Questions tagged [gm.general-mathematics]

Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.

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229 votes
46 answers
87k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
211 votes
40 answers
37k views

Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
167 votes
50 answers
55k views

17 camels trick

The following popular mathematical parable is well known: A father left 17 camels to his three sons and, according to the will, the eldest son should be given a half of all camels, the middle son ...
164 votes
22 answers
25k views

When should a supervisor be a co-author?

What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few ...
163 votes
46 answers
31k views

Every mathematician has only a few tricks

In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
163 votes
9 answers
28k views

Endless controversy about the correctness of significant papers

In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest ...
160 votes
37 answers
15k views

Conceptual reason why the sign of a permutation is well-defined?

Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
Tim Campion's user avatar
  • 60.5k
114 votes
96 answers
16k views

What would you want to see at the Museum of Mathematics? [closed]

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
108 votes
20 answers
18k views

Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public. Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...
106 votes
32 answers
14k views

Special rational numbers that appear as answers to natural questions

Motivation: Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. ...
101 votes
15 answers
17k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending through ...
93 votes
20 answers
10k views

Short papers for undergraduate course on reading scholarly math

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.) Today, I was reminded of the existence of this ...
90 votes
11 answers
13k views

What are possible applications of deep learning to research mathematics?

With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
90 votes
8 answers
12k views

Books on music theory intended for mathematicians

Some time ago I attended a colloquium given by Princeton music theorist Dmitri Tymoczko, where he gave a fascinating talk on the connection between music composition and certain geometric objects (as ...
83 votes
24 answers
8k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
81 votes
6 answers
11k views

Is data science mathematically interesting?

I have seen a plethora of job advertisements in the last few years on mathjobs.org for academic positions in data science. Now I understand why economic pressures would cause this to happen, but from ...
80 votes
22 answers
14k views

Are there proofs that you feel you did not "understand" for a long time?

Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...
78 votes
49 answers
43k views

Examples of interesting false proofs

According to Wikipedia False proof For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...
78 votes
21 answers
17k views

Is rigour just a ritual that most mathematicians wish to get rid of if they could?

"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was ...
78 votes
15 answers
9k views

Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
77 votes
15 answers
13k views

Each mathematician has only a few tricks

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
77 votes
13 answers
8k views

What computational problems would be good proof-of-work problems for cryptocurrency mining?

What computational mathematics problems that could be used as proof-of-work problems for cryptocurrencies? To make this question easier to answer, I want proof-of-work systems that work in ...
Joseph Van Name's user avatar
73 votes
6 answers
11k views

Still Difficult After All These Years

I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
70 votes
22 answers
10k views

Small ideas that became big

I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research ...
67 votes
6 answers
4k views

Good ways to engage in mathematics outreach?

Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...
65 votes
21 answers
9k views

Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones

I am looking for examples of the following situation in mathematics: every object of type $X$ encountered in the mathematical literature, except when specifically attempting to construct ...
65 votes
6 answers
8k views

Does a referee have to check carefully the proof ?

I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths ...
63 votes
27 answers
8k views

Golden ratio in contemporary mathematics

A (non-mathematical) friend recently asked me the following question: Does the golden ratio play any role in contemporary mathematics? I immediately replied that I never come across any mention of ...
63 votes
10 answers
6k views

Fascinating moments: equivalent mathematical discoveries

One of the delights in mathematical research is that some (mostly deep) results in one area remain unknown to mathematicians in other areas, but later, these discoveries turn out to be equivalent! ...
60 votes
10 answers
12k views

What do you do when you're stuck?

I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there ...
59 votes
9 answers
5k views

Examples of back of envelope calculations leading to good intuition?

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
59 votes
9 answers
7k views

Has anyone thought about creating a formal proof wiki with verifier?

Mathematics has undergone some rather nice developments recently with the adoption of new techologies, things like on-line journals, the arXiv, this website, etc. I imagine there must be many further ...
58 votes
7 answers
9k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
Randy Qian's user avatar
54 votes
7 answers
9k views

Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?

String theory (and related areas of purely theoretical quantum gravity, like loop quantum gravity) has a unique position within the academic physics community. Many academic physicists don't really ...
53 votes
10 answers
7k views

Changes forced by the pandemic

The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
53 votes
9 answers
11k views

How does a mathematician choose on which problem to work?

Main question: How does a mathematician choose on which problem to work? An example approach to framing one's answer: What is a mathematical problem - big or small - that you solved or are ...
52 votes
37 answers
83k views

What is your favorite "strange" function? [closed]

There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you ...
52 votes
4 answers
8k views

Changing field of study post-PhD

I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be ...
50 votes
37 answers
6k views

Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...
50 votes
13 answers
13k views

Is amateur research in mathematics viable?

After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job. I will therefore ...
50 votes
1 answer
8k views

What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?

I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...
Ryan Hendricks's user avatar
49 votes
8 answers
4k views

Published results: when to take them for granted?

Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority). Checking the result. Of course, one should ...
47 votes
2 answers
5k views

Well known theorems that have not been proved

I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
45 votes
4 answers
4k views

How to invoke constants badly

In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways ...
44 votes
9 answers
4k views

Journals and other sources with "easy reading" papers?

Some time ago the journal "Algebra and Analysis" (English translation is published in "St. Petersburg Mathematical Journal") had a special section which was called "easy readings for professional ...
44 votes
2 answers
5k views

Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...
Martin Brandenburg's user avatar
43 votes
18 answers
5k views

Results in linear algebra that depend on the choice of field

Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers). I am looking for a list of concepts, and results, in ...
43 votes
5 answers
15k views

How to deal with an advisor that offers you nearly no advising at all?

I am a young PhD student (24) at a Germany university and I am not sure whether this is the right place to ask this kind of question. If not feel free to move it elsewhere or delete it completely. ...
43 votes
7 answers
4k views

Can pure mathematics harness citizen science?

Having just finished Michael Nielsen's book "Reinventing Discovery", I find myself wondering if there are ways that pure mathematics research can engage the public in the way that GalaxyZoo or Foldit ...
42 votes
26 answers
8k views

Where can square roots come from when they are not distances?

In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...

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