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Several fundamental problems in math are undecidable; see the sphere recognition problem for example. To make it worse, quantum computers will not help. Therefore, I wonder what we can do about undecidable problems, assuming no fundamentally better computing devices are available.

Certainly, limiting the questions to some easier cases is a choice, but that does not really tackle with the undecidable part (this is not to say that the easier cases are not interesting). Also, if humans can tackle with the undecidable part, does that mean our brains are strictly more powerful than a Turing complete machine?

EDIT

My question might seem pointless, as why would we bother to deal with things out of our control? I wish to establish my point through a naive example. But this would require some imagination.

Imagine that we don't have a theory of polynomials. How do we prove that $$x + x = 2x,$$ as functions from $\mathbb{N}$ to $\mathbb{N}$? By "$x$" I mean the identity function, and by equality I mean the functions on both sides evaluate to the same value for all inputs. Computationally, one will never be able to check if both functions $(x+x)$ and $2x$ have the same value for each input, as there are infinitely many cases to check. However, the equation can be proved from the axiom of contradiction.

Student
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    What should quantum computing have to do with decidability? -- While human brains may sometimes show weird behaviour, IMHO any agreeable mathematical result must be communicatable and thereby cannot transcend the capabilities of a Turing machine – Hagen von Eitzen Dec 13 '20 at 18:42
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    Even undecidable problems like the the correspondence problem of Post can often be solved for the majority of instances occuring in practice. The situation is not necessarily hopeless for a particular instance, so it can still have a merit to try to solve such problems. – Peter Dec 13 '20 at 18:47
  • @HagenvonEitzen quantum computing has nothing to do with decidability, as shown in a link in OP. I pointed that out just to clarify that such technology is not an answer to my question. – Student Dec 13 '20 at 18:48
  • Negative results in mathematics are also useful ! Since we know that there is no formula to solve polynomials with degree $5$ or more , we need not waste time anymore to try to find one. Finding principle limits is an important part of mathematics. And to expect that some tools will eventually pass those limits is not much less optimistic than expecting time machines somewhere in the future. – Peter Dec 13 '20 at 18:55
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    What should we do about it? Nothing. What can you do about things beyond your control? Are you suggesting that we limit mathematics to things we can compute? That's not the domain of mathematics, that's barely even the domain of physics. – Asaf Karagila Dec 13 '20 at 18:58
  • @AsafKaragila Excellent comment ! – Peter Dec 13 '20 at 18:59
  • @AsafKaragila I added a naive example and hopefully it clarifies why I think/hope there could be something we can do. – Student Dec 13 '20 at 19:26
  • Some undecidable problems can become decidable with a small change, like "is there a proof of X" is undecidable, but "is there a proof of X less than N characters long" is NP complete. – DanielV Dec 13 '20 at 19:43
  • I want to say "nobody computing anything", but that'd false. We have inference rules and we have axioms, and we use them to infer theorems. You don't need to compute anything, you can do symbolic manipulations, or you can use more complicated tools. It's not about computing stuff. You can't even compute all the digits of $2^{2^{2^{2^{2^{10000}}}}}-1$, and that's just a finite expression. – Asaf Karagila Dec 13 '20 at 22:41
  • @AsafKaragila That's scratching my concern! Why can symbolic manipulations, or more complicated tools, surpass the computational difficulties? Doesn't that imply human brains cannot be subsumed by a Turing machine? – Student Dec 13 '20 at 23:11
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    Because math is *not* computational. – Asaf Karagila Dec 14 '20 at 00:24
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    @Student But turing machines can also do symbolic manipulations, infer theorems from axioms, etc. The question of whether the human brain is more powerful than a Turing machine has not reached consensus. A few (e.g. Searle, Penrose) have argued forcefully for 'yes' but most haven't found them convincing. – spaceisdarkgreen Dec 14 '20 at 00:28
  • @AsafKaragila But a large part of math is computational. In this sense, what is the non-computational part? Why can't a turing machine achieve what a human brain can? That's essentially my question. – Student Dec 14 '20 at 00:37
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    The computational part is very different from just trying to prove arbitrary things by computing them. As for a Turing machine, we don't understand the brain, but we're really just mining for proofs which is a very computable process. – Asaf Karagila Dec 14 '20 at 00:43
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    A human brain can't decide undecidable problems either. – Misha Lavrov Dec 14 '20 at 03:55
  • @DanielV Would you mind providing an explicit example? – Student Dec 14 '20 at 03:59
  • @MishaLavrov Human brains cannot prove that $x+x$ and $2x$ have the same value for each input. But they came up a way to surpass it. I am hoping for something similar for undecidable problems. – Student Dec 14 '20 at 04:06
  • @Student To counter Misha's comment, this example is not suitable. $2x=x+x$ is provable , in fact a consequence of the definition of multiplication. You have to argue with problems having no algorithm to solve them. And if we accept the Church–Turing thesis (which has not been proven, but emprically very well approved) , then humans are helpless as well. – Peter Dec 24 '20 at 12:23
  • @Peter, by f = g I mean both functions f and g take the same value for each input. How would you prove that without introducing extra axioms (like the law of excluded middle)? By applying this example to my question, I hope that we can introduce extra axioms for resolving undecidable problems. – Student Dec 24 '20 at 16:23
  • @Student For proving $x+x=2x$ we neither need induction nor do we need the law of the excluded middle. $2x$ is DEFINED to be $x+x$ as I said. Concerning creating new axioms for undecidable problems : Goedel showed that this new theory with the new axioms will again be incomplete (assuming consistency), so we would never come to an end. – Peter Dec 25 '20 at 06:51
  • @Peter that is not how I defined $2x$. I define $2x$ to be the function whose output is two times of the input. So with this definition of $x$, $2x$, $=$, how do you prove the equation without invoking extra axiom? – Student Dec 25 '20 at 15:22
  • @Student I suggest you to read some articles about proof theory and in particular goedels incompleteness theorems. The matter is more difficult than it seems and the consequences of Goedels results are often completely misunderstood. Statements can often be proven although we cannot check every case seperately. – Peter Dec 25 '20 at 16:59
  • @Student: There is zero reason to arbitrarily introduce extra axioms for resolving undecidable problems. If ZFC is consistent, then so is ZFC+¬Con(ZFC). Should we add ¬Con(ZFC) to 'resolve' that? Obviously not. But that's because Con(ZFC) is an arithmetical sentence whose truth-value we have some intuition/hope about. What about undecidable sentences that we do not? We cannot resolve them by simply adding it or its negation based on our whim... – user21820 Dec 26 '20 at 19:55
  • @user21820 Do you imply that undecidable problems and independent statements are the same? If not, would you mind showing how it relates to what I asked? If yes, would you kindly provide some pointers? I'm indeed lacking basic knowledge. – Student Dec 27 '20 at 02:46
  • @Student: No, they are not the same. But your question seems to be about sentences that you cannot decide (their truth value) due to lack of axioms, not about undecidable problems. And no, humans are definitely not able to solve undecidable problems, no matter what anyone might have claimed to you. If you want to understand incompleteness, you should have at least good facility with FOL (first-order logic) and basic programming knowledge, and then you can read my post on it. – user21820 Dec 27 '20 at 03:41
  • Once you understand the incompleteness theorems and the proofs I gave, it will be clear why, for every practical formal system that can reason about programs, every undecidable problem yields at least one sentence that the formal system does not decide (prove or disprove). So they are related, even if not the same. – user21820 Dec 27 '20 at 03:46

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