Hardy would agree with the viewpoint that you espouse but it would be pushed back against by Arnol'd, Poincare, Gauss, Von Neumann, and even Grothendiek: Arnol'd and Poincare were vituperatively against the division between "pure" and "applied" mathematics; they considered mathematics and physics interchangeable, and Arnol'd lamented that the field had lost a large amount of funding/prestige/relevance due to groups like the Bourbaki that took a purely aesthetic view; Gauss had a critical view of problems like Fermat's last theorem (he felt that you could construct infinitely many such problems, and felt that attempting to prove it was a generally useless endeavor), along with outright calling pure mathematics worthless; but while Von Neumann and Grothendiek were more moderate, both were critical of the field losing motivation/quality as it strayed away from empirical science into—quoting Von Neumann—"abstract inbreeding".
Arnold's polemics are perhaps the most infamous and easily found online (see "On Teaching Mathematics"), but the written opinions of Poincare et seq. are also easy to find. Even today the vast majority of research funding for mathematics, at least in the United States, is dolled out for highly applied fields like partial differential equations. The field does not even close to unanimously (contemporarily or historically) "explicitly take pride" in working on problems that have no obvious application, or being a "jobs program for nerds": the notion of such "pure" or "nonapplied" mathematics is at the very least a highly fractious and controversial subject, with a number of big names taking opposing viewpoints (often vehemently).
I think your picture of the field is over-represented on the internet, much like the fixation on certain niche fields: Category Theory, Homotopy Type Theory or, worst of all, outright dubious fields like Geometric Algebra; fields with a large number of online promoters, but with much less funding and relevance in the actual academic space. Of course there are reputable people with PHDs that feel this way,—but I can only imagine that there's a legion of tyros, pop math consumers, and undergraduate students who disproportionately promote this viewpoint.
"The bulk of US college students attend colleges who do not have the resources to build high-quality, industry relevant curriculum, train teachers to teach with modern pedagogy, and efficiently manage dorms, student affairs, and other administrative infrastructure"
I would like to see a source on this: your claim appears ungrounded when considering American colleges.
Am I misunderstanding your post?: you're implying that HYPSM increase their matriculation by ten times? These "elite" colleges,—one of which I've attended for graduate school,—have serious issues already with becoming degree mills; degrees have depreciated enormously in value over the last several decades: consider the collapse in being able to find a tenure track research position, even from one of these colleges. If we wanted elite colleges to provide the benefits that they are supposed to; then we would, if anything, want to reduce matriculation.
Stanford,—and I would hazard a guess many other HYPSM schools,—are already minting out too many students; this is especially true when it comes to non-PHD masters degrees, which are essentially an unbecoming cash cow for departments. Actual "quality of education" mostly comes from a low staff/student ratio and direct access of students to elite researchers: this difference in education mostly takes the form of better research labs to work in, with some spillover into office hours; increasing matriculation would only lead to more auditorium-sized classes that are run by lecturers or postdocs—these classes are essentially at the same level as trudging through online material.
Your proposed "solution" would have a Procrustean effect: I can't speak for Chinese or Indian universities, but while schools like UC Berkeley, UT Austin, University of Michigan, et seq... have good reputations, they have a noticeably lower reputation than the ivy leagues and certain private colleges like Stanford, MIT, and Caltech—and a worse reputation for being degree mills.
If you think that Stanford having 180,000 students matriculated will give everyone a quality education, then I think that you fundamentally misunderstand the markers that make an in-person education higher quality. The only benefit that would come of it would be popping the degree bubble and prematurely ending the current moribund trajectory that universities are on; where they are already treating degrees as if they were artificial-scarcity NFTs, rather than providing the actual scarcity that is access to,—and direct training from,—high-level researchers.
Great article. I recently went through Crying of Lot 49 by Pynchon; the sequence of eccentric personalities in this article reminded me of a similar section that Pynchon has in the bay area. Unfortunately the personages interviewed here are not only real but climb beyond any fictional parody.
You would forget that this would cause exponential growth: in a couple decades, a single lab could produce more people seeking tenure track than an entire country's worth of positions; there need to be smarter ways to provide the requisite labor for science, since this is clearly unsustainable praxis. Running a pyramid scheme of this magnitude is only going to cause an implosion—which we may already be witnessing.
There's a big difference between cutting off all foreign-born talent—and addressing the serious issue of graduate school turning into an immigration racket; the current issue with graduate degrees is a very close mirror to the issue with H1b worker visas. The abuse of both systems has harmed Americans—and to some extent the long-term health of the tech industry and the academy.
