zyklu5

> For example, we can compare variance in song energy and danceability for U2, ABBA, and Katy Perry, then rank these artists against each other. According to this subset of variables, ABBA's discography offers greater sonic variety than U2's or Katy Perry's.

Stopped reading here

While I respect your point about rigour I should say one has to be careful not least because the way the subject is presented in textbooks are most often backwards -- axioms are really an end not the starting point.

Here's another view: Euclidean geometry is euclidean because the underlying transformation group (the group of those transformations which preserve what we want to preserve -- in this case the metric) is the euclidean group (the semi-direct product of the orthogonal group and translations). This is the symmetry that encodes our intuition -- the same intuition the kid is using to prove the above result. If we were to change the underlying space to the real projective space instead of R^2, and instead of choosing to preserve the metric we choose incidence and cross-ratio, we'd get a different group (GL(3,R)) and different geometry, viz. projective geometry.

This is an ancient dialectic that runs within mathematics -- embodied in modern math by Hilbert on one side (the formalist) and Poincare on the other.

I think you are mixing layers of abstraction. To make a crude but I think not unhelpful analogy: 'Understanding' is a natural language concept that is our way to describe whats happening in our heads, and like most other such concepts is resistant to any clear definition and will exhibit sorites type paradoxes when one is attempted. It belongs to the presentation layer of the stack. While the process of curve fitting, however it is implemented, with whatever NN structure (like transformers) or maybe something else entirely belongs to the physical layer of the stack -- akin to frequency modulation.

While I am unsure whether LLMs are really understanding, whatever that means, I think it is not difficult to believe that any form of understanding we implement will involve 'curve fitting' as a central part.

I don't know what a phd in "algebra" means but I don't understand how anyone with a phd in any part of mathematics can find this unintelligibility surprising. I have a phd in what could be said to be the langlands program itself and I am not at all surprised I can't follow this result.

If he could've done it in a week then he didn't really need to do it at all.

There's another Weyl. Hermann Weyl. A mathematician almost as great as Hilbert. In fact he is the person who rigorously introduced Riemann surfaces. He is also the originator of Gauge theory -- so fundamental to modern physics.

There is no general statement to be made here -- some tribal organisations are hierarchical other's aren't (including the ancestors from my fathers side as recent as 100 years ago). For decades now western ethnographers and cultural anthropologists seem to have projected their dissatisfactions of their own world onto their subjects -- which is something more insidious than the frank judgements you get to read in first-hand encounters during the early modern period. This has reached a certain pitch in current years by the overt activism of modern X-studies.

> today's hierarchy was invented by the modern Christian church.

On the contrary, the Church was responsible for destroying the hierarchical structure of my father's people via colonialism. Just to repeat: I am trying to point at a strange phenomenon that one notices in modern western academia: it is precisely the unaddressed universalising, really, Christian beliefs of, say, a current professor of cultural studies coupled with a dissatisfaction with her own culture that leads her to attribute what is considered good under her ideological commitments onto the natives and the other way round. For example, the idea that all men are equal would be flat out 'devilish' to my father's people even 100 years ago.

Americans have been brainwashed to conflate preference for prejudice.

If the terms: fundamental group, covering space etc are jargon then more so is the case for words like 'explanation', 'laden' and well, 'jargon' itself. They have precise definitions meant to capture mathematical concepts and ideas for which there can be no adequate words in natural language.

This was published as: Abel's Theorem in Problems and Solutions by VB Alekseev if you are interested in taking a look.

https://link.springer.com/book/10.1007/1-4020-2187-9

Not to mention the kind of mathematician that the Fields medal is poorer for not having as an awardee

The difference is, to put in Rumsfeldian fashion, between known unknowns (kids proving QR now) and unknown unknowns (euler sensing QR, gauss proving it).

That Euler knew most of the math of his time is irrelevant. If one had any serious learning at any time in most of human history one would know all the math of their time.

Your first sentence can be used for anything:

Etale cohomology was a low-hanging fruit because it depended on a bunch of mathematics and technology that Grothendieck benefitted from: algebra, the printing press, and modern transportation. Sure, Euler had horse drawn carriages but that is nothing compared to the speed of modern transportation that Grothendieck had available to him.

I find the notion of 'low-hanging fruit' in such contexts profoundly ahistorical. If graph theory was low-hanging why did it take thousands of years since Sumer or ancient Egypt? Or consider something from number theory: every other batch of students in a math camp I'm familiar with has someone who has 'proved' quadratic reciprocity for themselves -- and how could they not ? -- since childhood they have been immersed in a culture which points at it; while it took Euler roughly 40 years to even formulate the idea and then Gauss to prove it. It's not that - to use an anachronistic term - class field theoretic phenomena was not known to other cultures thousands of years ago.

(Btw there are many contemporary mathematicians at least at the level of Terence Tao but for some reason haven't been blessed by lay popularity -- mostly because Tao's math looks more school math like / familiar to non-math people than say Peter Scholze's)

No amount of 'progressive' re-writing of history or (ironically) whiggish re-interpretations of what some non-western cultures (including mine) did will change the fact that science as we understand it began in Europe around the early modern period and it had the most revolutionary effect on the the world. This is why the oft quoted trivia that Cleopatra is closer to us than to Imhotep is so striking.

Having said that, there's an excellent scholarly work of revisionist history written by Lucio Russo called the Forgotten Revolution that I can recommend for some pleasuraeble outrage.

Can't believe you are the only one to mention this. I thought concentration of measure would be at least mentioned to ML students right about the time they learn about the curse of dimensionality