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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.
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 …
195
votes
Most harmful heuristic?
This isn't really a heuristic, but I hate "functions are formulas". For most students it takes a really long time to think of a function as anything other than an algebraic expression, even though nat …
174
votes
What are the most misleading alternate definitions in taught mathematics?
Here's another algebra peeve of mine. The definition of a normal subgroup in terms of conjugation is pretty strange until it's explained that normal subgroups are the ones you can quotient by. Again …
138
votes
Accepted
What is entropy, really?
Here is a simple story one can tell about the entropy
$$H = -\sum_{i=1}^n p_i \log p_i$$
of a discrete probability distribution. Suppose you wanted to describe how surprised you are upon learning …
121
votes
What are the most misleading alternate definitions in taught mathematics?
In my experience, introductory algebra courses never bother to clarify the difference between the direct sum and the direct product. They're the same for a finite collection of abelian groups, which …
115
votes
What are your favorite instructional counterexamples?
A polynomial $p(x) \in \mathbb{Z}[x]$ is irreducible if it is irreducible $\bmod l$ for some prime $l$. This is an important and useful enough sufficient criterion for irreducibility that one might w …
99
votes
Your favorite surprising connections in mathematics
From an essay of Arnol'd:
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and t …
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
95
votes
Famous mathematical quotes
Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discove …
87
votes
What non-categorical applications are there of homotopical algebra?
As a student, I'm always looking for organizing principles in mathematics to help me keep track of all of the mathematics I learn. It's easy to get lost in a deluge of definitions unless I organize th …
76
votes
Why do so many textbooks have so much technical detail and so little enlightenment?
I absolutely agree that this is a question worth asking. I have only recently come to realize that all of the abstract stuff I've been learning for the past few years, while interesting in its own ri …
63
votes
Favorite popular math book
Title: Godel, Escher, Bach: an Eternal Golden Braid
Author: Douglas Hofstadter
Short Description: It's mildly debatable whether this is in fact a book about mathematics, but any mathematician who …
61
votes
What do named "tricks" share?
How about the following (which I think applies to some of these tricks but not others): a trick is something whose usefulness is not fully captured by any particular set of hypotheses, so it would lim …
59
votes
Accepted
59
votes
What are the most overloaded words in mathematics?
Normal
Normal distribution
Normal vector
Normal space
Normal extension
Normal subgroup
Normal operator
Normal convergence