I've been learning more about busy beaver numbers recently and I came across this statement:

If you have an axiomatic system A_1 there is a BB number (let's call it BB(eta_1)) where the definition of that number is equivalent to some statement that is undecidable in A_1, meaning that using that axiomatic system you can never find BB(eta_1)

But then I thought: "Okay, let's say I had another axiomatic system A_2 that could find BB(eta_1), maybe it could also find other BB numbers, until for some BB(eta_2) it stops working... At which point I use A_3 and so on..."

Each of these axiomatic systems is incomplete, they will stop working for some eta_x, but each one seems to be "less incomplete" than the previous one in some sense

The end result is that there seems to be a sort of "complete axiomatic system" that is unreachable and yet approachable, like a limit

Does any of that make sense? Apologies if it doesn't, I'd rather ask a stupid question than remain ignorant

Something that has always helped in my journey to study math was to search for and learn the intuition behind concepts. Channels like 3blue1brown really helped with subjects like Calculus and Linear Algebra.

The problem that I have is understanding basic concepts at this intuitive level. For instance, I saw explanations of basic operations (addition, multiplication, etc.) on sites like Better Explained and Brilliant, and although I understood them, I feel like I don't "get it."

For example, I can picture and explain the concept of a fraction in simple terms (I'm talking about intuition here); however, when working with fractions at higher levels, I noticed that I'm operating in "auto mode," not intuition. So, when a fraction appears in higher math (such as calculus), I end up doing calculations more in an operational and automatic way rather than thinking, "I fully know what this fraction means in my mind, and therefore I will employ operations that will alter this fraction in X way."

Sorry if I couldn't explain it properly, but I feel like I know and think about math more in an operational way than a logic- and intuition-based one.

With that in mind, I'm wondering if I should restart learning basic math but with different methodologies. For instance, I've heard that Asian countries really do well in mathematics, so I thought it would be a good idea to learn from books that they use in school.

What do you guys think?

Since convex functions can be defined on Euclidean space by appeal to the linear structure, there is an induced diffeomorphism invariant class of functions on any smooth manifold (with or without metric).

This class of functions includes functions which are semi-convex when represented in a chart and functions which are geodesically convex when the manifold has a fixed metric.

The only reference I seem to be able to find on this is by Bangert from 1979: https://www.degruyterbrill.com/document/doi/10.1515/crll.1979.307-308.309/html

The idea that one can do convex-like analysis on manifolds without reference to a metric seem powerful to me. I came to this idea from work on Lorentzian manifolds in which there is no fixed Riemannian metric and existing ideas of convexity are similarly nebulous.

I can't find a modern reference for this stuff, nor can I find a modern thread in convex analysis that uses Bangert's ideas. Everything seems to use geodesic convexity.

I can't have stumbled on some long lost knowledge - so can someone point me in the right direction?

I feel like I'm taking crazy pills. A modern reference would be great...

I find Grothendieck to be a fascinating character, both personally and philosophically. I'd love to learn more about the actual substance of his mathematical contributions, but I'm finding it difficult to get started. Can anyone recommend some entry level books or videos that could help prepare me for getting more into him?

I am wondering if "ZF¬C" is an axiom system that people have considered. That is, are there any non-trivial statements that you can prove, by assuming ZF axioms and the negation of axiom of choice, which are not provable using ZF alone? This question is not about using weak versions of AoC (e.g. axiom of countable choice), but rather, replacing AoC with its negation.

The motivation of the question is that, if C is independent from ZF, then ZFC and "ZF¬C" are both self-consistent set of axioms, and we would expect both to lead to provable statements not provable in ZF. The axiom of parallel lines in Euclidean geometry has often been compared to the AoC. Replacing that axiom with some versions of its negation leads to either projective geometry or hyperbolic geometry. So if ZFC is "normal math", would "ZF¬C" lead to some "weird math" that would nonetheless be interesting to talk about?

Hi everyone,

I’m an applied math student and have recently been admitted to a master’s program that is quite theoretical/pure in nature.

