Questions tagged [ho.history-overview]

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

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52 votes
11 answers
6k views

What is an important mathematical question?

$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste....
52 votes
3 answers
5k views

Mathematical research in North Korea -- reference request

Question: Where can one find information on which areas of mathematics are represented at which of the more than 20 universities in the Democratic People's Republic of Korea (DPRK), and on which ...
42 votes
5 answers
11k views

Riemann's formula for the metric in a normal neighborhood

I would love to understand the famous formula $g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(\|x\|^3)$, which is valid in Riemannian normal coordinates and possibly more general situations. I'...
36 votes
4 answers
6k views

Is "Cartan's magic formula" due to Élie or Henri?

The formula $\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. ...
7 votes
4 answers
924 views

Origin of tropical mathematics

On Wikipedia, it is claimed without a source that Imre Simon founded tropical mathematics. The first work of his I was able to find on the subject is Limited subsets of a free monoid which uses the ...
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 ...
31 votes
2 answers
1k views

Group theory with grep?

While reading Bill Thurston's obituary in the Notices of the AMS I came across the following fascinating anecdote (pg. 32): Bill’s enthusiasm during the early stages of mathematical discovery was ...
1 vote
0 answers
179 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 ...
14 votes
3 answers
2k views

Where are Serre’s lectures at Collège de France to be found?

Having run into several references, at various places and occasions, to "Serre’s Course at Collège de France, 19xy-19xy+1" for various values of xy, I would genuinely like to know where these lectures ...
2 votes
1 answer
169 views

Terminology associated with mathematical induction

In "Number: The Language of Science" (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as "the induction step" (and refers to what we call the ...
9 votes
4 answers
2k views

The name for an assumption made for the sake of contradiction

What is the name (or adjective) for an assumption made for the sake of contradiction? To be clear, I'm in search of an expression in the form "a(n) $\underline{\quad \quad \quad \quad}$ ...
18 votes
4 answers
15k views

The Ramanujan Problems

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here. In the Wikipedia page on Ramanujan (current ...
40 votes
5 answers
7k views

Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences?

Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences? Background for the question: Park, M., Leahey, E. & Funk, R.J. Papers and patents ...
25 votes
3 answers
1k views

Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?

30 years ago, Yves Colin de Verdière introduced the algebraic graph invariant $\mu(G)$ for any undirected graph $G$, see [1]. It was motivated by the study of the second eigenvalue of certain ...
40 votes
5 answers
6k views

Were Bourbaki committed to set-theoretical reductionism?

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...
6 votes
2 answers
504 views

Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing theory....
7 votes
2 answers
588 views

Reference request for recurrence relation of division polynomials

The recurrence relations for division polynomials of elliptic curves are well known: $$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$ $$\Psi_{2n+1} = \...
5 votes
1 answer
921 views

Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?

In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
13 votes
1 answer
4k views

Is there another controversial statement by Grothendieck apart from 57 being prime?

There is a well-known story about Grothendieck being asked to explain concretely some result involving prime numbers and of his answering "You mean an actual number? All right, take 57". ...
31 votes
2 answers
3k views

The Erdős-Turán conjecture or the Erdős' conjecture?

This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics: Conjecture: If $A\subset \mathbb{N}$ and $$\...
17 votes
2 answers
2k views

Why did Gödel name his constructible universe $L$?

It seems like Gödel didn't use the letter $L$ for his model before his book "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory", which is ...
4 votes
1 answer
365 views

Etymology “Kulkarni–Nomizu product”

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...
37 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 ...
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 ...
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 ...
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 ...
13 votes
1 answer
704 views

Emanuel Lasker, Max Noether, and Emmy Noether

In 1900, Emanuel Lasker (world chess champion from 1894 to 1921) received his Ph.D. under Max Noether. In 1905, Lasker published a theorem that Emmy Noether generalized in 1921, now well known as the ...
44 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 ...
4 votes
0 answers
189 views

History of tropical mathematics

This is a follow-up to this question about the origin of tropical mathematics. Are there any articles, websites or books which deal with the history of tropical mathematics? I have been trying to find ...
4 votes
1 answer
132 views

The origin and use of the term "equianharmonic" (elliptic function)

This question has been posted on History of Science and Mathematics stack exchange, but there was no answer or comments there. In Weierstrass notation, the principal elliptic function $\wp$ is a ...
4 votes
1 answer
481 views

Origin of 'Analytic' Geometry?

My impression is that the name analytic geometry, which I understand roughly to be geometry in Euclidean space using coordinates, is not used that much anymore. We would probably classify the subject ...
8 votes
9 answers
2k views

Papers of the masters translated to English in one location

Surely someone has collected these papers and translations and has them in a single location for download? For music there is musipedia. Surely there is a mathepedia equivalent? If I search gauss ...
32 votes
4 answers
2k views

Authorship of Grothendieck universes

Universes seem to first enter Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is ...
5 votes
1 answer
388 views

Motivating unpublished statements of Gauss about congruences and quaternions

Background Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this ...
8 votes
0 answers
498 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 ...
22 votes
2 answers
2k views

History of differential forms and vector calculus

Who and when was it realized that the classical operators of vector calculus (grad, rot, div) can be expressed in a unified form using the exterior differential? I have searched a little bit on the ...
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 ...
48 votes
8 answers
13k views

Who wrote up Banach's thesis?

Sometime ago I read somewhere (and I don't remember where it was) that Stefan Banach--a highly creative and great mathematician--did not always write down his ideas. Allegedly, he did not write his ...
14 votes
2 answers
2k views

How did the Baker-Gill-Solovay paper come to be?

How did the Baker-Gill-Solovay paper come to be? Why were those three people talking together about "Relativizations of the $P=?NP$" question, and what was their collaboration like for the ...
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 ...
26 votes
2 answers
3k views

What is the origin/history of the following very short definition of the Lebesgue integral?

Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
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 ...
64 votes
2 answers
8k views

Who is the "young student" André Weil is referring to in his letter from the prison?

I am reading a nice booklet (in Italian) containing the exchange of letters that André and Simone Weil had in 1940, when André was in Rouen prison for having refused to accomplish his military duties. ...
36 votes
9 answers
4k views

Why is it still common to not motivate results in publications? [closed]

This is a question about practice and publication of research mathematics. On the Wikipedia Page for Experimental Mathematics, I found the following quote: Mathematicians have always practised ...
9 votes
1 answer
917 views

True origin of the term "Spline"

In mathematical contexts the term spline essentially refers to interpolating or approximating piecewise functions with continuity constraints. According to the history of mathematical splines In the ...
2 votes
0 answers
280 views

Cartier and the continuity of the early history of schemes

If you allow me I would divide the early history of schemes this way _ Weil, Zariski, Bourbaki, Nagata, Van der Waerden,... up to Chevalley (you can find an interesting blog here) J P Serre varieties ...
5 votes
0 answers
232 views

Seeking identity of mathematician in photo on the cover of "The Honors Class" by Ben Yandell

All of the photos on the cover of Ben Yandell's book The Honors Class appear in the book, except the one in the upper left. I'd wager a beer that the mathematician in the upper left corner is A.N. ...
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, ...
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 ...
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 ...

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