Questions tagged [soft-question]

Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.

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408 votes
89 answers
141k views

Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
372 votes
22 answers
59k views

Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
348 votes
52 answers
115k views

Widely accepted mathematical results that were later shown to be wrong?

Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly ...
313 votes
22 answers
98k views

Why do so many textbooks have so much technical detail and so little enlightenment? [closed]

I think/hope this is okay for MO. I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that ...
298 votes
16 answers
128k views

What's a mathematician to do? [closed]

I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of ...
279 votes
122 answers
85k views

What are some examples of colorful language in serious mathematics papers?

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ...
276 votes
47 answers
106k views

Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
257 votes
67 answers
125k views

Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
253 votes
29 answers
87k views

Mathematical games interesting to both you and a 5+-year-old child

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me... How to make both of us to do what they want ? I guess ...
249 votes
41 answers
94k views

A single paper everyone should read? [closed]

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues. Do ...
238 votes
29 answers
155k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
226 votes
46 answers
82k 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 ...
226 votes
14 answers
71k views

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
223 votes
87 answers
40k views

Your favorite surprising connections in mathematics

There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
222 votes
11 answers
24k views

Refereeing a Paper [closed]

I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling. I'm wondering what is actually expected and what people tend to do... Some things ...
217 votes
20 answers
34k views

How can a mathematician handle the pressure to discover something new?

Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering ...
216 votes
140 answers
47k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
195 votes
13 answers
55k views

Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?

To begin with, I am aware of these questions, which seems to be related: How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of ...
194 votes
72 answers
46k views

What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical. In many branches of mathematics, it seems to me that a good counterexample can be ...
190 votes
88 answers
49k views

Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow. How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...
186 votes
18 answers
13k views

Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...
185 votes
79 answers
40k views

Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
184 votes
94 answers
100k views

Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wiki rules apply: one ...
182 votes
47 answers
98k views

Magic trick based on deep mathematics

I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...
180 votes
62 answers
88k views

Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...
178 votes
78 answers
63k views

Best online mathematics videos?

I know of two good mathematics videos available online, namely: Sphere inside out (part I and part II) Moebius transformation revealed Do you know of any other good math videos? Share.
176 votes
125 answers
59k views

Most memorable titles

Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
175 votes
11 answers
27k views

Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an ...
168 votes
39 answers
40k views

Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
167 votes
8 answers
17k views

How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research.

As an undergraduate we are trained as mathematicians to be universalists. We are expected to embrace a wide spectrum of mathematics. Both algebra and analysis are presented on equal footing with ...
164 votes
37 answers
200k views

Too old for advanced mathematics? [closed]

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ...
155 votes
47 answers
22k views

Generalizing a problem to make it easier

One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you ...
152 votes
54 answers
20k views

Old books you would like to have reprinted with high-quality typesetting

There are some questions on mathoverflow such as What out-of-print books would you like to see re-printed? Old books still used with answers that tell us things such as: Mathematicians prefer to use ...
152 votes
28 answers
27k views

How To Present Mathematics To Non-Mathematicians?

(Added an epilogue) I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching). In the last ...
150 votes
6 answers
15k views

Proofs shown to be wrong after formalization with proof assistant

Are there examples of originally widely accepted proofs that were later discovered to be wrong by attempting to formalize them using a proof assistant (e.g. Coq, Agda, Lean, Isabelle, HOL, Metamath, ...
147 votes
25 answers
40k views

What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
144 votes
31 answers
65k views

What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
144 votes
18 answers
12k views

Suggestions for special lectures at next ICM

(I am posting this in my capacity as chair of the ICM programme committee.) ICM 2022 will feature a number of "special lectures", both at the sectional and plenary level, see last year's ...
143 votes
21 answers
19k views

How does one justify funding for mathematics research?

G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be ...
143 votes
5 answers
24k views

What makes dependent type theory more suitable than set theory for proof assistants?

In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he ...
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141 votes
26 answers
24k views

Good "casual" advanced math books

I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks ...
141 votes
10 answers
14k views

What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.) More ...
141 votes
10 answers
17k views

Why are flat morphisms "flat?"

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason! Is ...
139 votes
20 answers
21k views

Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence: "The unorientable surfaces are never discussed ...
137 votes
59 answers
29k views

Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the "abelian grape" variety. I take my cue from the following passage in A Mathematician's Miscellany by J.E. Littlewood (Methuen 1953, p. 79): ...
131 votes
26 answers
26k views

What are some famous rejections of correct mathematics?

Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner Theorem: There are only a finite number of imaginary quadratic fields that have unique factorization. They are $...
127 votes
74 answers
20k views

Most helpful math resources on the web

What are really helpful math resources out there on the web? Please don't only post a link but a short description of what it does and why it is helpful. Please only one resource per answer and let ...
125 votes
19 answers
66k views

Periods and commas in mathematical writing

I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after ...
125 votes
22 answers
8k views

Books that teach other subjects, written for a mathematician

Say I am a mathematician who doesn't know any chemistry but would like to learn it. What books should I read? Or say I want to learn about Einstein's theory of relativity, but I don't even know much ...
124 votes
63 answers
25k views

Counterexamples in algebra?

This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in analysis and ...

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