Post #2920364

2026-05-17 16:36 UTC

Tarski's papers provide many excellent examples of how a paper might be structured in this way. In both On Definable Sets of Real Numbers and The Concept of Truth in Formalized languages, he states rather explicitly that his primary objective is to find definitions (in the former case, a definition of definability, and the latter case, a definition of truth). To Tarski, the formal results are meant to convince the reader that the definitions are correct. As far as we know, this is something that an AI simply cannot do, because saying what is a 'correct' definition is provably not computational. He also focuses on aligning his reader to the 'right questions' which he often states explicitly as having been neglected under his view. The main takeaway is that these may require a paradigm shift: not simply a change in workflow, but a restructuring of the philosophical underpinnings of our field. What is required is a widespread acceptance that there are aspects of mathematics that are not computational, together with a paradigm shift that codifies or formalizes these aspects of mathematics, in the same way that Frege and Hilbert codified and formalized classical aspects of logic. I have left out a discussion of the third point: philosophical curiosity, this is more difficult to describe. But the point is that we have gotten used to motivating others to read our work by connecting them to 'famous' ideas and questions together with providing some certificate of authenticity (formally correct proofs). (7/n)

Replies (1)

@wikiemol@mathstodon.xyz 2026-05-17 16:37

Here 'famous' does not necessarily mean millennium prize level famous, but also includes famous enough for a small group of people who don't know each-other to come to learn about the same ideas and questions independently. The trouble is, this kind of motivation is not very useful in an age of AI, because this method renders all aspects of mathematical practice essentially decidable. Instead, the burden we will face in the future is that our work will have to "light someone's soul on fire" in the same way the blue eyed islander puzzle does, or otherwise hope that someone else's soul has already been lit, and that they stumble upon your work on their own due to shared curiosity. This ability for humans to 'light up each-others souls' is the essence of the alignment problem. Pure mathematics papers will no longer be able to start with an introduction with a structure consisting only of "in so and so et al, xyz was proved, then Smith et al expanded on xyz by proving lmnop. In the present paper, we expand further by generalizing lmnop to all fantasy creatures, not just unicorns, we do so by..." This is because this leaves the burden of justifying the present work as a philosophically edifying to the reader, and as you discuss in this talk, this is no longer pragmatically possible. Pure mathematics has never been pursued for only philosophical reasons, but it seems to me that modern mathematicians have essentially banished philosophical discussion entirely, and the totality of this banishment seems to have only fully taken over in the 21st century as a result of the development of computers making the formalist paradigm seem entirely natural. (8/n)

Open ##2920365