All Questions
Tagged with big-list ho.history-overview
97 questions
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, ...
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 ...
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 ...
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 ...
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, ...
- 531
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....
4
votes
1
answer
369
views
Examples of rich theories that started out as an infinite-dimensional inquiry
It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it ...
- 485
52
votes
14
answers
9k
views
Modern results that are widely known, yet which at the time were ignored, not accepted or criticized
What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It ...
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 ...
6
votes
3
answers
558
views
Anomalous phenomena [closed]
What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?
This is most interesting when ...
14
votes
29
answers
7k
views
Which great mathematicians had great political commitments? [closed]
Some mathematicians claim that their field has nothing to do with political concerns; others are deeply involved in political life.
Are there many great mathematicians with great political commitments?...
22
votes
17
answers
5k
views
Which great mathematicians were also historians of mathematics?
As the question title suggests, which great mathematicians were also historians of mathematics?
We all know plenty of great mathematicians, but not many historians of mathematics. Not to mention that ...
11
votes
4
answers
513
views
Autobiographies and correspondences of mathematicians [duplicate]
Lately I have enjoyed reading several autobiographies and correspondences of mathematicians. I'd like to find more, so I thought I'd ask here which others you have come across and enjoyed.
P.S. I have ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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,...
6
votes
3
answers
720
views
Lesser known examples of perseverance with a successful ending [closed]
The stories of Wiles, of Perelman, and of Zhang, are very well-known to illustrate that sometimes great results are achieved through particularly long perseverance.
What are lesser known-examples ...
38
votes
17
answers
8k
views
Examples of "unsuccessful" theories with afterlives
I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and ...
35
votes
30
answers
6k
views
Examples of simultaneous independent breakthroughs
I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. ...
15
votes
1
answer
1k
views
Telling right from left
I know a lot of people, some of them mathematicians, who have trouble telling right from left. This can lead to problems when you are composing functions, for example.
When did this seemingly ...
53
votes
11
answers
6k
views
What definitions were crucial to further understanding?
Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
15
votes
2
answers
910
views
Terminology: Lost in translation or multiple-meanings
I was reading Uniformization of Riemann Surfaces by Henri Paul de Saint Gervais (not a real person, but a group of French mathematicians), and the translator kindly points out that the name of "the ...
56
votes
3
answers
11k
views
Work of plenary speakers at ICM 2018
The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from 2014 and 2010. Can you, please, ...
67
votes
22
answers
10k
views
When has discrete understanding preceded continuous?
From my limited perspective, it appears that the understanding
of a mathematical phenomenon has usually been achieved,
historically, in a continuous setting
before it was fully explored in a discrete ...
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 ...
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 ...
63
votes
7
answers
8k
views
Theorems demoted back to conjectures
Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered.
I am ...
10
votes
5
answers
919
views
Important results with one or more than one proof [closed]
Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How ...
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 ...
3
votes
1
answer
3k
views
Famous examples of PhD advisors younger than their student [closed]
What are the most famous examples of PhD advisors in mathematics, younger than their student?
(if possible put the date of birth and/or the difference in age).
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 ...
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
...
29
votes
25
answers
7k
views
Mathematicians who made important contributions outside their own field? [closed]
It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand.
I am looking for mathematicians who did just that. ...
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 ...
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 ...
73
votes
17
answers
9k
views
Mathematical research published in the form of poems
The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...
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 ...
76
votes
19
answers
18k
views
What are some deep theorems, and why are they considered deep?
All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
89
votes
27
answers
12k
views
Modern Mathematical Achievements Accessible to Undergraduates
While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...
35
votes
13
answers
5k
views
Great mathematics books by pre-modern authors
Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...
67
votes
19
answers
14k
views
Mathematicians whose works were criticized by contemporaries but became widely accepted later
Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...
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 ...
91
votes
70
answers
18k
views
Old books still used
It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...
15
votes
4
answers
3k
views
Statements which were given as axioms, which later turned out to be false.
I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom....
23
votes
7
answers
7k
views
What are some Applications of Teichmüller Theory?
I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:
No-wandering-domain Theorem (Sullivan)
Theorems of Thurston (...
9
votes
9
answers
1k
views
Examples where adding complexity made a problem simpler
I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples:
$S^n$ is never contractible, but $S^{\infty}$ is.
The ...