Questions tagged [ho.history-overview]

History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.

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53 votes
9 answers
76k views

What is the shortest Ph.D. thesis? [closed]

The question is self-explanatory, but I want to make some remarks in order to prevent the responses from going off into undesirable directions. It seems that every few years I hear someone ask this ...
12 votes
1 answer
2k views

Is there evidence whether undergraduate math courses improve problem-solving?

The most commonly stated reason for why mathematics should be a required condition for graduating is }to improve problem-solving skills". Usually it's taken for granted that taking a mathematics ...
Anna Varvak's user avatar
16 votes
1 answer
2k views

Fulfilling Pythagoras' Dream using Nonstandard Models of Arithmetic and/or Surreal Numbers

Pythagoras and his followers believed that the Universe was made of numbers. Specifically, they thought that if you compared any magnitudes of the same kind, say the lengths of two objects, you would ...
Keshav Srinivasan's user avatar
70 votes
3 answers
17k views

The story about Milnor proving the Fáry-Milnor theorem

This question is similar to a previous one about "urban legends", but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, ...
Greg Kuperberg's user avatar
421 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 ...
10 votes
1 answer
631 views

History of the Lagrange Inversion Theorem

I'm doing research on the history of the Lagrange inversion theorem. The earliest predecessor I've found is the one referenced by De Morgan; viz. Jo. H. Lambert's construction in Observationes Variae ...
Scott Guthery's user avatar
3 votes
1 answer
969 views

Whence the k-tuple conjecture?

What is the source of the $k$-tuple conjecture, that every integer tuple $(k_1,\ldots,k_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst $(k_1+c,\...
Charles's user avatar
  • 8,974
55 votes
14 answers
10k views

Does any research mathematics involve solving functional equations?

This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those ...
Qiaochu Yuan's user avatar
10 votes
2 answers
2k views

The different Branches of Arithmetic

... "and then the different branches of Arithmetic-- Ambition, Distraction, Uglification, and Derision." (Alice in Wonderland, chapter IX: the Mock Turtle's story) As a child I wondered for ...
Pietro Majer's user avatar
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134 votes
69 answers
220k 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,...
11 votes
7 answers
6k views

Where can you find Grothendieck's "Récoltes et Semailles"?

Where can you find Grothendieck's "Récoltes et Semailles"? Is it available anywhere?
user4's user avatar
  • 911
6 votes
2 answers
2k views

What is the shortest proof of the existence of a prime between $p$ and $p^2$ ? other examples? [closed]

1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)? (One can say that we can have it as a collorary of ...
12 votes
7 answers
2k views

History of classifying spaces

Where did the idea and formal definition of the "classifying space of a (small) category" first appear? Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...
Dr Shello's user avatar
  • 1,160
19 votes
4 answers
12k views

How did Bernoulli prove L'Hôpital's rule?

To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
John Palmieri's user avatar
43 votes
3 answers
8k views

Why the name 'separable' space?

It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
minimax's user avatar
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40 votes
7 answers
15k views

How might M.C. Escher have designed his patterns?

I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it ...
Dan Tao's user avatar
  • 461
21 votes
3 answers
5k views

Did André Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis?

Did André Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?
Jonathan Sondow's user avatar
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 ...
9 votes
0 answers
595 views

Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
19 votes
2 answers
2k views

History of Irrationality results

The Greeks knew that numbers of the form $\sqrt{n}$ for nonsquare integers $n$ are not rational. Much later, Lambert (1768) proved that the values of $e^x$ and $\tan x$ are irrational for nonzero ...
Franz Lemmermeyer's user avatar
40 votes
1 answer
3k views

First correct proof of FLT for exponent 3?

It is well known that Euler gave the first proof of FLT ($x^n + y^n = z^n$ has no nontrivial integral solutions for $n > 2$) for exponent $n=3$, but that his proof had gaps (which are not as easily ...
Franz Lemmermeyer's user avatar
17 votes
4 answers
3k views

Did Grothendieck write about modular forms?

This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, ...
James D. Taylor's user avatar
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 ...
72 votes
3 answers
9k views

Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?

This question is partly motivated by Never appeared forthcoming papers. Motivation Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance ...
Jonathan Chiche's user avatar
35 votes
4 answers
6k views

Why are differential forms called closed and exact?

