The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Abstract

The Ideal Mathematician (IM) is sitting in her of ice and hears a metallic knocking at the door. She inds this peculiar as the door of her of ice is made of wood. When she opens the door, she inds the Arti icial Mathematician (AM), a large bulky computer, running various automated mathematics software programs, playing door-knocking-sounds out of its speakers. AM: Could I interrupt you for a minute? IM: You already are, so go ahead. AM: I'd like to be part of the mathematical community. IM: You already are, so go ahead. AM: Oh, I know you employ me as a tool in the practice of mathematics, but my dream is to be a full-ledged mathematician. IM: That doesn't sit very well with me. AM: Why not? IM: Well, you are a computer and mathematicians are human. AM: That is ironic. Yesterday I overheard you say to the skeptical classicist that mathematics 1 is free of the speci ically human and now you are disqualifying me for not being human. IM: Well, it's not that being human is a necessary condition for being a mathematician. But there are unsatisfactory differences between you and humans that are not in your favor. AM: Like what? IM: Take your famous contribution to the 4CT for instance. You go through over more than a thousand cases of testing and then you tell me "it checks out", but how do I know it does? AM: Because it checks out, I've checked it. IM: I know you've checked it, but a mathematician hasn't checked it. AM: If you accept me as a mathematician, then a mathematician has checked it. IM: This is not just a matter of de initions. Why should I believe you? How do I know you haven't made a mistake, didn't have some bug or hardware failure? AM: By checking my code, running my program multiple times and on multiple systems. IM: But regardless of all these things, it'll always lack perfect rigour. I'd have to put some degree of trust in, or perhaps put a degree of probability on, the result. This effectively makes your result more of an empirical corroboration than a mathematical proof. AM: So the difference is that humans don't make mistakes, is that it? IM: No, they do make mistakes, but that's why we have peer-review. AM: Oh, it's the peer-reviewer that never makes any mistakes and always spots all the ones made by the prover? IM: Not all, always, no. AM: It sounds to me as if human-generated mathematics is just as empirically fallible, just differently so. IM: Very differently so! You don't seem to realise how reliable human provers and peer-reviewers are.