77
votes
16answers
18k views
What’s a mathematician to do? [closed]
I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with …
77
votes
31answers
19k views
Widely accepted mathematical results that were later shown wrong?
I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significan …
77
votes
12answers
10k views
Have any long-suspected irrational numbers turned out to be rational?
The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surpr …
77
votes
0answers
7k views
Ultrafilters and automorphisms of the complex field
It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have th …
77
votes
56answers
15k views
Your favorite surprising connections in Mathematics
There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the conne …
75
votes
59answers
13k views
Jokes in the sense of Littlewood: examples? [closed]
First, let me make it clear that I do not mean jokes of the
"abelian grape" variety. I take my cue from the following
passage in A Mathematician's Miscellany by J.E. Littlewood
(M …
73
votes
66answers
12k views
Most helpful math resources on the web
What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource …
72
votes
24answers
10k views
Extremely messy proofs
Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, …
72
votes
7answers
7k views
If $f$ is infinitely differentiable then $f$ coincides with a polynomial
Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ c …
72
votes
16answers
9k views
Geometric Interpretation of Trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
…
72
votes
55answers
15k views
Which math paper maximizes the ratio (importance)/(length)?
My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7 …
72
votes
43answers
18k views
Where does a math person go to learn quantum mechanics?
My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've bee …
71
votes
26answers
8k views
How To Present Mathematics To Non-Mathematicians?
(Added an epilogue)
I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teac …
71
votes
7answers
4k views
How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research.
As an undergraduate we are trained as mathematicians to be universalists. We are expected to embrace a wide spectrum of mathematics. Both algebra and analysis are presented on equa …
71
votes
10answers
7k views
Is there an introduction to probability theory from a structuralist/categorical perspective?
The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory ( …
