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

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### Why does Loday call the permutohedra “zylchgons”?

Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He ...

**14**

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172 views

### Young's natural representation of the symmetric group

The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story.
One of the standard ways to ...

**9**

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**0**answers

181 views

### English translation of Wilhelm Killing's 1889 paper

In the paper The greatest mathematical paper of all time, by A. J. Coleman (The Mathematical Intelligencer 11 (1989) pp29-38, https://doi.org/10.1007/BF03025189) the author argues that Wilhelm Killing'...

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votes

**1**answer

142 views

### Looking for an erratum (reference request)

Note: Since what I am asking about below touches on a potentially controversial subject, let me emphasize that I am only asking for a specific reference, and I am not asking for a discussion of the ...

**28**

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**2**answers

1k views

### Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?

In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...

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votes

**5**answers

991 views

### Early examples of mathematicians publishing (from home) in a foreign language?

Today this is common, but how exactly did it start? I am looking for examples in various languages, and suggest:
Exclude Latin (as more “ancient” or “international” than “foreign”)
Exclude French ...

**10**

votes

**1**answer

545 views

### Quantum functional analysis

Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...

**6**

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168 views

### History of the definition of completeness of algebraic varieties (and properness of morphisms)

An algebraic variety (say over an algebraically closed field) $X$ is complete if for every variety $Y$, the projection $X \times Y \to Y$ is closed (with respect to the Zariski topology).
As it is ...

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**4**answers

685 views

### History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...

**6**

votes

**0**answers

112 views

### What (if any) was known about null sets before Lebesgue?

The notion af a null set, i. e., a set of Lebesgue measure zero, does not require a full blown construction of Lebesgue measure:
A set is $E\subset \mathbb{R}$ is called a null-set if it can be ...

**7**

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**3**answers

1k views

### “Mächtigkeit” versus “Kardinalität”?

In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?

**8**

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**1**answer

477 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 ...

**73**

votes

**5**answers

11k 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**

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**2**answers

542 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 ...

**9**

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**1**answer

185 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 ...

**36**

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**6**answers

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,...

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**2**answers

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?

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**0**answers

464 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

**1**answer

355 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

**1**answer

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

**2**answers

427 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**

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**0**answers

136 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**

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**0**answers

360 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 ...

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**2**answers

4k 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

**1**answer

140 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.
...

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**5**answers

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 ...

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**1**answer

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

**0**answers

79 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 ...

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**0**answers

506 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

**1**answer

738 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 ...

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**1**answer

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 ...

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**2**answers

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 ...

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votes

**1**answer

436 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 ...

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votes

**1**answer

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 ...

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votes

**1**answer

487 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 ...

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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" ...

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**1**answer

866 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 ...

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**1**answer

316 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 ...

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**0**answers

465 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**

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**2**answers

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**

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**0**answers

77 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 ...

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**1**answer

3k 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

**1**answer

178 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 ...

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37 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?
...

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411 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 ...

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**1**answer

249 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

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**1**answer

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 ...

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votes

**1**answer

471 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 ...

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**1**answer

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 ...

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**1**answer

243 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 ...