinals, then the proof is simple!

So I lost my fascination, as it seemed just a clever game where you can find “mysteries” but you solve them by additional rules to cover the “mystery”

side note: I attended a public lecture at the Cambridge Newton Institute shortly after Andrew WIles had claimed a proof of FLT, the proof was still not verified. John Coates gave the lecture and at the end Stephen Hawking asked a question – “Does the proof use the axiom of choice?” – Poor John Coates was very uncomfortable giving an answer, he said no, it was all very straightforward constructions that didn’t employ AC. To be fair to professor Coates, the proof was not fully understood at the time, and Wiles had yet to provide the crucial Euler system to fix a big error. As it turns out, it is now accepted that the axiom of choice IS required in Wiles’ proof.

So how do people feel about a proof of FLT requiring a slightly “dodgy” (“unnatural”?) axiom – is that satisfactory? Should we require a proof in ZF alone (not ZFC)?

]]>In particular, it is natural to sharpen the question to

Which of the Millennium Prize problems can be stated as a postulate that can be falsified by a $\Pi^0_1$ counterexample?

There are two classes of answers.

The first class of answers accepts the problem statements in their present form: then if we further assume P≠NP, then the Riemann Hypothesis is the sole Millennium Prize problem that can be falsified by an $\Pi^0_1$ counterexample whose form can presently be envisioned.

The second class notes that Millennium Prize problem statements *can* be amended (at the discretion of the Scientific Advisory Board (SAB) of the Clay Mathematics Institute (CMI)).

Then the following question is (seemingly) entirely open

Can any of the Millennium Prize problem statements be reasonably amended — PvsNP in particular— so as to be stated as a postulate that can be falsified by a $\Pi^0_1$ counterexample?

Opinions vary widely in regard to the second question … this variation signifies a healthy and vigorous mathematical community (needless to say).

**Conclusion** It is reasonable to consider whether amended postulates associated to the Millennium Prize statement of the PvsNP problem might simultaneously evade proof obstructions *and* still address practical questions like “do there exist dynamical systems — including both physical systems and ciphers — that provably *cannot* be simulated/solved in with resources in P?”

I think the possibility of the undecidability of open problems should be considered a serious possibility precisely because, for many of the problems you mention, unprovability of undecidability is or is nearly a logical consequence of undecidability. So I’m uncomfortable saying the Godelian gremlin hasn’t reared its head when there are plenty of problems where a) it could have and b) if so we’ll almost certainly never know that it has.

To make my point more concrete let’s consider the possible universes for P vs NP:

1. P = NP and this can be proven.

2. P = NP and this is impossible to prove.

3. P != NP and this can be proven.

4. P != NP and this is impossible to prove.

P = NP seems less likely precisely because most of the proof for it has been written — all we need is a polynomial time algorithm for an NP-complete problem, and if P = NP then such an algorithm certainly exists. That makes #2 almost impossible. I say almost, because there is a remote possibility that algorithms exist that have a polynomial runtime bound, but it is impossible to prove that their runtime is polynomially bounded. So to prove the undecidability of P vs NP you would need to show that either P != NP or that it will be impossible to prove that the runtime bound of any polynomial algorithm to an NP-complete problem is actually polynomially bounded. That would be a truly miraculous proof; I would be quite surprised to discover that it is possible to prove that and not possible to establish P vs NP. I am not holding my breath for that proof. But the remaining possibilities are, we will either some discover a proof one way or the other, or we will beat our heads endlessly against this problem never knowing that it is impossible and we should just add P != NP as an axiom. I see no reason to doubt that last possibility; it seems quite likely. It is even more likely that, among the various open questions you mentioned, there is at least one problem like that. We’ll never know that to be true; we’ll only maybe see a few of them knocked off the open questions list and figure the problems we can’t solve seem no different from the problems we had a lot of difficulty solving but ultimately did.

]]>There shouldn’t be any puzzlement that complex stuff arises, but also we wouldn’t expect a HUGE number of surprises – we are aren’t performing magic by creating a few rules/axioms in the first place

Nature, also, surely has a ruleset – and discovering that is a much bigger deal.

Unfortunately us dumb humans find approximations to nature’s rules and then extrapolate this in fantasy worlds that are much more complex than anything in nature.

There’s mystery in mathematics, because it’s unnatural, if we ever meet an alien species I’m sure they will have discovered weird mathematics beyond number theory etc that will astound us, just as they may be astounded by inter universal teichmuller theory and other weird stuff that we’ve managed to create.

But nature has only one possible correct interpretation, and it surely won’t involve a lot of the more esoteric areas of mathematics.

]]>There a few issues here going on, and Scott has touched on most of them, but I’d like to make one of them more explicit: if I prove in some broad sense that say RH is undecidable in some axiomatic system (say ZFC), then I haven’t proven in ZFC that RH is true, but I’ve only proven it true in some larger metamathetical sense.

]]>– You play around with things, notice a pattern, and wonder whether always holds or whether there’s a deeper explanation for it

– You’re trying to do something very concrete (say, solve an equation), notice a difficulty, and wonder whether the difficulty represents an inherent limitation of the tools you’re using

– Someone else proves something, and you wonder whether it can be generalized, or whether the converse is also true (“but is it an if and only if?”), or whether it’s still true if you drop an assumption, or whether an analogous result holds for something you personally care about

– You’re trying to compute something, and you wonder whether there’s a faster way to do it

– You know from experience or history that asking a certain kind of question tends to be fruitful—so when a new mathematical object is proposed, you immediately ask that question about it

– (My personal favorite ) Someone else says something you think is wrong, and you wonder how to *prove* it’s wrong