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1500 questions
46
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5 answers
Set theory and Model Theory
This question probably doesn't make any sense, but I don't see why, so I ask it here hoping someone will illuminate the matter:
There is this whole area of study in Set Theory about the consistency, independence of axioms, etc. In some of these you…
Enrique Acosta
- 1,209
46
votes
2 answers
When to postpone a proof?
One possible practice in writing mathematics is to prove every theorem and lemma right after stating it.
A long, technical proof — and sometimes even a short one — can interrupt the flow of the presentation, so postponing the proof can improve…
Joonas Ilmavirta
- 8,046
46
votes
4 answers
What was Gödel's real achievement?
When I first heard of the existence of Gödel's theorem, I was amazed not just at the theorem but at the fact that the question could be made precise enough to answer: how on earth, even in principle, could one show that it was impossible to prove…
gowers
- 28,729
46
votes
7 answers
What is the time complexity of computing sin(x) to t bits of precision?
Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference?
Long version of the question:
I'm sort of surprised to be asking this, because it's such an extremely basic sounding question. …
Ryan O'Donnell
- 6,571
46
votes
11 answers
What is the Cayley projective plane?
One can build a projective plane from $\Bbb R^n$, $\Bbb C^n$ and $\Bbb H^n$ and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as $\Bbb OP^2$, the Cayley projective plane.
What are the…
skupers
- 7,923
- 2
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46
votes
3 answers
Category theoretic interpretation of matroids?
First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category theoretic definition of a matroid should be. The only definition I know…
Mikola
- 2,352
46
votes
4 answers
Using linear algebra to classify vector bundles over ℙ¹
There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let $\mathcal{E}$ be a vector bundle of rank $r$, with…
Ila Varma
- 523
46
votes
4 answers
What is the source of this famous Grothendieck quote?
I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck.
It is better to have a good category with bad objects than a bad category with good objects.
Question: Does anyone know…
Daniel Moskovich
- 21,887
46
votes
7 answers
Conway's game of life for random initial position
What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). Specifically suppose that to start with every cell is alive with…
Gil Kalai
- 24,218
46
votes
5 answers
Are the two meanings of "undecidable" related?
I am usually confused by questions of the type "could such and such a problem be undecidable", because as far as I know there are two distinct possible meanings of "undecidable". I regard the following as the standard definition:
Let $P(n)$ be a…
John Pardon
- 18,326
46
votes
15 answers
Strong induction without a base case
Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication
"If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true."
for every nonnegative integer $n$. There is no need for…
Bjorn Poonen
- 23,617
46
votes
4 answers
Can we ascertain that there exist an epimorphism $G\rightarrow H?$
Let $G,H$ be finite groups. Suppose we have a epimorphism $G\times G\rightarrow H\times H$. Can we find an epimorphism $G\rightarrow H$?
A fellow graduate student asked me this question during TA sessions. Baffled, I asked this question on…
Kerry
- 543
46
votes
4 answers
Which topological spaces admit a nonstandard metric?
My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological space X is a nonstandard metric space, if there is…
Joel David Hamkins
- 224,022
46
votes
13 answers
Why is it so cool to square numbers (in terms of finding the standard deviation)?
When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why do we need to square and then square-root the…
user668
- 643
46
votes
10 answers
effective teaching
Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, i.e. professors lecture to students and sketch…
user577