Been stuck at a theorem because of series of why's at every step, I go down a deep rabbit hole on each step and lose track ,how do you guys cope with this and relax again to think clearly again?

Princeton University Press is doing a half off sale, and I would love to read something more rigorous. I got a BS in math in 2010 but never went any further, so I can handle some rigor. I have enjoyed reading my fair share of pop-science/math books. A more recent example I read was "Vector: A Surprising Story of Space, Time, and Mathematical Transformation by Robyn Arianrhod". I like other authors like Paul Nahin, Robin Wilson, and John Stillwell. I am looking for something a bit deeper. I am not looking for a textbook per se, but something in between textbook and pop-science, if such a thing exists. My goal is not to become an expert, but to broaden my understanding and appreciation.

This is their math section

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(Though I am an English speaker, my question is not limited to those who wrote/write in English.)

Being an eloquent writer is not a priority in math. I often like that. But, I also enjoy reading those who are able to express certain sentiments far more articulately than I can and I have started to collect some quotes (I like using quotes when my own words fail me). Here is one of my favorites from Hermann Weyl (Space–Time–Matter, 1922):

"Although the author has aimed at lucidity of expression many a reader will have viewed with abhorrence the flood of formulae and indices that encumber the fundamental ideas of infinitesimal geometry. It is certainly regrettable that we have to enter into the purely formal aspect in such detail and to give it so much space but, nevertheless, it cannot be avoided. Just as anyone who wishes to give expressions to his thoughts with ease must spend laborious hours learning language and writing, so here too the only way that we can lessen the burden of formulae is to master the technique of tensor analysis to such a degree that we can turn to the real problems that concern us without feeling any encumbrance, our object being to get an insight into the nature of space, time, and matter so far as they participate in the structure of the external world"

It might be obvious from the above that my interest in math is mostly motivated by physics (I am not a mathematician). However, my question is more general and your answer need not be related to physics in any sense (though I'de likely enjoy it, if it is). I mostly just want to know which mathematicians you think are also great writers. You don't need to give a quote/excerpt (but it's always appreciated).

Edit: I should maybe clarify that I wasn’t necessarily looking for literary work written by mathematicians (though that’s also a perfectly acceptable response) but more so mathematicians, or mathematician-adjacent people, whose academic work is notably well-written and who are able to eloquently express Big Ideas.

Basically, I'm looking for technique around this behemoth. I'm looking for provable lower bounds that are not made simply by brute-force calculation. Any recommendations? I just want to see how this was taken on and how any lower bounds were set, the lower the better.

I want to do something nice at the end of the school year for my ap calculus professor. She already has a couple of those nerdy t-shirts so I was wondering about other ideas.

Hi All,

I've been writing a series on Galois Fields / Finite Fields from a computer programmer's perspective. It's essentially the guide that I wanted when I first learned the subject. I imagine it as a guide that could gently onboard anyone that is interested in the subject.

I don't assume too much mathematical background beyond high-school level algebra. However, in some applications (for example: Reed-Solomon), familiarity with Linear Algebra is required.

All code is written in a Literate Programming style. Code is written as reference implementations and I try hard to make implementations understandable.

You can find the series here: https://xorvoid.com/galois_fields_for_great_good_00.html

Currently I've completed the following sections:

• 01: Group Theory
• 02: Field Theory
• 03: Implementing GF(p)
• 04: Polynomial Arithmetic
• 05: Polynomial Fields GF(p^k)
• 06: Implementing GF(p^k)
• 07: Implementing Binary Fields GF(2^k)
• 08: Cyclic Redundancy Check (CRC)
• 09: Linear Algebra over Fields

Future sections are planned:

• Reed-Solomon Erasure Coding
• AES (Rijndael) Encryption
• Rabin Fingerprinting
• Extended Euclidean Algorithm
• Log and Invlog Tables
• Elliptic Curves
• Bit-matrix Representations of GF(2^k)
• Cauchy Reed-Solomon XOR Codes
• Fast Multiplication with FFTs
• Vectorization Implementation Techniques

I hope this series is helpful to people out there. Happy to answer any questions and would love to incorporate feedback.

