136
votes
Accepted
Why do we have two theorems when one implies the other?
Some mathematicians seem to agree with you, and strive only to state and prove the most general versions of their theorems. I've had co-authors express that view. And I've sometimes had referee ...
125
votes
Every mathematician has only a few tricks
$$
\sum_{i=1}^m\sum_{j=1}^n a_{i,j}=\sum_{j=1}^n\sum_{i=1}^m a_{i,j}
$$
(and its variants for other measure spaces).
I still get misty-eyed whenever I read something that capitalizes on this trick in ...
91
votes
Every mathematician has only a few tricks
A very useful generic trick:
If you can't prove it, make it simpler and prove that instead.
An even more useful generic trick:
If you can't prove it, make it more complicated and prove that instead!
77
votes
Every mathematician has only a few tricks
Dennis Sullivan used to joke that Mikhail Gromov only knows one thing, the triangle inequality. I would argue that many mathematicians know the triangle inequality but not many are Gromov.
76
votes
Every mathematician has only a few tricks
In combinatorics: shove it into OEIS, and see what's up.
Also, add more parameters!
Note: the Macdonald polynomials were introduced by adding more parameters to the Jack and the Hall-Littlewood ...
73
votes
Accepted
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?
First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial ...
- 42.4k
63
votes
Each mathematician has only a few tricks
The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it'...
61
votes
What recent discoveries have amateur mathematicians made?
Aubrey de Grey, The chromatic number of the plane is at least $5$ (on the arXiv in April 2018)
Apparently, de Grey is a famous biogerontologist and he attacked the mathematical problem in his spare ...
60
votes
The use of computers leading to major mathematical advances II
There is the recent computer-assisted verification of some key statements by Scholze and Clausen about "condensed mathematics". The task has been accomplished by Buzzard, Commelin, and ...
57
votes
Every mathematician has only a few tricks
Integration by parts has allegedly earned some people big medals.
55
votes
Breakthroughs in mathematics in 2023
Marcelo Campos, Simon Griffiths, Robert Morris, Julian Sahasrabudhe made a breakthrough in Ramsey theory. The abstract is a good enough summary to just quote it here:
The Ramsey number $R(k)$ is the ...
54
votes
The use of computers leading to major mathematical advances II
Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kashaev's Volume Conjecture (which relates the order of growth of the values of Jones ...
54
votes
Breakthroughs in mathematics in 2023
The solution by D. Smith, J.S. Myers, C. S. Kaplan, C. Goodman-Strauss, posted in arXiv in March 2023, of the einstein problem:
In plane geometry, the einstein problem asks about the existence of a ...
52
votes
Breakthroughs in mathematics in 2021
Advancing mathematics by guiding human intuition with AI, Nature 600, 70 (2021), stands out because it represents the first significant advance in pure mathematics generated by artificial intelligence....
50
votes
Every mathematician has only a few tricks
For a finite set of real numbers, the maximum is at least the average and the minimum is at most the average.
Of course this is just the real version of the Pigeonhole Principle, but Dijkstra had an ...
49
votes
Theorems that impeded progress
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
48
votes
Every mathematician has only a few tricks
Although Erdős was mentioned in the comments as perhaps having prompted this whole discussion, I'm surprised not to see the basic trick of "try a random object/construction" posted as an ...
46
votes
What recent discoveries have amateur mathematicians made?
An anonymous poster of a 4chan messaging board, in thinking about how long it would take to watch a 14-episode nonlinear anime program in any order, improved the lower bound for a length of a ...
46
votes
Accepted
Is algebraic geometry constructive?
If you forget about all the layers of abstraction, algebraic geometry is, ultimately (and very roughly speaking), the study of polynomial equations in several variables, and of the geometric objects ...
- 32.4k
44
votes
Why is the definition of the higher homotopy groups the "right one"?
I think that obstruction theory is one of the most important reasons to study homotopy groups. If you are interested in studying the possible homotopy classes of maps $X \to Y$ of spaces where $X$ has ...
41
votes
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?
The other answers are quite good and nothing is wrong with them, but lest a wrong impression be given (e.g. by the implicit suggestion that the idea of "categories as sets in the next dimension" in ...
- 66.7k
40
votes
Breakthroughs in mathematics in 2021
Strictly speaking this is not a new mathematical result (meaning no new proof), but let me mention the Liquid Tensor Experiment, the verification in Lean of a very recent theorem by Clausen and ...
37
votes
Every mathematician has only a few tricks
If an integer-valued function is continuous, it has to be constant.
This trick shows up in many places, such as the proof Rouché's theorem, and basic results about the Fredholm index.
37
votes
Do empirical studies have a place in contemporary mathematics research?
You can browse the journal Experimental mathematics, which publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting ...
36
votes
Accepted
Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme
The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) ...
- 42.4k
36
votes
Every mathematician has only a few tricks
Whenever you find yourself trying to implement inclusion–exclusion by hand ... stop immediately and start over using the Möbius $\mu$-function.
35
votes
Theorems that impeded progress
I would vote for the classification of finite simple groups as an example of a (hopefully correctly proved) theorem which impeded progress in the field since it was announced to be proved.
And a ...
34
votes
The use of computers leading to major mathematical advances II
Here is an interesting one. Reinforcement learning to generate counter-examples to several open conjectures in combinatorics and graph theory.
https://arxiv.org/abs/2104.14516
33
votes
Theorems that impeded progress
I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject ...
Only top scored, non community-wiki answers of a minimum length are eligible