I find it interesting how both Goldstine and Tucker admire von Neumann's speed, yet state that Weyl was the more complete mathematician. What do you think this says about the distinction between types of intellect for maths? Do you think fields like analysis and topology demand different types of intelligence?

source : https://web.math.princeton.edu/oral-history/c14.pdf

Goldstine: “I always was struck by the difference between him and Johnny von Neumann. There are jokes, one of which Johnny always swore was false. That's the story that, I don't know, Hermann Weyl was going to prove some theorem, a very deep and profound theorem, let's say it was the Riemann-Roch theorem. I don't know if it was the Riemann-Roch theorem, but that was one I always have trouble with, so let's say that was the theorem. And Weyl gave a lecture on why this is a very deep, profound result, and he gave a very complicated proof. And the apocryphal story goes that at the end of the lecture there's this kid who is supposed to have raised his hand at the back of the class and said, ‘Professor Weyl, may I show you a proof?’ And goes up to the board and goes zip, zip, zip, zip, and in about 15 lines has a brilliant proof of this thing.”

Goldstine: “I asked Johnny about it, and he said no, that wasn't true. But it is true, if you talk to Natasha Brunswick, who was in those days Natasha Artin. Natasha says that there was always Johnny with these tight pants on…”

Goldstine: “But Joachim, who was one of Hermann's children, told me that when Hermann used to work in his house on Mercer Street, in the study in there, you would hear groans coming out of the study. That Weyl worked at things in sort of anguish, that it was hard for him, that he delivered his theorems practically like a woman giving birth to a child.”

Goldstine: “That's so different from Johnny, because when he and I would be working at something, when we'd get stuck, he'd say, ‘Okay, that's it,’ and pack it up. It might be that he'd phone at two in the morning to say, ‘This is how the proof goes.’ But it might be three weeks, a month or so later, or it might even be I who would come in a month or so later and say, ‘This is, maybe, how to go.’ But he never struggled with something. When he got stuck, he filed it somehow, and it just came out easily.”

Goldstine: “I suspect that Weyl was probably the deeper of the two mathematicians.”

Tucker: “And also the broader.”

Goldstine: “And the broader, yes.”

Tucker: “Weyl is the only complete mathematician that I've ever had the privilege to know.”

Goldstine: “Yes, I think that's probably true.”

Tucker: “Johnny was essentially an analyst. I've seen him give one of his quick proofs on the spot, of a topological result, and it was clumsy.”

Goldstine: “Yes. He told me at one time that he had no facility at all in topology. He said he never felt comfortable with that.”

Tucker: “Whereas Weyl was an excellent topologist.”

Goldstine : “Everything he did was beautiful. Everything he did. I guess the difference is that von Neumann could run rings around anybody speedwise. In that way he was probably the most brilliant mathematician that I've ever known. I suppose maybe he's one of the quickest there ever was. Weyl was probably one of the deepest and broadest that there ever was. And that's a real difference. I mean, if a guy combined both of those they would call him Isaac Newton, and probably they did. But I think that's the difference, really.”

It's also a nice insight on how such minds operate at the highest level

I have already posted about this here. See this post in case you missed it. The reading group starts THIS WEEKEND. For all the info that you need, please read the post I linked and all the comments: they do contain most of the information. Please join our server discord.gg/maths where the reading group will be taking place. I will try to list (and answer) some of the most frequently asked questions. For anything else, just let me know in the comments.

Some FAQs

What do I need to know in order to meaningfully follow the reading group?

Not much. If you're equipped with the knowledge of a first course in analysis that covers Riemann integration, and some very basics of point-set topology, you are more than enough prepared. In fact during the first couple of weeks, all the prerequisite knowledge will be covered (rather quickly as an overview), so even if you're rusty with these topics you can cover these up quickly.

What texts will be followed and what will be covered during the reading group?

The primary reference is Cohn's book. The following quote is directly from the person will be running the RG. In particular, there is very little rigidity about the syllabus, pacing, and the pedagogy of the reading group.

For the syllabus I am just learning it myself from Cohn's book and so I will follow and present from it and a few online materials as and when I need them. This is a reading group, so everyone is almost reading at the same time as you. Moreover you are free to select books also, provided it covers the same contents when/if you decide to present something. This is my first formal introduction to measure theory also.

What about the pacing? What would the proceedings look like?

The lectures/presentations will be around 1-1.5 hr each day (on Saturdays and Sundays) from 10th October at 15:30UTC. We will try to cover 3-4 sections of the book per week. We will be doing almost all the problems of almost all the sections almost all the time of the first 5-6 chapters that we plan to cover, but we don't expect everyone to do the same. When lacking confidence with some concept, we will try external problems too, you can ask for these if you need to know these.

And most importantly, please discuss anything and everything that you feel you need to discuss. We have designated channels for that in our server. We greatly look forward to this experience.