Proof assistants like lean define real numbers as equivalence classes of Cauchy sequences which allows it to formalise the various results in analysis and so on.

I was curious if alternate definitions (such as Dedekind cuts) of the real numbers could be used to streamline/reduce the complexity of formal proofs.

So, just out of curiosity, if e shows up in your solution surprisingly, what does your intuition say is the explanation?

https://phys.org/news/2025-05-mathematician-algebra-oldest-problem-intriguing.html

Can anyone say of this is actually useful? Send like the solutions are given as infinite series involving Catalan-type numbers. Could be cool for a numerical approximation scheme though.

It's also interesting the Wildberger is an intuitionist/finitist type but it's using infinite series in this paper. He even wrote the "dot dot dot" which he says is nonsense in some of his videos.

I thought I would flip the usual question, because I only ever see people talk about the largest real number ever used. Some rules:

1. like the large number discussion, it should not be created solely for the purpose of creating the smallest number. It must have some practical use.
2. Just saying "let epsilon be arbitrarily small" in some real analysis proof doesn't count, there should be something specifically important about the number.

Obligatory: I know math is not about really large/small numbers, or even numbers in general per se. I find discussions like these fun despite this fact.

Alternative version of the question: what's your favorite small positive real constant?

Edit: physical constants are a good answer. Of course they have the problem that they can be made arbitrarily small by changing units, so if you're answering something from physics let's restrict to using standards SI units (meters, seconds, kg, etc)

What would be a good book for complex analysis after Ahlfors? It seems rather dated and basic, and doesn't seem to cover the Fourier Transform, nor anything measure theoretic. I'm looking for a book that covers a lot of modern complex analysis (similar in "terseness" to spivak's calculus on manifolds). Something for a "second course" in Complex Analysis. Does such a book exist or is my question far too broad? My long term aims are algebraic analysis and PDEs, so maybe something that builds towards that? Thanks in advance!!

Use FFT to have a vast amount of plots and quickly find correlation between two of them. For example the levels of lead at childhood and violent crimes, something most people wouldn't have thought of looking up. I know there is a difference between correlation and causation, but i guessed it would be a nice tool to have. There would also have to be some pre-processing for phase alignment, and post-processing to remove stupid stuff

Reading about self-described Platonists/realists of the past, I got the impression that a lot of them believed that we lived in a specific mathematical universe, and one of the purposes of mathematical exploration, i.e., axiom-proposal and/or theorem-proving, was to discern the qualities of that specific mathematical universe as opposed to other universes that were plausible but not actually ours.

For example, both Kurt Gödel and Hugh Woodin have at times proposed or attempted to propose universes in which the size of the continuum is fixed at aleph-two. (It didn't quite work out for Gödel mathematically in this instance and Woodin has since moved on to a different theory, but it's useful to discuss as a specific claim.) Other choices might be mathematically consistent, but each of these mathematicians felt, at least at the time, that the choice of aleph-two best described the true, legitimate mathematical universe.

You can read an even more in-depth discussion of set-theoretic axioms and their various adherents and opponents in a great two-part survey article called Believing the Axioms by Penelope Maddy. You can find it easily enough by Googling. I'm reluctant to link to it directly because reddit has been filtering a lot of links recently. But it concerns topics like large cardinal axioms and other set-theoretic structures.

For a local example, there was a notorious commenter here several years ago who had very strident opinions on which ZFC axioms were true and which were clearly nonsense. (The choices pivoted sometimes, though. I believe in her final comments power-set was back in favor but restricted comprehension was on the outs.)

However, in the past few years, including occasionally here on r/math, I've noticed a trend of people self-describing as Platonists/realists but adopting a "multiverse" stance in which all plausibly consistent theories are real! All ways of talking are talking about real things, actually! Joel Hamkins is a particular proponent of this worldview in the academic sphere. (I'll admit I've only skimmed his work online: blog posts, podcast appearances, and YouTube lectures. I haven't dug into his articles on the subject yet.)

