How close are computers to automating mathematical reasoning?

Re:How close are computers to automating mathematical reason... by tsimionescu on Saturday August 29, 2020 @12:51PM (#24317425)

The Church-Turing thesis claims (and is believed to be right) that a Turing machine can solve (compute) any problem that can be solved (is computable).

For now at least, we don't know of any counter-example to the Church-Turing thesis. Humans are certainly not able to tell whether a program in general will halt for any input it recieves. If you believe otherwise, please tell me whether the following program halts:

    Foo():
        if P = NP:
            return true
        else:
            Foo()

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Re:How close are computers to automating mathematical reason... by qayxc on Saturday August 29, 2020 @01:32PM (#24317800)

> If you believe otherwise, please tell me whether the following program halts

That's a logical fallacy - just because I can't show whether P=NP doesn't mean nobody can!

Turing-Church is a conjecture, not a theorem. There have been arguments made against it [1].

[1] https://arxiv.org/ftp/cs/papers/0207/0207055.pdf

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Re:How close are computers to automating mathematical reason... by sudosysgen on Saturday August 29, 2020 @03:11PM (#24318666)

There is no reason to believe that the human mind can solve a superset of the problems solvable by a Turing machine, and a lot of evidence to the contrary.

Also, the link you replied provides an argument of efficiency, not an argument of possibility. It is arguing that we might use things that are not Turing machines because they can solve the problem more efficiently, not because they can solve problems Turing machines could never hope to solve.

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Re:How close are computers to automating mathematical reason... by tsimionescu on Sunday August 30, 2020 @10:20AM (#24323166)

Solving the halting problem means being able to tell, for ANY Turing machine and ANY input, whether it will halt. I know of absolutely nothing that would make us believe humanity or any individual human can do this.

If there is a single Turing machine for which there is no human that can tell whether it will halt, the conjecture that human reasoning is not more powerful than Turing machines is still not disproven.

Also, if you were right that humans can do things that Turing machines can't, it would be an interesting excersise to try to find that example of super-Turing human reasoning and try to see why it can't be encoded as a Turing machine. No one has successfully done this yet, so there is no reason to believe that it is possible.

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