Post #2920362

2026-05-17 16:31 UTC

I have selected 1 and 2 because, in terms of the formalist paradigm, neither Questions nor definitions have been properly 'integrated' into our current mathematical practice in the same way that the rest of logic has. The question of what is the 'right' question to ask or what is a 'wrong' definition seems to be entirely absent from our systems of formal logic, despite the fact that there seems to be very wide agreement on when a question is a 'great' question in some cases, or when a definition is clearly 'wrong' by mathematicians in some cases. The formalist paradigm provides no distinction between definitions that are 'wrong' and definitions that are 'right'. Nor does it provide a distinction between 'great' questions and 'good'/'bad' questions. Both questions and definitions have some chance of having something along the lines of a formal treatment, it is just that these formal treatments may need to be essentially entirely outside of the computationalist paradigm. So, my feeling of what a solution should look will come from this area of inquiry. (5/n)

Replies (1)

@wikiemol@mathstodon.xyz 2026-05-17 16:34

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)

Open ##2920363