Why is there still a lot of unsolved math problems, despite having really advanced computers?
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The answers so far are a bit off: yes, there are a lot of statements about infinitely many things, so you cannot just test them all; but that isn’t the only thing a computer can do. And the real answer is: our computers and algorithms are simply too slow. So far. Given extremely long run-times (not truly infinite, just absurdly large), a computer could solve all the problems mathematicians can.
So how can a computer theoretically solve every problem? First we have to notice that mathematical “evidence” is all about _proofs_, formal logical reasoning from the given axioms. Those are often presented in a human-readable format (but some authors are worse at this than others), but ultimately, a proof uses only the following constructs:
– Variable and constant names such as a, b, c, e, pi, Hamster or any other name. Restricting them to be somewhat sane is usually a good idea.
– Function names such as f(…), +, -, ·, /, and more, including the information how many entries (and of what type) it wants.
– Symbols for basic constructs such as = (“equal”) or ∈ (“element of”).
– Logical constructs such as “for all” (“∀”), “there exists at least one” (“∃”), “implies” (“⇒”) and “and” or “or”.
– Maybe a few more, but lets end the list here.
The key point is that ultimately there are only finitely many different proofs of a given length. What a computer can now do is literally try one-by-one every possible sequence of such symbols to check if it
(a) makes sense (unlike “apple being ⇒ hamster up down +”),
(b) ends with a formalized version of what we want to prove.
All this is perfectly doable. But even for proving the simplest math from over 2000 years ago, this would fail. It simply takes way too long. Our computers are sooooo many orders of magnitude off to get to the end.
Now obviously that also means this is not what mathematicians do. Indeed, they use their experiences, intuition and knowledge to try those avenues to the goal which are much more likely to actually work. And results show this often works really really well.
Recently and for the first time, we were able to make basic artificial intelligence. Assuming some further advancement, which is likely, we may see the day when computers have a comparably good intuition than us, and thus will be able to tackle real mathematical problems.
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