I know absolutely nothing about string theory, or the culture of high-energy physics, but I don't buy the pecuniary argument you are making. You aren't considering the downwind effects of allowing academic rot. The Bourbaki—and their acolytes—also sponged up only a tiny amount of academic funding, but a fever in the pulpit can spread out into the pews; we've seen the "New Math" paradigm damage a generation of primary-and-secondary-school students. Even today, we have issues with engineers not understanding that a derivative is a slope and an integral is an area—due in no small part to a cartel of bad actors in mathematical research. Allowing bad behavior in high-value and influential positions has consequences beyond a waste of government expenditure; a president could turn a democracy into a banana republic, and we would have issues beyond his salary of a few hundred thousand dollars being wasted.
I was driving from Sacramento to Reno, and there was a bad snowfall up in the mountains. I ended up getting struck in a small country town, and couldn't get up an iced hill. I thought I was going to be stuck there the entire day (at least), but a stranger pulled up in their pickup truck—with four wheel drive—and towed me up the hill and to safety. That's the most significant act of kindness I can remember—from a complete stranger.
This is cool to look at, but isn't this just obtained by taking the absolute value of the first equation minus the second? These are very pretty visualizations—but trying to present them as some kind of "sea change" in perspective feels unhelpful.
I'd like to know about your experience in modern academe. I've gone to top-ranked schools in America (UT Austin, Stanford). My experience with the average foreign graduate student is not "top research talent." Most of the time you have a mid-level grifter that wants a green card, a work visa, or something else that simply lets them immigrate here. The work that they produce in exchange for that is low quality. The decline in the quality of graduate degrees may in many ways mirror the issues that tech workers have had with H-1Bs: they were intended to attract high quality talent, but became a corrupt racket.
Wonderful, now we're at a point where Hacker News has people defending a fundamentalist antisemitic hate group that's quite literally performed suicide bombings and launched rockets at civilians.
Maybe you haven't felt terror from them because you aren't one of the innocent Jewish civilians who have been deliberately targeted. Can you go back to reddit where your similarly deranged opinions are tolerated, and stay off of what is meant to be a relatively sane forum?
To add to what the other guy said, rotors (by extension Clifford Algebra) is better. A fatal issue with Quaternions is that while they handle rotations perfectly, things like the norm or cross product of two vectors is messy.
This is unsurprising when several of your coordinates become -1 when multiplied. That's why historically Gibbs Heaviside (dot and cross product) became the dominant vector algebra over quaternions.
Clifford Algebra is the better than both, as you can seamlessly do dot, cross (wedge in CA) and can also embed quaternions within the system. I've heard that it can also accommodate some of the nonmetrical aspects that make differential forms appealing for manifold integration, but that's currently outside of my range of knowledge.
> Even engineering undergrads have to learn formal logic in their first semester.
I got my engineering B.S. from a top 5 college in the US, and have known many people who have gone to similar schools. None of us have had to take a class that goes into first order logic or proof writing. I don't know what college you go to where that is a thing, but it would be exceedingly rare.
Yes, but I think this presents the core of the problem in modern pedagogical methods when it comes to mathematics. The Bourbaki attempted to reduce math to a highly axiomatic foundation, while disregarding the intuition and visualization that used to be a part of mathematics. The issue is that this sort of "code only" or "language only" approach really works when mathematics is a true "perfect language", the likes of which philosophers were attempting to construct, but is likely in fact impossible to create. Unfortunately, not only did the ideas of the Bourbaki fail, as modern research advances mostly still work with intuition instead of their ideas, but their approach polluted and ruined education. Many "textbooks" are terribly written reference books that have gaps and ambiguities that only people already knowledgeable in the field know about. Rudin's Analysis textbooks are probably the classical example of this. I would argue that any notation or abuse of notation is fine within insular fields and private practice, but there does need to be a leaning towards universal notation within all pedagogical works, at least up through all the core Algebra, Analysis, and Geometry and Topology work that you would see within a PHD qualifying exam.
Arnold's polemics are perhaps the most infamous and easily found online (see "On Teaching Mathematics"), but the written opinions of Poincare et seq. are also easy to find. Even today the vast majority of research funding for mathematics, at least in the United States, is dolled out for highly applied fields like partial differential equations. The field does not even close to unanimously (contemporarily or historically) "explicitly take pride" in working on problems that have no obvious application, or being a "jobs program for nerds": the notion of such "pure" or "nonapplied" mathematics is at the very least a highly fractious and controversial subject, with a number of big names taking opposing viewpoints (often vehemently).
I think your picture of the field is over-represented on the internet, much like the fixation on certain niche fields: Category Theory, Homotopy Type Theory or, worst of all, outright dubious fields like Geometric Algebra; fields with a large number of online promoters, but with much less funding and relevance in the actual academic space. Of course there are reputable people with PHDs that feel this way,—but I can only imagine that there's a legion of tyros, pop math consumers, and undergraduate students who disproportionately promote this viewpoint.