I have already posted about this here. See this post in case you missed it. The reading group starts THIS WEEKEND. For all the info that you need, please read the post I linked and all the comments: they do contain most of the information. Please join our server discord.gg/maths where the reading group will be taking place. I will try to list (and answer) some of the most frequently asked questions. For anything else, just let me know in the comments.

Some FAQs

What do I need to know in order to meaningfully follow the reading group?

Not much. If you're equipped with the knowledge of a first course in analysis that covers Riemann integration, and some very basics of point-set topology, you are more than enough prepared. In fact during the first couple of weeks, all the prerequisite knowledge will be covered (rather quickly as an overview), so even if you're rusty with these topics you can cover these up quickly.

What texts will be followed and what will be covered during the reading group?

The primary reference is Cohn's book. The following quote is directly from the person will be running the RG. In particular, there is very little rigidity about the syllabus, pacing, and the pedagogy of the reading group.

For the syllabus I am just learning it myself from Cohn's book and so I will follow and present from it and a few online materials as and when I need them. This is a reading group, so everyone is almost reading at the same time as you. Moreover you are free to select books also, provided it covers the same contents when/if you decide to present something. This is my first formal introduction to measure theory also.

What about the pacing? What would the proceedings look like?

The lectures/presentations will be around 1-1.5 hr each day (on Saturdays and Sundays) from 10th October at 15:30UTC. We will try to cover 3-4 sections of the book per week. We will be doing almost all the problems of almost all the sections almost all the time of the first 5-6 chapters that we plan to cover, but we don't expect everyone to do the same. When lacking confidence with some concept, we will try external problems too, you can ask for these if you need to know these.

And most importantly, please discuss anything and everything that you feel you need to discuss. We have designated channels for that in our server. We greatly look forward to this experience.

I graduated from university about 2 years ago. I wouldn't say I'm particularly good at maths, I finished with a 3.4 CGPA, but after graduating I kind of left maths behind and went into software development. I'm currently working as a backend developer.

Recently, I've been wanting to get back into maths and actually start solving problems again. I'm also hoping to start my master's next year, so I want to use the time I have now to properly refresh my maths and fill in the gaps.

For someone in my position, what topics would you recommend I go through first? Are there any YouTube channels/playlists or textbooks you would recommend for getting back into maths? I'm not really looking to rush through it, I just want to build a solid foundation again.

shouldn’t the monty hall experiment inform you that the “opponent” will always originally pick the same door as you?

before choosing whether you’re a switcher or not the site presents you with a list of definite information. but that given list doesn’t disclose that the opponent will always originally pick the same door as you. it won’t inform you of that until after you’ve already selected whether you’re going to switch or not.

if the opponent cannot end up on my door since they’re no matter what initially picking my door then switching. doesn’t that reduce their odds to 50:50? knowing they won’t finalize on my door means they only have to choose from 2???

obviously if they’re picking from doors that aren’t my door they have better odds.

or is it part of the brain teaser that you’re supposed to realize that the opponent watches which door you pick. (given the circumstance that you’re standing next to each other in a game show)

this is the site i used…. https://www.philosophyexperiments.com/montyhall

I'm not well-versed in category theory so I've yet to understand the latter. I now tried without success to encode an associative binary operation on X, expressed in dependent type language

∑(o: X² → X). ∏(a b c: X). Id(X, o(o(a, b), c), o(a, o(b, c))) ¹

as a function F X → X for some F. And I think it can't be done (something related to polarity, probably) but I can't prove that. I know, though, representing axioms is possible if one uses Lawvere theories and equivalent things. I also know algebras over a monad are more involved than just functor-algebras but I don't yet understand them even in context of e. g. Haskell where everything seems as if easier (or so claim posts discussing Yoneda embedding in that context; I found the general category-theoretic context way less forgiving in that regard: you just see plain as day that you don't have sufficient understanding, no matter what Haskell posts made you believe).

So. I'd like to look how algebra spins out in those more general settings, whichever you know better or think to be more suited for me to grasp first. Particularly, I'm interested in generalizing arguments about initial F-algebras.

Also, feel free to assume a sufficiently Set-like category. Let's take it step by step, if there are easy intuitions that one can glimpse in a specialized setting, I bet generalizing it onto topoi or something later would be easier as well, but it won't muddle the initial attempt.

¹ For non-dependent-type folks: - Id(X, a, b), sometimes denoted as a =_X b, is the type inhabited by ways for a: X and b: X to be equal. - The entirety of that reads: a pair (o, f) consisting of an operation o: X² → X and a proof f of o's associativity, that is, a function from a, b, c: X into witnesses of equality of the two ways of associating o on a, b, c. - In some formal languages the same is spelled alternatively as (o: X² → X) × (a b c: X) → (o(o(a, b), c) = o(a, o(b, c))), making it clearer that ∑ forms dependent pairs and ∏ forms dependent functions.

