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Results tagged with soft-question
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user 290
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.
8
votes
Are there situations when regarding isomorphic objects as identical leads to mistakes?
Expanding on Andy Putman's comment, beginners in linear algebra often think of all finite-dimensional vector spaces (over $\mathbb{R}$, say) as the same, in particular as $\mathbb{R}^n$ with the stand …
- 118k
9
votes
Learning new mathematics
I read blog posts about it and then I blog about it. The main thing this does is supplement reading a textbook, especially one that doesn't provide motivation or connection with other branches of mat …
- 118k
28
votes
Memorizing theorems
The only way I ever really understand a theorem is by blogging about it; this is what I did with the Polya Enumeration Theorem, for example, whose proof I hadn't followed very closely in my algebraic …
49
votes
20
answers
33k
views
What are good non-English languages for mathematicians to know?
It seems that knowing French is useful if you're an algebraic geometer. More generally, I've sometimes wished I could read German and Russian so I could read papers by great German and Russian mathem …
1
vote
Between abstract and concrete: What's the right way to think of specific categories?
Position 2 is only tenable because the categories you describe automatically come with forgetful functors to $\text{Set}$. But in order to think about more general categories (say, homotopy categorie …
- 118k
5
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
Here's a basic but important example. In the modern approach to things like algebra we study objects like groups and rings in two stages: first we study their abstract structure, then we study their …
4
votes
Most helpful heuristic?
A sort-of heuristic in combinatorics is that if you can't figure out what to do with a set, take the free abelian group / vector space on that set and work with linear transformations instead of funct …
35
votes
What should be offered in undergraduate mathematics that's currently not (or isn't usually)?
I think undergraduates should take problem-solving classes. I don't think such classes are widely available, but bright students who didn't do a lot of problem-solving in high school would definitely …
6
votes
Learning to Think Categorically
My own personal exposure to category theory mostly consists of reading the following blog entries:
The entire beginning of The Unapologetic Mathematician,
Todd Trimble's series on basic category the …
5
votes
Categorification request
There are related examples at this MO question, but most power series identities can be categorified to natural isomorphisms between combinatorial species, which are functors $\text{FinSet}_0 \to \tex …
- 118k
4
votes
How can I really motivate the Zariski topology on a scheme?
Here's an idea related to Tim Carstens' answer. As in Ben's answer we start from the point of view that it makes sense to think of $\text{Spec } R$ as a set. Given an ideal $I$ we have a homomorphis …
- 118k
20
votes
How to mentor an exceptional high school student?
Tell him about his other opportunities (although perhaps being on AoPS he is already aware of them). Summer programs like
Ross
PROMYS
the Canada/USA Mathcamp
HCSSiM
and others come highly recomme …
18
votes
Particular problem solved by solving a more general problem.
Frequently in mathematics the best way to determine the value of a sequence at a particular index is to compute its value at every index, even though the latter seems on the surface like a harder prob …
12
votes
Theorems with unexpected conclusions
Here's one I was reminded of recently. Recall that a projective plane is a triple $(P, L, I)$ where $P$ is a set of "points," $L$ is a set of "lines," and $I$ is a subset of $P \times L$ describing t …
17
votes
Most harmful heuristic?
Also not really a heuristic, but "differentiation is easy," as encoded in the following two sub-heuristics:
Differentiation is just repeated application of the product and chain rules, and
Most func …