All Questions
Tagged with big-list ho.history-overview
97 questions
85
votes
19
answers
15k
views
Each mathematician has only a few tricks
The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
- 8,435
170
votes
47
answers
34k
views
Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
- 6,349
110
votes
10
answers
15k
views
Analogues of P vs. NP in the history of mathematics
Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...
CommunityBot
- 1
71
votes
34
answers
12k
views
Trichotomies in mathematics
Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...
- 4,713
5
votes
2
answers
668
views
Recent breakthroughs with applied origins
Historically, the boundary between pure mathematics and its applications was much less defined. However, with the increasing complexity of modern mathematics and the resulting need for specialization, ...
- 101
45
votes
10
answers
10k
views
Has the mathematics research community ever been led astray by a dumb mistake?
This is a highly subjective question, but here goes.
Has anyone ever published a result that was "taken seriously" by the research community, but was then discovered to be incorrect because ...
- 111
424
votes
93
answers
149k
views
Video lectures of mathematics courses available online for free
It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
- 4,425
36
votes
35
answers
11k
views
Titles composed entirely of math symbols
I apologize for burdening MO with such a vapid, nonresearch question, but
I have been curious ever since
Suvrit's popular October 2010
Most memorable titles MO question
if there were any "$E=mc^2$...
- 35.5k
217
votes
28
answers
53k
views
The most outrageous (or ridiculous) conjectures in mathematics
The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics.
An outrageous conjecture is qualified ONLY if:
1) It is most likely false
(Being hopeless is ...
- 433
125
votes
31
answers
16k
views
Papers that debunk common myths in the history of mathematics
What are some good papers that debunk common myths in the history of mathematics?
To give you an idea of what I'm looking for, here are some examples.
Tony Rothman, "Genius and biographers: The ...
- 236k
41
votes
67
answers
14k
views
What are examples of mathematical concepts named after the wrong people? (Stigler's law)
It's a common observation in Lie theory that Cartan matrices and the Killing form are named after the wrong people; they were discovered by Killing and Cartan, respectively. I remember learning about ...
- 12.9k
401
votes
53
answers
151k
views
Widely accepted mathematical results that were later shown to be wrong?
Are there 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 significant amount of time later, possibly ...
- 101
185
votes
127
answers
65k
views
Most memorable titles
Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
- 12.9k
67
votes
16
answers
9k
views
What do named "tricks" share?
There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous
tricks, a term which in this context is in no sense derogatory.
Here is a list of 11 such tricks (the ...
33
votes
24
answers
7k
views
Early two-author math papers
The middle of the twentieth-century featured several famous papers with two authors. For example, Eilenberg and Mac Lane's papers introducing categories and Eilenberg-MacLane spaces appeared in 1945. ...
- 2,185
92
votes
74
answers
27k
views
Pseudonyms of famous mathematicians
Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
- 13.4k
75
votes
29
answers
13k
views
(Preferably rare) Audio/Video recordings of famous mathematicians?
Terence Tao's homepage has a link to a collection of quotes, and one among them was Hilbert's famous "We must know, we will know" quote. This quote also had an audio link to it. Now although I'm not ...
- 10.4k
84
votes
11
answers
12k
views
What are examples of (collections of) papers which "close" a field?
There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
A total characterisation,...
- 30.3k
399
votes
23
answers
69k
views
Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
51
votes
30
answers
8k
views
Taking a theorem as a definition and proving the original definition as a theorem
Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are ...
- 1,422
19
votes
11
answers
4k
views
Approachable French masters
Similar to this topic, what are the easiest foundational French texts for someone learning the language? My intuition would be Cauchy and Lebesgue, but I have no idea where to start or which of their ...
- 63.9k
107
votes
32
answers
15k
views
The half-life of a theorem, or Arnold's principle at work
Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that ...
- 19.6k
103
votes
15
answers
17k
views
Have you solved problems in your sleep?
I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending through ...
- 3,068
1
vote
0
answers
198
views
A zoo of derivations
Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...
- 167
36
votes
65
answers
13k
views
Fiction books about mathematicians? [closed]
What are some fiction books about mathematicians?
It seems to me rather difficult for writers to create good books on this subject.
Some years ago I thought there were no such books at all.
There ...