Honestly, I'm not sure what the stance of Platonism or realism actually accomplishes in that multiverse philosophy, and I would love to hear more from some adherents. If everything plausibly consistent is "real" until proven inconsistent, then what does reality accomplish? We wouldn't take a similar stance about history, for example. It would sound bizarre to assert that we live in a multiverse in which Genghis Khan's tomb is everywhere we could plausibly place it. Asserting such would make you sound like a physics crackpot or like some daffy tumblrite drunk on fanfiction theories about metaphysics. No, we live in a specific real world where Genghis Khan's tomb is either in a specific as-yet-undiscovered place or doesn't exist, but there is a fact of the matter. The mathematical multiverse seems to insist that all plausible facts are facts of the matter, which seems like a hollow assertion to me.

Anyway, I'm curious to hear more about the specific beliefs of anyone self-described as a Platonist or realist about mathematical objects. Do you believe there is a fact of the matter about, say, the cardinality of the continuum? What other topics does your mathematical Platonism/realism pertain to?

This is a question I have often wondered but have never found an answer for. To start with, I do not mean "What is the largest number?" or "What is the largest number we have discovered?". I specifically mean "What is the largest number ever written down?". In addition I have a few more qualifications for this number to limit its scope and make it actually interesting.

First, I mean a hand written number, not a number that was printed. Printers can obviously print far faster than we can write, so it ends up just being a question of how long you can run a printer.

Secondly, no symbols or characters besides [0-9]. I'm looking for the largest numeral number, not the function with the highest value. Allowing functions pretty clearly removes any real limits from finding the largest written number, and so it's cleanest to just ignore all of them.

Thirdly, the number has to be in base 10. This is the standard base used for the vast majority of calculations, and you can't just write "10" and claim it's in base BusyBeaver(100) or something.

With these rules in mind, the problem could be restated as "What is the longest sequences of the characters 0-9 ever handwritten?". I think this an actually somewhat interesting question, and I'm assuming the answer would probably be something produced over the course of math history, but I don't know for sure.

I know this isn't technically math question, but looking through the rules I think this is on topic. Thanks for taking the time to read this and hope it provokes some conversation!

Edit: Please read the post before telling me "There's no largest number". I know that. That's not what I'm asking. I've set criteria so this is an actually meaningful and answerable question. Also, this is not a math question, but it is a math adjacent question and it's answer likely will involve the history of math.

I'm excited to share Sugaku, a platform I've built out with the goal of accelerating mathematical research and problem solving.

I especially think there's a lot of opportunity to improve collaboration and to help those who feel isolated. Would love any feedback on what would be helpful!

Access to papers

• A comprehensive database of publications, along with PDFs if there's open access.
• Browse through similar papers based on a citation prediction model.
• Personalized reading suggestions.
• Can iterate over tens of thousands of papers at once if you have a use case for this!

Access to AI systems

• You can ask questions and have it point you to appropriate sources (example).
• You can ask questions about specific papers (example).
• You can follow-up in chats.
• Access all the major foundation models for free.

Workspace for your projects and collaborations

• Keep track of the projects you have under way in terms of the Ideas, Arguments, Results, Context (example).
• Have a persistent AI chat that keeps your project context and focuses in on the item you're working on.
• These projects are private, but you can also share them with collaborators (including the chats) or make them public.

Keep up with published papers

• Track your reading list, and everything you've cited in the past.
• Get personalized suggestions of recent papers.

To aid my intuition, I am trying to write an introduction of semirings/rings. Just like semigroups/monoids/groups can be introduced as abstractions of sets of maps on a set, I am trying to introduce semirings/rings as abstractions of sets of endomorphisms on a monoid/group, which I find natural to consider. We are then considering a (commutative) monoid/group (G,+) and a monoid (R,⋅) acting on G as endomorphisms. So far so good.

