Questions tagged [ho.history-overview]
History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.
1,410
questions
422
votes
91
answers
146k
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 ...
395
votes
23
answers
66k
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, ...
382
votes
53
answers
144k
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 ...
235
votes
36
answers
34k
views
Conway's lesser-known results
John Horton Conway is known for many achievements:
Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-...
220
votes
140
answers
48k
views
Fundamental Examples
It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...
215
votes
67
answers
45k
views
Proofs that require fundamentally new ways of thinking
I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
208
votes
26
answers
50k
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 ...
207
votes
14
answers
58k
views
Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?
To begin with, I am aware of these questions, which seems to be related:
How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of ...
192
votes
94
answers
105k
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 ...
183
votes
127
answers
62k
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 ...
164
votes
9
answers
28k
views
Endless controversy about the correctness of significant papers
In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest ...
163
votes
46
answers
31k
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 ...
161
votes
23
answers
29k
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 ...
150
votes
26
answers
38k
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 ...
150
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 ...
142
votes
7
answers
14k
views
Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?
Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...
- 77.5k
137
votes
7
answers
32k
views
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
- 149k
136
votes
14
answers
28k
views
Careers advice for Ph.D.s without current postdocs or university jobs
Hi,
I'm sure I'm not the only Ph.D. mathematician on MO in serious need of career advice. I'm sure there will be other readers in similar situations, who will find any good advice very helpful. Can ...
135
votes
26
answers
28k
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 $...
134
votes
69
answers
221k
views
Mathematical "urban legends"
When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee,...
133
votes
6
answers
22k
views
what mistakes did the Italian algebraic geometers actually make?
It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations (...
- 40.8k
122
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
...
120
votes
41
answers
28k
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 ...
119
votes
33
answers
14k
views
Examples of theorems misapplied to non-mathematical contexts
For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments ...
117
votes
29
answers
15k
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 ...
116
votes
5
answers
32k
views
How did Cole factor $2^{67}-1$ in 1903?
I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193{,}707{,}721\times 761{,}838{,}257{,}287$. There doesn't seem to be ...
- 151k
114
votes
96
answers
16k
views
What would you want to see at the Museum of Mathematics? [closed]
EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
113
votes
1
answer
10k
views
What happened to Suren Arakelov? [closed]
I heard that Professor Suren Arakelov got mental disorder and ceased research. However, a brief search on the Russian wikipedia page showed he was placed in a psychiatric hospital because of political ...
- 6,141
111
votes
34
answers
84k
views
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
109
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 ...
109
votes
19
answers
37k
views
Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
108
votes
20
answers
18k
views
Mathematical habits of thought and action which would be of use to non-mathematicians
Once again I come to MO for help with something I'm writing for the public.
Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...
107
votes
32
answers
14k
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 ...
107
votes
10
answers
28k
views
What is the oldest open problem in mathematics?
What is the oldest open problem in mathematics? By old, I am referring to the date the problem was stated.
Browsing Wikipedia list of open problems, it seems that the Goldbach conjecture (1742, every ...
- 18.5k
106
votes
26
answers
14k
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 ...
106
votes
10
answers
14k
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 ...
103
votes
19
answers
14k
views
Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
101
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 ...
99
votes
5
answers
10k
views
New arXiv procedures?
Recently I encountered a new phenomenon when I tried to submit a paper to arXiv. The paper was an erratum to another, already published, paper and will be published separately. I got a message from ...
98
votes
16
answers
28k
views
What if Current Foundations of Mathematics are Inconsistent? [closed]
The title of the question is also the title of a talk by Vladimir Voevodsky, available here.
Had this kind of opinion been expressed before?
EDIT. Thanks to all answerers, commentators, voters, ...
92
votes
74
answers
26k
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 ...
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 ...
91
votes
8
answers
15k
views
Has incorrect notation ever led to a mistaken proof?
In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...
91
votes
6
answers
17k
views
Why didn't Vladimir Arnold get the Fields Medal in 1974?
As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)
In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, the ...
90
votes
24
answers
21k
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 ...
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 ...
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 ...
86
votes
2
answers
4k
views
History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$
Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...
- 29.5k
85
votes
6
answers
17k
views
Are there any serious investigations of whether "mathematicians do their best work when they're young"?
There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious (...
84
votes
38
answers
21k
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}{...