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The tools are not ready. Using proof assistants isn't possible in many areas of maths as the foundational concepts have not yet been built up from fundamental
by mkl 4y ago
The tools are not ready. Using proof assistants isn't possible in many areas of maths as the foundational concepts have not yet been built up from fundamental primitives. Even when it is possible, using proof assistants takes far more work, and the proofs are much more verbose and hard for humans to understand. The tools will be ready when the concepts are available for any subfield, when the results are easy to read, and when it takes a reasonable amount of work (ideally it will be faster than doing it by hand).
- martincmartin 4y agoI agree the tools are not ready because, as you say, the foundational concepts are not yet encoded. But for OP, the point isn't having humans understand. It's having the computer check the proof. You still use traditional journal papers for communicating the proof to humans. You're just also able to say "The proof has been verified and really is true."
- mkl 4y agoYes, but humans using the proof assistants need to be able to read and understand the proofs while they are constructing them.
- lacker 4y agoideally it will be faster than doing it by hand That's a good point... I agree the tools are not ready. Most computer-assisted tools make it easier to do your work. It's easier to type a five page paper in Word than it is to write it longhand. But for now it's a lot more work to encode a proof in a proof assistant than to just write it up in LaTeX (which I think a lot of people don't realize). It's not just that the foundational concepts are missing, it's also that proof assistants currently require you to fill in much more detail than human proofs do. You can't just write something like... "this corollary is trivial", or, "since x is a power of 10, obviously cos(x^2) is irrational"; you have to spell out a proof.