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.

"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.

Comments URL: https://news.ycombinator.com/item?id=49577167

Points: 3

# Comments: 1

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.

IME a vastly more common sentiment among mathematicians regarding mathematical talent leaving the nest to apply their skills in other fields is that those other fields are lucky to get them!

[1] https://arxiv.org/html/2605.20695v1

Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable.

Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.

I think the machine learning community was largely over overfitophobia by 2019 and people were routinely using overparametrized models capable of interpolating their training data while still generalizing well.

The Belkin et al. paper wasn't heresy. The authors were making a technical point - that certain theories of generalization are incompatible with this interpolation phenomenon.

The lottery ticket hypothesis paper's demonstration of the ubiquity of "winning tickets" - sparse parameter configurations that generalize - is striking, but these "winning tickets" aren't the solutions found by stochastic gradient descent (SGD) algorithms in practice. In the interpolating regime, the minima found by SGD are simple in a different sense perhaps more closely related to generalization. In the case of logistic regression, they are maximum margin classifiers; see https://arxiv.org/pdf/1710.10345.

The article points out some cool papers, but the narrative of plucky researchers bucking orthodoxy in 2019 doesn't track for me.