math_dandy

How do we measure if someone is a good mathematician or not?

Letters of recommendation from trusted colleagues have always been essential for evaluating candidates. Hopefully, human recommendations will remain a strong, faithful signal as the utility of other metrics rapidly deteriorate.

Entirely correct. The math establishment needs fundamentally overhaul its incentive structure-irretrievably broken-to function under the assumption that AI involvement in research is completely ubiquitous.

Regarding 6., that we shouldn't throw AIs at unsolved problems in bulk and thus lose human expertise. I think that ship has well and truly sailed. AI will continue to be thown at all manner of unsolved problems, and at an increasing rate - by commercial labs, by individual researchers, and by academic research teams. The mathematical community needs to establish paths to meaning, progress, and growth of human expertise in a world where AIs will by default be thrown at every unsolved problem.

From the slide Beyond Problem Solving:

"Reaching these lighthouses [resolutions of open problems] prematurely by automated tools can disrupt the exploration of the paths not taken, and sterilize the surrounding field."

This crucial issue is centered in mathematician psychology and the incentive structure of academic/institutional mathematics worldwide. For mathematics to flourish going forward, we will need to realign our brains to think differently about the nature of mathematical progress. And we need to reorient our institutional incentive structures towards the promotion of meaningful mathematical progress itself rather than targeting proxies that are no longer faithful.

Regardless of the precise nature or the causes of the "sterilization" Tao refers to, we (the mathematics community) can only rely on ourselves to repair it. Though, since it will involve fundamental change at the level of ossified academic institutions with many stakeholders and divergent vested interests, any such repair will be slow, frustrating, controversial, and lacking any guarantee of success.

An example of how mathematicians and AI can complement one another in probing the frontiers of knowledge: Number theorist Youness Lamzouri has constructed a new, conceptually simpler proof of Claude's Theorem that 2/3 of the zeros of Riemann's zeta function lie on the critical line and are simple.

> Levent Alpöge and Ralph Furman, two of Anthropic’s own mathematicians, examined Claude’s work to understand the new results and how they related to the prior work mentioned above.

My wife did her mathematics PhD under an analogous structure in Germany where it’s fairly common. She was supervised by a professor at the University of Munich and a lab at Bosch, wrote and defended a thesis as usual. Employed in industry immediately after graduating. A great experience.

I guess what I’m saying is that since there’s always a context, ambiguity stemming from uninspired naming is never an issue in practice.

Certainly overloaded but rarely ambiguous. Context will determine which notion of “normal” applies.

tl;dr: Being a homomorphism from a multiplicative structure into an additive structure isn't enough to grant it the logarithm title.

Although logarithms are certainly ubiquitous in mathematics, I don't think that the mappings that the article's author identifies as logarithms are appropriately viewed as such.

I can't endorse viewing dimension as a logarithm. It appears superficially logarithm-like because we typically (and somewhat unfortunately) write the direct sum of n copies of a vector space V as V^n rather than nV. Writing nV, we simply get the dimension identity dim(nV) = n dim(V). Writing nV instead of V^n also conveniently frees up V^n for the tensor product of n copies of V, with corresponding dimension identity dim(V^n) = dim(V)^n. So I don't think there's any "multiplicative-to-additive" business going on here at all.

Also, I don't think it's advisable to view the p-adic valuation ord_p as a logarithm, even though it's a homomorphisms from the multiplicative group of the rational or p-adic field into the additive group of the rational field. In fact, in many number theoretic contexts, the ratio log_p/ord_p is of particular interest.

I think a good rule of thumb for viewing a mapping as some kind of logarithm is that it has to have some relation with the Taylor expansion of log(1 + x) around x=0. Being a homomorphism from a multiplicative structure into an additive structure isn't enough to get the logarithm title.

I think the OpenAI model that resolved the Unit Distance Problem would be capable of solving a significant proportion of mathematics PhD thesis problems.