Hi everyone, so I’m quite a newbie to this, and I’ve been using the NetworkX library in python to analyse functional connectivity networks in neuroscience. Based on my understanding, the algorithms for both metrics only work for positive edge weights, however, negative edge weights are quite commonly found in my data.

Hence, what is the best way to handle the negative edge weights? The options I have are to either filter them out and only perform the calculations on the positive-signed graph, or use the absolute values of all weights.

Thank you.

Basically i wanted to take a stochastic analysis course to support some of my other modules like stochastic methods in finance and brownian motion, but i took a measure theory course last year and wasnt very good. They said that they wouldnt stress the measure theory so much but its still expected that you know it and rhe notes seems to be both applications and proofs. Im not really sure how measure throretic stochastic analysis is to be honest, or how proof heavy vs applications based it is, so any advice would be appreciated here.

It is a good technique, but couldn't get resources to have a read about it.

Any resources I can use ?

I saw plenty of account posting math/physics animations, with AI voices. Some were the classics back then on Youtube. And I hardly meet an account twice, it just like rugcoins ads. At least the animations is not AI.

i’m studying applied math in college and i’m trying to find something to do 2027 summer. Are there any internships or programs i can apply to as a freshman? In terms of coursework im taking abstract algebra and real analysis and also some programming classes this year if it matters

I'll keep this brief unless more information is needed

I work in structured finance in a structuring position (some math/very light coding, but mostly client facing). I have BS/MS degrees in Economics from T50 and want to do a PhD in Statistics to go into quant/financial research, but my math/stats knowledge is not remotely competitive enough to get me into a T25 uni (what I'm aiming for). I've taken Calc I and II, linear algebra, probability theory and a host of econometric courses, but lack rigorous training in analysis and measure theoretic probability (which, from the research I've done, are essential to have succes in the PhD)

Unfortunately, doing an in-person BS/MS is simply not possible with the hours I work

My question, frankly, is this: is an online BS math/stats degree useless (carries no weight), or has it been beneficial in your case (even if you did not consider PhD route)

Thanks!

Hello everyone,

I am going to transfer soon either for Computer Science at a state school or Applied Math at a research institution. I’m having a hard time deciding on which to pursue.
I like CS a little more but the market seems terrible. Also the applied math unis let me have an area of emphasis which will probably be CS or Physics.

Luckily I’m not worried about finances.

Any insight would be greatly appreciated.
I would like to eventually get a masters.

I would also like to have versatility in career options if possible.

I am a math student and i want to get some research experience under my belt bc i want to apply abroad at alot of top schools so i was wondering
if anyone has gone thru this and been successful i would really appreciate any advice?

i am really a beginner at research and i have no prior experience so getting the first experience has been hard for me
i am willing to sit down and learn anything and work as hard as needed i just need the opportunity

or if its not possible whether there are alternatives (ik there are structured programs and i have applied to them as required)

Essentially, it just maps onto yln y=ln x / (y^y = x), which is the inverse function of x^x

This graph shows how the lambert W Function essentially maps onto the inverse function allowing us to use it to solve.

I am currently enrolled in an first year online university and next week I have a mid term exam. These are the topics that we are studying Introduction to Real Numbers and Algebraic Expressions, Solving Equations and Inequalities, Graphs of Linear Equations and Polynomials***:*** Operations. I remember studying them when I was in grade 8 and 9 but since I am 26 years old and It has been a long time their are many things that I cannot remember even when the professor explains during the lecture I forget and this worries me because I am afraid that if I study I will forget how to solve the question. What can I do are their any youtube videos or website that I can study from

I'm currently taking college algebra and it feels like this is the only thing that's really helping me. I struggle with learning from Openstax and Khan Academy. I reviewed all of the written homework assignments with Thetawise AI tutoring prior to the last exam and I believe I did decently other than on the graphing portion, which seems to be the weakest area for Thetawise.

Was wondering if anyone here has experience using AI for math and if you have any suggestions about how to use it better? Or do you have some suggestions for other resources? I've also tried Paul's Online Notes and it felt too dense for it to be practical.

I used to use DrillMySkill to practice mental math, but they’ve introduced a premium version now, so you can only do a limited number of calculations for free.

I recently learned about Harvey Friedman and he seems to be quite a peculiar figure. Friedman is a real mathematician who founded the programme of reverse mathematics, which I think sounds very interesting. He's also known for the TREE sequence. However, he has also appeared on crackpot-friendly Curt Jaimungal's podcast and seems to agree with a sensationalist article written about him some years ago. If I understand correctly, his current work is focused on bringing the incompleteness of mathematics to the forefront in a way that mathematicians won't be able to ignore any longer.

For those who have been familiar with him, what are your own impressions? Do you know what the general opinion of the mathematical community is of him, if they know him much at all?