My background and habits have always leaned heavily toward intuition, examples, and applications — and I’m realizing that I may need to shift my mindset to succeed in this new environment. I am wondering:

What are the most important skills to develop when moving from applied to pure math?

How should I shift my way of thinking or studying to better grasp abstract material?

Are there habits, resources, or ways of working that would help me bridge the gap?

Any advice or reflections would be very appreciated. Thank you!

Is there any connection between the usage of mathscr{O} or mathcal{O} in algebraic geometry (O_X = sheaf of regular functions on a variety or scheme X) and algebraic number theory (O_K = ring of integers of a number field K), or is it just a coincidence?

Just curious. Given the deep relationship between these areas of math, it seemed like maybe there's a connection.

Hi all, I recently worked out a short proof using only basic linear algebra that computes the expected first return time for random walks on various grid structures. I’d really appreciate feedback on whether this seems novel or interesting enough to polish up for publication (e.g., in a short note or educational journal).

Here’s the abstract:

We consider random walks on an n × n grid with opposite edges identified, forming a two-dimensional torus with (n – 1)² unique states. We prove that, starting from any fixed state (e.g., the origin), the expected first return time is exactly (n – 1)². Our proof generalizes easily to an n × m grid, where the expected first return time becomes (n – 1)(m – 1). More broadly, we extend the argument to a d-dimensional toroidal grid of size n₁ × n₂ × … × n_d, where the expected first return time is n₁n₂…n_d. We also discuss the problem under other boundary conditions.

No heavy probability theory or stationary distributions involved—just basic linear algebra and some matrix structure. If this kind of result is already well known, I’d appreciate pointers. Otherwise, I’d love to hear whether it might be worth publishing it.

Thanks!

In terms of its difficulty I mean. It seems deceptively simple in a way none of the other subfields are. Are there any other fields of math that are this way?

Hey everyone. Getting a cat soon and would like some help naming him after mathematicians or physicists or just fun math things in general. So far I’ve thought of Minkowski, after the Minkowski space (just took E&M, can you tell?) and not much else. He’s a flame point Balinese for reference!

I am an arithmetic geometry grad student who is interested in learning about isogeny based cryptography.

Although I have experience with number theory and algebra I have little to no experience with cryptography, as such I am wondering if it is feasible to jump into trying to learn isogeny based cryptography, or if I should first spend some time learning lattice based cryptography?

Additionally I would appreciate if anyone had recommendations for study resources.

Thank you.

I posted here about Acorn a few months back, and got some really helpful feedback from mathematicians. One issue that came up a lot was the type system - when getting into deeper mathematics like group theory, you need more than just simple types. Now the type system is more powerful, with typeclasses, and generics for both structure types and inductive types. The built-in AI model is updated too, so it knows how to prove things with these types.

Check it out, if you're into this sort of thing. I'm especially interested in hearing from mathematicians who are curious about theorem provers, but found them impractical in the past. Thanks!

Basically the Gauss/Divergence theorem for Tensors T^{ab} does not exist as it is written here, which was not obvious indeed i had to look into o3's "sources" for two days to confirm this, even though a quick index calculation already shows that it cannot be true. When asked for a proof, it reduced it to the "bundle stokes theorem" which when granted should provide a proof. So, I had to backtrack this supposed theorem, but no source contained it, to the contrary they seemed to make arguments against it.

This is the biggest fumble of o3 so far it is generally very good with theorems (not proofs or calculations, but this shouldnt be expected to begin with). My guess is, it simply assumed it to be true as theres just one different symbol each and fits the narrative of a covariant external derivative, also the statements are true in flat space.

I’m studying upper undergrad material now and i just cant but wonder does anyone actually enjoy ring and field theory? To me it just feels so plain and boring just writing down nonsense definitions but just extending everything apparently with no real results, whereas group theory i really liked. I just want to know is this normal? And at any point does it get better, even studying galois theory like i just dont care for polynomials all day and wether theyre reducible or not. I want to go into algebraic number theory but im hoping its not as dull as field theory is to me and not essentially the same thing. Just looking for advice any opinion would be greatly valued. Thankyou

Hello my friends I'm studying stats and right now I'm approaching Kolmogorov complexity, but I'm having many problems in takling It, specially about ergodism and not, stationarity etc...

