Post #2920363

2026-05-17 16:34 UTC

For an example of a question that is widely agreed to be 'great' consider the 'blue eyed islander' puzzle you helped popularize. It is a perfect example of a question that is as interesting and difficult to find (if not more so) than the answer itself. An AI may or may not have been able to generate such a question, but the question must have been fully 'digested' by a human being in order for it to be known to have any worth. The blue eyed islander puzzle is clearly seen by anyone who attempts to solve it to be an incredibly valuable question, even without having it be connected to any other previous work. The reason that there is such wide agreement about the value of the question itself remains a mystery from a formal perspective, but I do not think it is entirely outside of rational examination. With regards to definitions: the distinction between uniform continuity and continuity is one such simple example. In many cases, the notion of uniform continuity is clearly the 'right' definition of continuity for that context. For a human, one can realize that classic continuity is wrong and uniform continuity is right even when one has not seen uniform continuity before, but how exactly we do this is somewhat of a mystery from a formal perspective. For more interesting examples, one can see Imre Lakatos book Proofs and refutations, in which the historical development of the definition of a polyhedron is explored at length. There is also a math overflow post with more modern examples. (6/n)

Replies (1)

@wikiemol@mathstodon.xyz 2026-05-17 16:36

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)

Open ##2920364