Structuralism, a brief essay on what is perhaps the philosophical position most widely held amongst mathematicians. This is an extended excerpt from my book, Lectures on the Philosophy of Mathematics, MIT Press 2021.

What does it mean to adopt structuralism in mathematics? Are you structuralist? What are the reasons for or against this philosophical position? Does it lead to mathematical insight?

Hi everyone,

This is a long shot but I'm following through on a post I made here searching for various books.

I am trying to locate a copy of Calculus Made Easy from Macmillan and Co (1931) and was hoping that maybe a reader here has a copy of this edition who is willing to part with it.

If that is the case, please let me know what it is worth to you. I am otherwise distraught to have lost a bidding war on a copy recently on ebay and they are rare enough one only pops up every couple of months.

Any help in obtaining this book is also welcome.

Thanks all, Happy figuring!

Dear all,

I am a Master student in Germany and have a choice of taking either PDE or "gradient flows" (I'm inclined towards gradient flows) next semester, and my general research interests are extremal combinatorics, probability, and theoretical computer science.

I have read relevant questions on MSE and MO, and it seems that "major" applications of PDE in (pure) math are in geometry and topology; even in the context of probability, it's usually about stochastic PDE and/or stochastic process. Optimal transport seems like a hot topic right now.

So I was wondering that for people who used/applied PDE and/or gradient flows in the context of probability, TCS, and data science, what kinds of problems are you working on? Would you say that one needs to know a lot about PDE (say, grad-level PDE course with Evans) to apply them in research, or "just enough"?

Thank you for your time!

i would say mine is f: (0, 1) -> ℝ f(x) = tan(πx-π/2)

what about you guys?

It seems to me that in projective plane geometry, it is often cumbersome or difficult to draw pictures that accurately represent the relationships between objects. Things like lines intersecting too far away if one isn't careful enough. Is there a deep reason for this other than "shapes are distorted in projective geometry"?

My son (9) loves math & sciences and is always asking me to tell him math/science facts and anecdotes. I got him Randall Munro's "What If", "What If 2", and "How To Do It", and he loves them. These books approach science with humor and imagination, and are really exceptionally good. I'd appreciate any recommendations for books that focus on math, written in a similar vein. Thanks.

Does constructive mathematics give us new application / different insights into anything? Or is it all just a subset of classical mathematics?

If it doesn’t offer anything new except constructive algorithms why bother?

Please educate me, thank you

I’m going to be starting grad school this fall. I have interests across pure and applied math, and ideally I would want to do something supremely “pure” (like operator algebras or complex geometry). However, to pursue such a focus would imply that I should focus only on going into academia, and of course academic jobs are really tough to secure.

Given this dilemma, I was curious how the people on this subreddit working on more “pure” topics keep away the anxieties of the academic job market, or pursue their passions despite those anxieties. I would of course love to do something like that, but the thought that I might be able to sustainably pursue such a line of research solely for grad school is highly discouraging.

Mod, please delete this if this is not allowed.

I am a third year undergraduate going to a pretty good program in math. I love math and on paper I think I have achieved good mathematical achievements for my level with 3 REUs, multiple grad courses, and publications. I have also made good relationships with my teachers and next year I hope to apply for math PhD programs as well.

But the truth is I have realized the inherent nature of math (at least in my school) is making me depressed. I’m depressed about constantly feeling stupid. I’m depressed about feeling lonely because math majors tend to be more reserved and the need to work through problems independently in mathematics. I’m depressed about the constant subtle one upping people who wish to enter math academia are towards each other.

Am I setting myself for worse depression by going to graduate school? I genuinely love mathematics but is it possible that something I’m passionate for causes me depression?

