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Results tagged with soft-question
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user 1946
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.
10
votes
When are $\mathbb{R}^n$ and $\mathbb{R}^m$ essentially similar?
The qualitative intuition that you mention — that $\mathbb{R}^n$ begin to be similar in sufficiently high dimension — is the same intuition underlying my question Does the truth of any statement of re …
4
votes
Are there examples of families of objects which are canonically isomorphic, but where diagra...
Here is an example. Consider the family of rotated copies of the unit square in the plane $\mathbb{R}\times\mathbb{R}$, rotated by some angle about the origin. If one such copy is rotated only a littl …
18
votes
joint papers/collaboration, take two.
You are correct that the nature of one's collaborative publications is an issue that is considered during hiring and promotion, and this seems to me to be to be a perfectly reasonable career question. …
18
votes
What are good non-English languages for mathematicians to know?
As the great mathematician Groucho Marx said, English is the lingua franca.
Ha, ha, ha. :-)
67
votes
How has "what every mathematician should know" changed?
As mathematics grows and diversifies beyond belief, surely the collection of topics that every mathematician must know is shrinking fast. One can carry out serious mathematical research in one area wh …
9
votes
Interesting examples of generic behavior of mathematical objects being either unreasonably s...
Generic Turing machine programs are unreasonable: the computation head will fall off the beginning of the tape.
Basically, the situation is that on the usual one-way infinite tape model, a random pr …
8
votes
Accepted
Decidability of Frankl's union-closed sets conjecture
The statement has complexity $\Pi^0_1$, which means that it has a
single universal quantifier, quantifying over the possible
union-closed sets, and then making a simple assertion about those
objects.
…
- 236k
4
votes
Isn't a graph to be considered isomorphic to its complement, actually?
Perhaps what Hans means is simply that any graph has exactly the same information as the complement graph, because if we know completely where there are no edges, then we also know completely where ar …
- 236k
25
votes
How do you approach your child's math education?
Teach him to use language precisely. The idea that words can have hard, exact meanings is fundamental to mathematical thinking, and will also serve him well in most other subjects.
And I mean all lan …
21
votes
Making sure that you have comprehended a concept
I like to take a playful attitude towards things. I try out the concept in various elementary or even trivial contexts at first, and see what happens. This is particularly useful when the result disag …
61
votes
Naming in math: from red herrings to very long names
Let me mention as a counterpoint that there is less need for
new terminology than one might expect. Mathematical exposition
is often more successful and clearer without new terminology, and
one should …
17
votes
Particular problem solved by solving a more general problem.
Cantor proved the existence of transcendental real numbers by proving that most numbers are transcendental. The set of algebraic real numbers is countable.
2
votes
What is a satisfactory way to format definitions in Latex?
I set my definitions in roman font with the defined term in italic. Just switch to roman font inside the definition environment.
This way avoids the visual confusion caused by reverse emphasis of ro …
46
votes
Theorems true but wrong.
I remember years ago sitting in Leo Harrington's office in Berkeley explaining my dissertation to him (he was on my committee), and he spent some time just scanning through the dissertation seeking ou …
32
votes
What are some examples of colorful language in serious mathematics papers?
In this MO answer, I mentioned Arnold Miller's lecture
notes, where he gives
an entertaining account of the MM proof system (for Micky Mouse), having as axioms all validities and modus ponens as the o …