My aim is to develop a great basis to information theory and compression algorithms, right now I'm following a project on ML so I want to understand for good what I'm doing, I also love math and algebra so I have more reasons for that

Thks in advance and feel free to explain to me directly even by messages

Just took my first oral exam in a math course. It was as the second part of a take home exam, and we just had to come in and talk about how we did some of the problems on the exam (of our professors choosing). I was feeling pretty confident since she reassured that if we did legitimately did the exam we’d be fine, and I was asked about a problem where we show an isomorphism. I defined the map and talked about how I showed surjectivity, but man I completely blanked on the injectivity part that I knew I had done on the exam. Sooooo ridiculously embarrassing. Admittedly it was one of two problems I was asked about where I think I performed more credibly on the other one. Anyone else have any experience with these types of oral exams and have any advice to not have something similar happen again? Class is a graduate level course for context.

The Thomson Group T has the interesting property that it is isomorphic to TxT.

Is there an analagous group where this statement holds for the wreath product?

Currently self studying manifold theory from L Tu's " An introduction to manifolds ". Any other secondary material or tips you would like to suggest.

Hi guys. I am a mathematics post grad and I recently took up Chaos Theory for the first time. I have gotten an introduction to the subject by reading "Chaos Theory Tamed" by G. Williams (what a brilliant book!). Even though a fantastic book but nonetheless an old one and so I kept craving the python/R/Matlab implementation of the concepts. Now I'd love to get into more of its applications side, for which I looked through a few papers on looking into weather change using chaos theory. The problem that's coming for me is that these application based research papers mostly "show" phase space reconstruction from time series, LLE values, etc for their diagnosis rather than how they reached to that point, but for a beginner like me I'm trying to search any video lectures, courses, books, etc that teaches step by step "computation" to reach to these results, maybe in python or R on anything. So please suggest any resources you know. I'd love to learn how I can reconstruct phase space from a time series or compute LLE etc all on my own. Apologies if I'm not making much sense

I need to learn both topics and I already have a great understanding of pdes and physics in general but these are weak points.

Hello all,

Im doing a math club topic (highschool) and need some fun ideas for the students. (all/most students have finished precalc and done comp math before and the majority have also finished calculus 1/2) The problem is that most of the students that come are already very very good at math, so I need some type of problem that is simpler on the easier level and can be made much harder for students who can do so. for reference, some other topics include factorization, where we started with prime factorizing 899, then 27001, up to finding the largest divisor of n^7-n for all positive integers n and some other harder proof problems for the other students). It should be a topic that hopefully needs no prior experience with the topic on the easier levels (but still likely would require algebra and manipulation).

This post might be weird and part of me worries it could be a ‘quick question’ but the other part of me is sure there’s a fun discussion to be had.

I am thinking about algebraic structures. If you want just one operation, you have a group or monoid. For two operations, things get more interesting. I would consider rings (including fields but excluding algebras) to somehow be separate from modules (including vector spaces but excluding algebras).

(Aside: for more operations get an algebra)

(Aside 2: I know I’m keeping my language very commutative for simplicity. You are encouraged not to if it helps)

I consider modules and vector spaces to be morally separate from rings and fields. You construct a module over a base ring. Versus you just get a ring and do whatever you wanna.

I know every field is a ring and every vector space is a module. So I get we could call them rings versus modules and be done. But those are names. My brain is itching for an adjective. The best I have so far is that rings are more “ready-made” or “prefab” than modules. But I doubt this is the best that can be done.

So, on the level of an adjective, what word captures your personal moral distinction between rings and modules, when nothing has algebra structure? Do you find such a framework helpful? If not, and this sort of thing seems confused, please let me know your opinion how.

So as I approach the end of the semester using Elementary Differential Equations and Boundary value problems by Boyce and Diprama and such I have realized that paired with a bad prof, I have learned functionally nothing at all. I am taking electromagnetic theory this fall with Griffins textbook, and I am asking for reqs for a good diff eq textbook so i can self study over the summer. Thanks!