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

8
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1answer
443 views

Whence “Durchschnitt” and “Vereinigung”?

Today the set-theoretic operations of intersection $\cap$ [German: Durchschnitt] and union $\cup$ [German: Vereinigung] are standard. The modern notations are present in the first edition of van der ...
67
votes
5answers
10k 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, ...
9
votes
2answers
489 views

What did the Intuitionists want to do with applied mathematics?

Oversimplification: Newton & Leibnitz &c build the calculus and other methods that solve a vast number of practical problems. Weierstrass, Dedekind, Cantor &c build a foundation under it ...
8
votes
1answer
175 views

Origin of the concept of “homomorphism”? [duplicate]

When was the concept of a "homomorphism" of algebraic structures first introduced? Steinitz' 1910 paper Algebraic Theory of Fields is often pointed to as the first true work of abstract algebra, yet ...
0
votes
0answers
139 views

Did Srinivasa Ramanujan have a surviving sibling? [migrated]

Wiki (https://en.wikipedia.org/wiki/Srinivasa_Ramanujan) says 'After his death, his brother Tirunarayanan chronicled Ramanujan's remaining handwritten notes consisting of formulae on singular moduli, ...
34
votes
5answers
6k views

“Long-standing conjectures in analysis … often turn out to be false”

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis,...
13
votes
2answers
2k views

Who first chose the names Alice and Bob for players A and B? [closed]

Who first chose the names Alice and Bob for the players (or observers) A and B?
18
votes
0answers
447 views

Eckmann-Hilton argument / Grothendieck

In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...
4
votes
1answer
348 views

Constructibility of the regular 17-gon [closed]

There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was Gauss who found out that it is possible to do ...
26
votes
1answer
3k views

Why did Euler consider the zeta function?

Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
4
votes
2answers
421 views

Priority for lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). There seems to have been ...
3
votes
0answers
131 views

In history of algebra, who was the first to add one equation to another equation?

In history of algebra, who was the first to add one equation to another equation? Someone gave me the name of an Italian mathematician of Renaissance period, but I lost the email.
3
votes
0answers
340 views

Kodaira's Fields medal citation

In 1954, Kodaira won the Fields medal. His citation was Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to algebraic ...
50
votes
2answers
3k views

How were modular forms discovered?

When modular forms are usually introduced, it is by: "We have the standard action of $SL(2,\mathbb Z)$ on the upper half-plane, so let us study functions which are (almost) invariant under such ...
4
votes
1answer
126 views

Citations graphs what is known?

There have been much research related to webgraphs and social graphs. They can be thought of a kind of random graphs, but the point is that they are different from the well-known Erdős–Rényi model. ...
50
votes
5answers
5k views

The Logic of Buddha: A Formal Approach

Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features ...
13
votes
1answer
2k views

Some clarifications on Connes' approach to RH

How serious/promising is Connes' work on the Riemann Hypothesis? Connes is mostly known these days for his work in non-commutative geometry, having previously earned a Fields medal* for his work on ...
4
votes
0answers
76 views

History of the relation between $p$-adic measures and power series

In 1964, Kubota and Leopoldt defined the $p$-adic $L$-function by means of some $p$-adic sums (now called the Volkenborn integral which is a $p$-adic distribution). Later, Mazur (in his secret ...
1
vote
0answers
497 views

On the basis of a finite dimensional vector space (revised)

Revision in response to the comments to earlier version: To introduce the notion of a basis of a finite dimensional vector space over an arbitrary field $\Lambda$, without performing any computation ...
16
votes
1answer
728 views

New articles by Errett Bishop on constructive type theory?

Recently two formerly unknown articles by Errett Bishop (1928-1983) were posted online by Martín Escardó. One is entitled "A general language", deals with constructive type theory, and ...
2
votes
1answer
92 views

History of an open problem on partial tilting modules

The following is an open problem: Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
13
votes
2answers
1k views

“This category obviously leads to paradoxes of set theory.” What is the paradox?

Eilenberg and Mac Lane formally defined categories in their 1945 paper General Theory of Natural Equivalences. Their definition of a category starts as follows: "A category {A,a} is an aggregate of ...
10
votes
1answer
431 views

Ehresmann's approach to differential geometry

I have come accross this brief description of Charles Ehresmann's life given by his wife: http://www.cs.le.ac.uk/people/ah83/cat-myths/myth0002.html I quote the part from the text relevant to my ...
3
votes
1answer
1k views

Who was the first to capitalize Real?

For example in Atiyah's $KR$-theory there is the notion of a Real vector bundle in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^*$-algebra and there are ...
7
votes
1answer
478 views

Who was in the Fields committee for ICM 1962 (the first appointed by IMU)?

Traditionally, at the presentation of the Fields medals at the ICM opening ceremony, the composition of the Fields medal committee is disclosed. This information can be found in the first volume of ...
24
votes
4answers
2k views

What “metatheory” did early set theory/logic researchers use to prove semantic results?

Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic. The modern approach seems to be, usually, to interpret a "model" ...
12
votes
1answer
856 views

Historically, which came first: the Lie algebras or their classification?

