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Results tagged with soft-question
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user 2926
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.
117
votes
What's a mathematician to do?
I appreciate the OP's question, but it seems to me that
One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant p …
63
votes
Accepted
Why do categorical foundationalists want to escape set theory?
I don't agree that this is what (most) categorists who are interested in foundations are doing.
It is true that Lawvere in the mid-60's (and perhaps to this day) wanted to develop a theory of catego …
- 53.3k
32
votes
Accepted
Lawvere theories versus classical universal algebra
My own experience is that Lawvere theories help one "think outside the box" in ways that I really don't think are too likely with classical universal algebra. Qiaochu has already pointed to what is th …
- 53.3k
29
votes
Is a come back to mathematical research possible?
I hesitate to posit myself as an example, but I was out of academia from 2001 to 2019, when I decided to become a stay-at-home dad while my wife became the breadwinner. (I won't go into the details of …
29
votes
An example of a proof that is explanatory but not beautiful? (or vice versa)
A proof that many people say they find beautiful, but in my view is not at all explanatory, is Zagier's one-sentence proof of the sum of two squares theorem.
25
votes
Is the empty graph a tree?
The whole discussion seems to devolve on whether the empty graph (or empty space) should be considered "connected". Angelo and I are of the school that it should not, but this should be explained sinc …
- 53.3k
24
votes
Most memorable titles
You'd think that with John H. Conway around, this should be like shooting fish in a barrel. One title that comes to mind is
The Sensual (Quadratic) Form
and there are more goodies if you look at his …
23
votes
'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math
I would upvote Keerthi's comment multiple times if I could. Just find an area of mathematics that makes you smile and brings you happiness. If in the course of doing research you find that you need to …
19
votes
Applications of the Cayley-Hamilton theorem
Cayley-Hamilton can be useful in commutative algebra. Related to its close connection with Nakayama's lemma as mentioned in a comment by Qiaochu (see also Wikipedia), see for example the development g …
19
votes
How to refer to a “theorem” that you have shown to be wrong
In my opinion, it would be a bad idea to label statements known to be false as theorems. If you really want to do this, maybe you could put inverted commas around the word "theorem", to indicate you e …
19
votes
Non-rigorous reasoning in rigorous mathematics
Have you reads Proofs and Refutations by Lakatos? It's all about the dynamic tension between "heuristics" (don't get mad at me, Andrew Stacey!) and rigorous proof, centering particularly on a classro …
19
votes
Why is Set, and not Rel, so ubiquitous in mathematics?
Taking up remarks near the end of the OP, and somewhat in line with Mike Shulman's answer, I'd like to underline the structural interplay between $\mathbf{Set}$ and $\mathbf{Rel}$ to indicate one poin …
17
votes
Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differe...
One way to understand $l^1(X)$ for a set $X$ with counting measure is that $l^1(-): Set \to Ban$ provides a left adjoint to the functor $\hom(k, -): Ban \to Set$. Here $k$ is the ground field and $Ban …
16
votes
A map of non-pathological topology?
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hier …
- 53.3k
15
votes
Categorification request
I tried to discuss this geometric series example of categorification in one of my answers to another MO question by Jan Weidner, here. I can't tell whether this reply was considered unsatisfactory, bu …
- 53.3k