- 4,713
110
votes
89
answers
29k
views
Tweetable Mathematics
Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.
I am looking for ...
CommunityBot
- 1
152
votes
31
answers
27k
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, largely, is what ...
- 39.9k
43
votes
9
answers
6k
views
What are some examples of theorem requiring highly subtle hypothesis?
I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are ...
- 2,370
8
votes
0
answers
560
views
Landau's century-old problems: Anything comparable?
Landau's four problems
are now over a century old (1912), and each still unsolved.
This seems remarkable, even though he was not the originating author all four
(maybe only the 4th?). Still, he ...
- 5,407
27
votes
4
answers
3k
views
What are examples of theorems which were once "valid", then became "invalid" as standard definitions shifted?
That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
- 1,446
122
votes
41
answers
29k
views
What are some very important papers published in non-top journals?
There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly ...
- 123
91
votes
24
answers
22k
views
Examples of major theorems with very hard proofs that have not dramatically improved over time
This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of
Major theorems in mathematics whose proofs are ...
- 123
86
votes
38
answers
11k
views
Books about history of recent mathematics
I draw on this question to ask something that has always been a pet peeve of mine. It is very easy to find books about the history of mathematics, much less so if one wants books about the recent (say ...
- 39.9k
79
votes
13
answers
21k
views
Nontrivially fillable gaps in published proofs of major theorems
Prelude: In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem, as presented in Nash's well-known paper "The Imbedding ...
- 13.4k
152
votes
26
answers
39k
views
Has philosophy ever clarified mathematics?
I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
- 8,842
197
votes
94
answers
107k
views
Famous mathematical quotes [closed]
Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...
- 1,446
35
votes
26
answers
5k
views
Examples of mathematics motivated by technological considerations
I would like examples of technological advances that were made possible only by the creation of new mathematics. I'm talking about technology that was desired in some period of history but for which ...
- 4,786
107
votes
26
answers
15k
views
Fields of mathematics that were dormant for a long time until someone revitalized them
I thought that the closed question here could be modified to a very interesting question (at least as far as big-list type questions go).
Can people name examples of fields of mathematics that were ...
- 4,713
64
votes
68
answers
16k
views
Mathematicians with both “very abstract” and “very applied” achievements
Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant ...
- 23k
84
votes
37
answers
22k
views
What are some correct results discovered with incorrect (or no) proofs?
Many famous results were discovered through non-rigorous proofs, with
correct proofs being found only later and with greater difficulty. One that is well
known is Euler's 1737 proof that
$1+\frac{1}{...
- 12.9k
60
votes
35
answers
15k
views
Notable mathematics during World War II
It seems fairly well known that Leray originated the ideas of spectral sequences and sheaves while being held in a prisoner of war camp in Austria from 1940 to 1945. Weil famously proved the Riemann ...
- 4,713
19
votes
1
answer
883
views
Importance of textbooks in health of a sub-discipline
I am interested in published articles, and also more informal writing (blog posts, talk slides etc.) which discuss the importance of textbooks (where this word encompasses research monographs etc.) in ...
- 4,713
165
votes
23
answers
30k
views
Do you read the masters?
I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a ...
- 8,842
32
votes
21
answers
16k
views
What are some applications of other fields to mathematics?
It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some applications ...
- 4,713
8
votes
1
answer
388
views
Formalisation of intuitive concepts in the language leading to mathematical progress
In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
- 82.7k
137
votes
26
answers
29k
views
What are some famous rejections of correct mathematics?
Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner
Theorem: There are only a finite
number of imaginary quadratic fields
that have unique factorization. They
are $...
- 4,713
60
votes
15
answers
11k
views
Abstract thought vs calculation
Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was the use of more abstract notions ...
- 4,713
40
votes
29
answers
8k
views
Autobiographies of mathematicians
According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator.
An autobiography offers the author the ability to recreate history....
- 91.8k
47
votes
7
answers
8k
views
Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians
I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
- 12.8k
123
votes
35
answers
18k
views
Rediscovery of lost mathematics
Archimedes (ca. 287-212BC) described what are now known as the 13
Archimedean solids
in a lost work, later mentioned by Pappus.
But it awaited Kepler (1619) for the 13 semiregular polyhedra to be
...
- 2,369