The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F_4, G_2, E_6, E_7$ and $E_8$. I am trying to find out ...
1
vote
1answer
290 views

Different definitions of category

It appears that there are two different definitions of category. Some authors require the Hom-sets to be pairwise disjoint. Eilenberg and Mac Lane in their original definition require each identity ...
7
votes
0answers
418 views

What is your favourite wrong proof of RH? [closed]

Some of the users here receive claimed proofs of the Riemann hypotheses on a regular bases. As fas as we know all of them have been wrong. But sometimes failure is also interesting. So for all cases ...
28
votes
2answers
1k views

When did people start thinking of elliptic curves as groups?

I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...
2
votes
0answers
72 views

Properties of inverse Cayley-Menger matrices

in the online article A formula for the N-circumsphere of an N-simplex dated April 2013, G. Westendorp provides an interpretation of the entries of inverse of Cayley-Menger matrices $\hat{B}$, that ...
39
votes
1answer
2k views

Did Hilbert laugh?

Prof. D. C. McCarty recently gave an interesting interview (published in January 2015, and easily found on a large video hosting site), entitled What are the limits of mathematical explanation? I ...
1
vote
1answer
173 views

Intuition behind the proof of key step in Minkowski's second inequality on successive minima

I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...
3
votes
0answers
36 views

Bound on local packing density of 2D Delaunay cell

What is the history of the result that in a packing of the plane by unit disks, no Delaunay cell can be occupied by disk-sectors whose total measure exceeds $\pi/\sqrt{12}$ times the area of the cell? ...
7
votes
0answers
257 views

Where can I find Rademacher's wrong disproof of the Riemann Hypothesis?

Mathematical folklore has it that the famous algebraist Hans Rademacher once came up with a wrong disproof of the Riemann Hypothesis, which was initially believed by another famous mathematician, Carl ...
10
votes
1answer
244 views

History of the classification of mathematical subjects

I would like to know if there are sources on the history of the classification of mathematical subjects. Gérard Lang
8
votes
1answer
1k views

What is the translation of this ancient Greek verb πυθαγοριζει

Here it is used in a sentence It is therefore a priori probable that Plato πυθαγοριζει in the passage where he says that between two planes one mean suffices, but to connect two solids, two means ...
2
votes
1answer
403 views

History of Fargues-Fontaine curve

In this paper, Pierre Colmez wrote about some history of the Fargue-Fontaine curve. In this schedule of London Number Theory Study Group, Fargues was said to give a talk on November 15th on " Where ...
19
votes
1answer
1k views

Who was Heinrich Hake?

Hake's Theorem, due to Heinrich Hake of Düsseldorf in 1921, says that an improper Henstock–Kurzweil integral (aka generalized Riemann integral, gauge integral, Perron integral, or Denjoy integral) on ...
8
votes
1answer
236 views

Sophus Lie's contribution to solution of problems of variational type as in Euler and Lagrange

The original impetus for Sophus Lie's work was apparently to streamline the solution of certain problems of variational type such as those treated in the work of Euler and Lagrange. This presumably ...
8
votes
2answers
577 views

On Street's “australian conspectus”

Skimming the australian conspectus of higher category theory I noticed I have a few questions, both mathematical and historical. At about the middle of page 6, Kock-Zoberlein monads are defined as "...
2
votes
1answer
337 views

Who first used the word “Homomorphism”? [duplicate]

Who first used the word "homomorphism" to describe a link between two similar structures ? Following this, who specialized this concept with the words: - "Isomorphism"; - "Endomorphism"; - "...
16
votes
2answers
2k views

Who first used the word “Simplex”?

Who first used the word "Simplex" to describe the considered geometric figure?
11
votes
1answer
166 views

History of publication of von Neumann's characterization of orthogonally invariant matrix norms

Von Neumann has a result (rather well-known in convex analysis circles) which states that every orthogonally invariant matrix norm (meaning $\| P M Q\| = \| M \|,$ for any orthogonal $P, Q$) is a ...
2
votes
1answer
209 views

Cantelli's inequality: the original source

Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.
12
votes
1answer
758 views

Lalouvère's activities as censor

Fermat had a friend at Toulouse named Lalouvère. Lalouvère was censor, jesuit, and mathematician (in alphabetical order). Antonella Romano writes on page 512 of her book La Contre-Réforme ...
11
votes
2answers
363 views

Who first defined locally convex topological vector spaces?

Who first defined the class of locally convex topological vector spaces?
3
votes
0answers
80 views

Was Martin-Löf inspired by Peirce when he introduced the dependent sum and dependent product types?

In the following article: https://plato.stanford.edu/entries/peirce-logic/ it is mentioned that Peirce's introduced the use of the symbols $\Sigma$ and $\Pi$ to express logical sums and products, ...
41
votes
18answers
9k views

Mathematical research interrupted by a war

I am not sure that this is appropriate at MO, so if not, please, delete this. This is inspired by David Hansen's question where he asks about mathematics done during the WWII. I would like to ask the ...
9
votes
0answers
831 views

A quote by Lev Landau about prime numbers

I was talking with a student of mine about Goldbach's conjecture, and a certain point he asked why this apparently simple statement is so difficult to prove. Half-joking, I answered "well, because ...