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.
How many primary school students can't add fractions because string theory may be a less promising approach to a ToE versus loop quantum gravity or geometric unity? I know nothing about this stuff. You know nothing about this stuff. Since we both do know about the Bourbaki school of mathematics despite having different opinions on the value of building mathematics upward from foundational principles I'd say we are in the top .5% of the planet re general mathematical/scientific literacy. So I don't buy that even if string theory is wrong there is some massive spillover effect.
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.
>I don't know what college you go to where that is a thing, but it would be exceedingly rare.
German technical university. I was a tutor there. AFAIK this is normal.
As part of the first semester you take linear algebra and analysis, starting out with the basics of formal logic. Of course the courses are less focused on proof writing than the mathematics "major" courses.
I should also point out that German universities have very loose entry standards (except when places are very rare compared to applicants) and use the first two semesters to filter out students. These courses are often designed to have around a 50% failure rate.
> As part of the first semester you take linear algebra and analysis, starting out with the basics of formal logic
There is not enough time to learn the basics of formal logic and linear algebra and/or analysis in a single class, but I think what you’re referring to is an introduction to proof techniques like induction, modus tollens, quantifiers, etc.
Every math and computer science department in the US that I’ve ever heard of teaches these topics, but I wouldn’t call it a formal logic class.
For me basic formal logics means learning the symbols (conjunction, disjunction, implication, equivalency, not, etc.) and the rules of inference to maniuplate these symbols and using these rules to prove new things.
How can you teach analysis without that anyway. It is absolutely essential for set theory and how would you e.g. define the reals (in a "proper" math course, not engineering) without a good understanding of set theory?
> For me basic formal logics means learning the symbols (conjunction, disjunction, implication, equivalency, not, etc.) and the rules of inference to maniuplate these symbols and using these rules to prove new things.
That's really only baby logic. Which, probably, is what will be sufficient for most mathematician most of the time.
A real first introduction to formal logic would introduce an actual formal proof system and go at least as far as proving completeness of first order logic.
I don't want to quibble about names (and I wasn't the person to come up with the term "baby logic"), but the point is that just introducing a couple of connectives and proof strategies doesn't even constitute the basics of what mathematical logic really is about. Which btw I'm perfectly fine with, most people don't need more than that.
If you do want to study logic formally, the basics start with well-formed formulas, signatures, etc.
I guess what you call it doesn't matter a lot, but the discussion seems to have started with the assertion that most students, even in mathematics, never really learn formal logic, and I would agree with that (under my definition of "formal logic"), while also agreeing with you that you can't pursue a degree in maths without knowing how induction works or what a bijection is. But still, most people don't need to know exactly how to formalise induction and that it's actually (in its full form) a second-order axiom.
That level of logic taught to every mathematics and computer science student, and it’s really not what I was thought others in this thread were talking about.
Not really. First year courses usually have hundreds of students in large halls. There is some more effort as you need more tutors, but that is basically it. (Students usually do not live on campus)
The enormous upside is that all students are judged equally on their ability to academically succeed in their chosen field.
I think US university admissions are ridicolous for many reasons.
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.
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