Discussion
A thought about mathematics and genAI (sorry), but I haven't seen this point expressed yet, so wanted to share.
According to Platonism (in the sense of Gödel), humans discover mathematical truths by developing a kind of quasi-sensory relationship with their abstract object of study. This suggests an interaction that is quite similar to how we interact with physical objects: they're objectively there, independent of us, and we discover their properties through our senses. 1/2
Also, there have always been stricter schools (like intuitionism that doesn't accept non-constructive proofs, ultrafinitism that doesn't accept completed infinities, predicativism that only accepts certain types of definitions) that could often reproduce (and thus strengthen) a surprisingly large part of mainstream math results.
A diversity in approaches to AI in math could yield similar fruits. (But I'm rooting for the Human Mathematicians to become much bigger than some of these schools!) 3/2
Paradoxes (like Russell's) spurred new directions: Russell's type-theory (also led to Lean), 'standard' set theory & others.
In principle, current axiom system could be inconsistent (as Nelson believed of arithmetic).
Now, LLM-based systems seem very good at finding counterexamples. So, speculative scenario: they might find an inconsistency in a base theory, that would again require a fundamental revision, paving the way for new math outside of patterns current genAI was trained on.
4/2=fin 😅
@SylviaFysica Re: “a kind of quasi-sensory relationship with their abstract object of study” and FYI: C.S. Peirce (1839-1914, https://en.wikipedia.org/wiki/Charles_Sanders_Peirce) had a similar notion. He held that mathematicians work with “visual skeletons” (which would include, say, category-theory-type pictures but also “a system of equations written under one another so that their relations may be seen at a glance”). (1/2)
@SylviaFysica My (possibly too platonistic) interpretation of what happens is that mathematicians work with mental images of some mathematical world, while LLMs work in the realm of language and proofs. If LLMs suggest new axioms, why would we be willing to accept them? How can we predict that they will model our inner world / the mathematical reality?
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From that perspective, an LLM-based system can continue to build upon what mathematicians have written so far, exploring the convex hull of the space of their writings so far (which may yield new combinations that we could consider creative), but one could expect this to diverge significantly from what researchers, who are in contact with their object of study, would find. Much like you wouldn't expect physics to get ahead based only on what's written so far, without additional experiments. 2/2