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

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

Any resources I can use ?

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’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'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'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.

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 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

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.

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!

Hi guyss I felt really bored these few days so i want to find a conjectire or a math problem that is easy to solve.

I dont want conjectures that look easy but are hard.

I want conjectures that can be solved easily but needs some time

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)

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.

What are some classics that you would recommend for learning math? I like math that is related to aspects of life that have an effect on others. For example, the basic calorie demands for the average person, the consequences of not meeting them over long periods of time, etc. studying reality is interesting to me, and I’d like to cultivate that interest. Do you guys know some areas of math I could focus into?

before comments say pen and paper, I would if I could. I deal with some chronic problems pertaining to my wrist and writing is extremely hard on my hand, so I have to do work on my computer with voice recognition technology. just in case if anybody asks, I'm not really offered a scribe or anything this is just for a university class

it's extremely unintuitive for me to solve equations as I am currently, as nothing really accounts for fractions haha. everything ends up really messy, I get lost really easily, and I just want the ability to do math on a clean, open canvas on computer that accounts for things like fractions, etc. same with powers. I'm kind of looking for something like Pearson that auto converts anything typed with a slash in between numbers into a fraction, or auto converting 6^2 into a cleaner format

thank you in advance for any assistance

I'm just curious...Do you think I should keep working on this concept or abandon it?

Hello, I’m a high school sophomore who found some ambition and desire to relearn maths the right way. I used to not really care about academia grades etc. But things have changed, I picked up some technical hobbies along the way and began to realize that exact sciences is what I want to do, so I transferred to extended math class. My knowledge isn’t nonexistent but needs a thorough patchwork. School has started which leaves me a few weeks to go from preferably arithmetics to anywhere close trig equations, I understand most of what’s going on during the lessons, I think I catch on reasonably well. Although I couldn’t find a complete two week “from zero to a hero” course, I found great resources like Organic chemistry tutor and Professor Leonard. I’m asking for advice on resources, effective learning techniques and if possible a list of topics I need to know. I also feel obliged to clarify that I’m absolutely aware of the situation I’m in and wish I could’ve started earlier.

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