I'm trying to understand this paper and I'm stuck in section fucking 2 when they say:

In three dimensions, such an “exchange loop” is topologically equivalent to the trivial loop (which is no loop at all)... In two-dimensions, however, not all exchange loops are topologically equivalent to the trivial loop

This is not obvious to my puny mind, I've been trying to understand it for a couple of hours with no progress. I've also tried googling a bunch but I didn't find anything I could understand

I keep imagining something like a figure 8, and sure you can't transform that into a loop in 2D, but you can't do that in 3D either, here's my reasoning

1.- We can't separate what is together, thus the intersection in the middle isn't going anywhere even if we have three dimensions

2.- We can't join what was together, so we can't fold it in half to transform it into a single loop

If those two things are true I simply can't understand why are such loops trivial in 3R

But also, I don't understand why a non-trivial loop results in a phase that is not 0 or pi

Topology is based on open sets stable by union and finite intersection. Measure theory is based on measurable sets stable by countable union, countable intersection and complementary.

Are there other important theories working that way ?

Hello together,

after i struggled half of the day to install and run a .asy file on my Macbook, i decided to write a short guide on how to install Asymptote on MacOS.

I hope its okay to post this here, but i was not able to find a working guide online, so i hope this may help some people!

https://github.com/DomHaus/Install-Asymptote-on-MacOS-Silicon-GUIDE/

So we are currently covering the CLT in probability, and of course the usual proof is by Fourier transforms/ characteristic functions. As I go through problems, many exercises, some of which were fearsome before (for example, weak convergence of sums of independent random variables) are reduced to mere manipulation of CFs. It honestly feels like I am cheating and pushing symbols around at this point, although most of the work is underneath the hood and happens when proving the inversion formula and Levy's continuity theorem.

Does anyone feel the same way working with CFs? It's as if any time I am confronted with sums of independent RVs they will be the first thing I reach for.

and if so, how small does the pencil have to be for u to stop using it? and what do u do with the small leftover pencils?

By 'natural' I mean that the underlying question is very simple and it feels very intuitive to ask about something like that. In that sense, something like the Goldbach conjecture wouldn't really count as 'natural' because, although it is simple to understand, the question at hand doesn't really arise that naturally, it's not something that an amateur would come up with without spending some time analyzing what prime numbers do (at least that is my impression)

Here are mine:

The Hadwiger-Nelson problem: How many colors do you need to color every point in the plane such that no two points at unit distance to one another are colored the same? (Answer: 5, 6 or 7, we don't know!)

Or the moving sofa problem: What is the biggest shape that can be maneuvered through an L-shaped 'room' with walls a unit distance apart?

Or Moser's worm problem: What is the minimum area of a shape that can cover every unit-length curve?

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

Many pure maths textbooks and expositions fall into a 'bare minimum' trap, with dense definitions lemmas and theorems that get across the bare minimum info required to have technically described a topic, but without giving any reason why we should give a monkey's about any of them.

"We define a hypobloviating spelunkomorphism to be a spelunkomorphism that is both quasi-cooperative and ultra-faithful. [Flips back twenty pages] A cooperative spelunkomorphism is one given by the pseudo-colimit of endochromisms that are both happy and entertaining, and has Gambolputty index > 1... A happy endochromism is a semi-gleeful endochromism for which the following icosahedral diagram commutes..."

A good writer will say WHY we use that seemingly random '>1' condition, why that massive commutative diagram is important. Otherwise, most will lose track of the intuition behind these many variations of 'nice' a long time ago...

There are REASONS for all of these - but even simply saying 'If we don't insist on this condition, then this stupid pathology happens' or giving a motivating example, or explain what more approachable idea it's trying to generalise.

The same applies to theorems. So often there's a Bourbaki-esque, Serge Lang 'exposition'-style chain of lemmas that eventually leads to a theorem with a similarly long setup and where it might actually take a moment to understand why it's different from the lemma preceding it, and no explanation given, which makes the theorem seem a random combination of those confusingly defined words, and very unsatisfactory.

Why even bother writing an exposition if you could just refer people to the original published papers (which might be honestly clearer), if you're going to be this uninformative?

I don’t know if I worded my Q very well, but I ask this because yes, while research math is very different to olympiad math, there is no doubt that many of the “top” mathematicians today (fields medalists, or whatever) were also top performers at the IMO. Accordingly, most “top” mathematicians today are likely to be able to achieve gold with only light studying.