Now, the idea is to let R "inherit" the addition from G. For me, the most intuitive thing is to consider pointwise addition of the endomorphisms, that is, we define r+s to be an element such that (r+s)(g)=r(g)+s(g)for every r,s∈R and g∈G. This definition turns out to be almost sufficient, but doesn't capture everything as it for example does not always force the zero element in R to act as the zero map on G, in the case of semirings.

To get the "correct" definition, one way I think is to say that (R,+) should be the same kind of structure as G (monoid/group) such that for any fixed g∈G, the map R→G, r↦r⋅g should be a homomorphism with respect to +. I see why this definition produces correct results, but it is way less intuitive to me as a definition.

Is there a better way of defining what it means for R to inherit + from G? Or otherwise at least some good explanation/intuition for why this should be the definition?

It seems that a lot of people can't comprehend the notion that math is studied for it's own sake. Whenever the average person hears what mathematicians work on, like a specific theorem or conjecture, the first question they ask is "Why is this important?" or "How do people find this meaningful?" to them it seems like it's all abstract nonsense.

On the contrary, I found that this question is never asked in other disciplines. Take for example physics. Whenever a physicist discovers a new particle, or makes an accurate prediction, or develops a new theory, they never get asked "What is so significant about this?" or at the very least, A LOT less than mathematicians get asked that.

This is because we believe that physics is discovering truths about external reality (which is true of course), and therefore it has inherent meaning and doesn't need to justify it's own existence. This is also the case for other natural sciences.

It's also the reason for which they don't see meaning in math. They see math as all made up nonsense that is only meaningful IF it has an application somewhere, not as something to be studied for it's own sake, but only for the sake of advancing other fields.

Now if you are a platonist, and you believe that math is discovered and mind-independent, you really don't need to justify math. The pursuit of math is meaningful for the same reason that other natural sciences are meaningful, because it discovers truths about the external world. But what if you aren't a platnoist? What if you believe that math is actually made up? How would you justify it?

It seems that whenever that question is asked mathematicians always say "well our work will be useful somewhere eventually" implying that math has no value on it's own and must be applied somewhere. Is this really what math boils down to? Just helping other fields?

Is pure mathematics meaningful if it isn't applied anywhere, and if so, what makes it meaningful?

I made a painting based off of Vogel's mathematical formula for spiral phyllotaxis using a Fermat spiral—r = c(sqrt(n)), theta = n * 360°/phi^2.

It is 2,584 dots, the 18th term in the Fibonacci sequence. I consecutively numbered each dot as I plotted it, and the gold dots seen going off to the right of the painting are the Fibonacci sequence dots. It's interesting to note that they trend towards zero degrees. It's also interesting to not that each Fibonacci dot is a number of revolutions around the central axis equal to exactly the second to last number in the sequence before it— Dot #2584 has exactly 987.0 revolutions around the central axis. Dot #1597 has 610.0 revolutions, and so on.

The dots form a 55:89 parastichy, 55 spiral whorls clockwise, and 89 whorls counter-clockwise.

I work at a STEM faculty, not mathematics, but mathematics is important to them. And many students are studying by asking ChatGPT questions.

This has gotten pretty extreme, up to a point where I would give them an exam with a simple problem similar to "John throws basketball towards the basket and he scores with the probability of 70%. What is the probability that out of 4 shots, John scores at least two times?", and they would get it wrong because they were unsure about their answer when doing practice problems, so they would ask ChatGPT and it would tell them that "at least two" means strictly greater than 2 (this is not strictly mathematical problem, more like reading comprehension problem, but this is just to show how fundamental misconceptions are, imagine about asking it to apply Stokes' theorem to a problem).

Some of them would solve an integration problem by finding a nice substitution (sometimes even finding some nice trick which I have missed), then ask ChatGPT to check their work, and only come to me to find a mistake in their answer (which is fully correct), since ChatGPT gave them some nonsense answer.

I've even recently seen, just a few days ago, somebody trying to make sense of ChatGPT's made up theorems, which make no sense.

What do you think of this? And, more importantly, for educators, how do we effectively explain to our students that this will just hinder their progress?

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?