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Results tagged with soft-question
Search options questions only
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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.
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 …
25
votes
8
answers
6k
views
What is the "right" definition of a ring?
This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object …
- 118k
47
votes
33
answers
17k
views
What are the most overloaded words in mathematics? [closed]
This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or defini …
43
votes
77
answers
10k
views
What is the first interesting theorem in (insert subject here)? [closed]
In most students' introduction to rigorous proof-based mathematics, many of the initial exercises and theorems are just a test of a student's understanding of how to work with the axioms and unpack va …
96
votes
16
answers
18k
views
Why is it a good idea to study a ring by studying its modules?
This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules genera …
- 118k
23
votes
0
answers
651
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essential...
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize th …
- 118k
208
votes
72
answers
51k
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 w …