The Artificial Intelligence Mathematical Olympiad (AIMO) has launched on Kaggle. Its ultimate goal over the next few years is to create an open source model capable of achieving gold at the IMO.

Do you think the project will succeed? Is this a good first step towards an “AI mathematician”?

Thank you, all.

Now that there Is an accurate way to predict prime numbers, does this affect the security of ciphers that use prime numbers to encrypt?

Does it also mean anything for the riemann hypothesis with its heavy links to prime distribution?

I'm planning on starting grad school soon-ish and I've been thinking recently about what I might want to focus on in research. I have some ideas of things I definitely don't want to study but I still feel kinda open as far as everything else. There's a lot of things that I don't know about or understand that seem potentially kinda fascinating. How did you decide on what to specialize in?

I’m working on my undergrad degree, and I intend to pursue a PhD in mathematics. I’d really appreciate recommendations for foundational material in different subjects in math so that I can get some exposure to what’s out there.

Thanks!

Edit: I’m loving the recommendations I’m getting! Thank you all for your comments!

Let's say you have a set of n-experiments, each with their own unique probability of success, and all are independent..

If you were to take the average of the probability of successes, and plug that number into the formula for a Bernoulli Trial(BT), how well would you expect the resulting BT curve to fit observed results, for small, medium, and large number of trials run?

As I was typing this out, I realized that I could just write a python script to simulate running the trials with the exact probabilities and the average and see how close they were, but I'm still interested in anyone's thoughts who are willing to share them.

Edit: and by average, I was referring to something more akin to the weighted-mean. So, if the set of probabilities skews high or low, then the average would reflect that.

The idea is that if we subtract from any number the sum of the digits that make up that number, the result will always be a multiple of nine.

This idea is well rapresented by the formula: "n-S=9N", where "n" is any number, "S" is the sum between the digits that make up that number and "9N" is any multiple of nine. The inverse formula is also valid.

I also created a more extensive formula: (10^(m-k_1)*a1)+(10^(m-k_2)*a2)+...-(a1+a2...) Where "m" is the total number of digits that make up the number, "k" is the position of the digit within the number, and "a_1, a_2, a_3,..." are the numbers that make up the digits of the number.

An example with the number 456.

a_1=4 a_2=5 a_3=6 m=3 k_1=1 k_2=2 k_3=3

(10^(3-1)4)+(10^(3-2)5)+(10^(3-3)6)-(4+5+6) = (1004)+(105)+(16)-(4+5+6) = 400+50+6-15 = 441

I have never studied mathematics, I don't know how much sense it really makes, but I'm curious to know why every number follows this simple rule. Any clarification is deeply appreciated, thanks is advance.

Particularly focused on his works in calculus, statistics, and physics.

I realize that’s a bit specific but I’d love to hear your favorites.

Recently a colleague of mine mentioned to me Dirichlet regularisation and now I am pondering if there are tests for Dirichlet summability as there are for usual convergence. Reason I am asking this gere is because I cant find it on google with the same prompt

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

I run a math club in my school, and I'm gonna be doing math stuff with some 8th and 9th graders soon. I'm currently preparing a presentation on series - I started from the basics, about how series can be arithmetic, geometric, how they can converge or diverge, and introduced the harmonic series, which i hope will be interesting to the kids. Also I introduced telescoping series as they're pretty cool.

What are some other cool/surprising things about series that 8th-9th graders can understand (which aren't all that technical/requiring calc like the basel problem). Thanks!

Has anyone ever felt this before? You are in a semester, taking 2-3 classes, but you so wish you had the time to self learn/read another topic of interest? For me I’m a MS stats student I’m currently taking design of experiments and statistical inference and while the coursework is interesting, I have other more interesting topics I wish I could study, like causal inference or nonparametric regression, but I just don’t get the time since I’m constantly doing homework and assignments for my current classes.

Does anyone feel this way?