It seems to me that "exact" relates to exact differential equation. So, why are they called exact?
Nikita Kalinin's user avatar
26 votes
2 answers
2k views

William Rowan Hamilton and Algebra as Time

This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the admittedly-...
Cam McLeman's user avatar
  • 8,417
67 votes
16 answers
8k 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 ...
17 votes
1 answer
2k views

A synopsis of Adyan’s solution to the general Burnside problem?

Where can I find a high-level overview of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent? Additionally: If possible, would an expert please ...
Jon Bannon's user avatar
  • 6,977
6 votes
4 answers
2k views

The history of Proper Forcing

What were the initial motivations of the use of the proper forcing.?
13 votes
5 answers
1k views

"Let $x \in A$", beginning a proof of "$\forall x \in A$ ...", if A were empty [closed]

I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in A$...
Coward's user avatar
  • 139
17 votes
12 answers
780 views

What are some early examples of creation of lists / catalogues of (particularly) combinatorial objects?

A lot of effort in discrete maths / combinatorics is expended in the construction of lists, catalogues or census [sic] of combinatorial objects such as groups, graphs, designs etc. These catalogues ...
27 votes
2 answers
6k views

Countable connected Hausdorff space

Let me start by reminding two constructions of topological spaces with such exotic combination of properties: 1) The elements are non-zero integers; base of topology are (infinite) arithmetic ...
Fedor Petrov's user avatar
8 votes
3 answers
1k views

English or French translation of Gauss' "Summatio Quarumdam Serierum Singularium"

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...
Lea M's user avatar
  • 315
136 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 ...
Joseph O'Rourke's user avatar
6 votes
2 answers
632 views

English translation of Lambert's Theorie der Parallellinien?

Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's Theorie der Parallellinien? I was able to find it online in German by way of the ...
Justin Lanier's user avatar
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 ...
39 votes
7 answers
9k views

Why is "abelian" infrequently capitalized?

Posted with input from meta for improvement. I usually read, e.g. "Gaussian integers" and "Riemannian metrics", and occasionally "euclidean" or "cartesian" or even "lorentzian space", but the latter ...
7 votes
1 answer
722 views

Fibonacci = Leonardo Pisano?

Leonardo of Pisa is best known as Fibonacci; various stories found in books and on the web claim that the name Fibonacci was invented by Edouard Lucas or Guillaume Libri in the 19th century, and that ...
Franz Lemmermeyer's user avatar
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 ...
11 votes
3 answers
5k views

What are Central Limit Theorems and why are they called so?

I know two opinions: 1) "Central" means "very important" (as it was central problem in probability for many decades), and CLT is a statement about Gaussian limit distribution. If the limit ...
Fedor Petrov's user avatar
19 votes
1 answer
1k views

What's coherent about coherent sheaves?

In a recent answer to a recent question, BCnrd wrote [...] beyond the coherent case one cannot expect information about a fiber (e.g., vanishing, 6 generators, etc.) to "propogate" to information ...
Mariano Suárez-Álvarez's user avatar
20 votes
3 answers
3k views

Who is Kirszbraun?

Kirszbraun's theorem is one of my favorite theorems in mathematics. I always wanted to know something about Kirszbraun, or at least to see his picture. Do you have any information about him? (I know ...
Anton Petrunin's user avatar
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 ...
6 votes
2 answers
2k views

Birkhoff's theorem about doubly stochastic matrices

Birkhoff's theorem states: The set of $n \times n$ doubly stochastic matrices is a convex set whose extreme points are the permutation matrices This theorem seems to be commonly attributed to ...
Suvrit's user avatar
  • 28.4k
18 votes
2 answers
2k views

Missing exposes in SGA 5, and the composition of the SGA's

Over the past couple of years I had to look in SGA for various results, and I can't but marvel at how poorly constructed it is. In SGA1 expose VII "n'existe pas", SGA 1 references higher SGA's, and so ...
18 votes
4 answers
4k views

What was Weierstrass's counterexample to the Dirichlet Principle?

Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
Jeremy Shipley's user avatar
3 votes
1 answer
781 views

Where does the Chebyshev polynomial notation come from?

The $k$th Chebyshev polynomial is denoted by $T_k$ where $T_k(x) = \cos(k\cos^{-1}(x))$ I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the ...
alext87's user avatar
  • 3,167
62 votes
4 answers
10k views

who fixed the topology on ideles?

I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not ...
KConrad's user avatar
  • 49.5k
31 votes
4 answers
3k views

What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?

This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras: 37602. Like some others who started graduate study in ...
Jim Humphreys's user avatar
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, ...

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