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

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5
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1answer
270 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 ...
32
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0answers
854 views
+50

History: What was the Lemma? (Grothendieck Harvard Lectures; Mumford)

In an article about the life of Grothendieck, available here: http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf Allyn Jackson writes about how Mumford was profoundly impressed: Mumford ...
18
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1answer
634 views

On the history of the Artin Reciprocity Law

At the beginning of Milne's notes on class field theory, he has a quote by Emil Artin (as recalled by Mattuck in Recountings: Conversations with MIT mathematicians): I will tell you a story about ...
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0answers
47 views

Who was first to use reproducing kernals in order to try to solve interpolation problems?

I understand that Sarason generalized the interpolation problem by taking it into the operator theoretic setting via reproducing kernels, but whose idea was it to use reproducing kernels such as the ...
6
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2answers
779 views

Euclid vs Eratosthenes

Very little is known about Euclid's life--much less than about other famous ancient Greek mathematicians, which is puzzling. It is also strange to me that Euclid didn't write about the Eratosthenes ...
5
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1answer
320 views

History of the notation for substitution

One of the very common notations for syntactic substitution is $[\ /\ ]$. However, there seems to be an inconsistency in the literature about its usage. Many write $[t/x]$ for "substitute $t$ for $...
2
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1answer
181 views

Cambridge Mathematical Tripos papers from late 19th century

Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?
2
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2answers
940 views

Unreasonable application of mathematics to the other areas [closed]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science? I found ...
8
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1answer
319 views

Trinity College, Cambridge, circa 1896 maths scholarship papers [closed]

I've been searching around looking for the (maths component) of the scholarship papers to Trinity College (Cambridge) from around 1890. Can anyone provide a link to a pdf scan of these papers? Was ...
13
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2answers
2k views

A certain mathematical competition in the UK

There is a foreword, written by professor Snow, to the book A mathematician's apology. In the foreword, it is written some thing like the following: "Hardy was opposed to a certain mathematical ...
5
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0answers
181 views

Origin of the name ''momentum map''

Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?
7
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1answer
408 views

What was Gödel's Constitutional Problem? [closed]

It is well known that Kurt Gödel had doubts concerning the US constitution and believed that it somehow was inconsistent and opened up for a dictatorial grab. What was he thinking?
33
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4answers
4k views

Did Euler prove theorems by example?

In his 2014 book, Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7: Capitolo I Esempi e metodi dimostrativi Introduzione In The Calculus as Algebraic ...
18
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6answers
1k views

Mathematics contests before 1800

Aside from well known examples of mathematics contests in 1535 and 1548, what are some other examples before 1800? Background: In The History of Mathematics: an Introduction, 3rd edition (1995), ...
5
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5answers
582 views

Important results with one or more than one proof [closed]

Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How ...
3
votes
1answer
247 views

Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
7
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0answers
193 views

Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
4
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2answers
229 views

Historical reference request on Nilpotent groups

From Wikipedia: "Abelian groups were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the ...
11
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3answers
2k views

Has Dedekind's proof of existence of infinite sets been analyzed by historians?

This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set. The proof exploits the assumption that there exists a set $S$ of all ...
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0answers
66 views

Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15): Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the Gromov-...
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0answers
60 views

History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of - $\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...
3
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1answer
226 views

Have there been any claims of mathematical breakthroughs while in altered states of consciousness?

This certainly is a related question: Have you solved problems in your sleep? Has anyone seriously attempted to make a similar claim for other altered states, besides dreaming? I know the claim has ...
2
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2answers
371 views

What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper, Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495. the authors give the following quote of Frege, from his paper "&...
0
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1answer
202 views

Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true. Since posting the question, ...
5
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0answers
104 views

Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of non-...
3
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1answer
204 views

Early examples of problems that are easier in high dimension

In many areas of mathematics, there are problems that admit a natural formulation in any dimension. It often happens that such a problem is easier to solve in dimension $n>k$ as compared to ...
9
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0answers
329 views

Why do we study symplectic geometry? [closed]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form? The subject certainly originated from physics, but is there a deeper reason for why it is still an ...
3
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1answer
190 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
23
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3answers
2k views

Nelson's proof of Liouville's theorem

The paper "A proof of Liouville's theorem" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in $\...
3
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1answer
435 views

origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase. where does this ...
0
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2answers
285 views

The Zeta Function Before Riemann [duplicate]

Leonhard Euler studied the function that is now known as the Riemann zeta function. I have not found the notation $\zeta$ in any of the works of any mathematicians prior to Bernhard Riemann's paper On ...
1
vote
1answer
218 views

Have some works by Émile Borel ever been translated from French to English or another foreign language?

I plan to submit a couple of questions around Émile Borel's works in probability theory to MO. In this scope, I'd like to know if the following works have ever been translated from French to English ...
11
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2answers
780 views

Gauss proof of fundamental theorem of algebra

My question concerns the argument given by Gauss in his "geometric proof" of the fundamental theorem of Algebra. At one point he says (I am reformulating) : A branch (a component) of any algebraic ...
-4
votes
1answer
284 views

Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
2
votes
1answer
660 views

Reference for Connes Bourbaki membership or otherwise

Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?
10
votes
1answer
386 views

What was a cusp to Hurwitz in 1892?

Let $d\in\mathbb{N}$ be squarefree. Let $\mathcal{O}_d$ be the ring of integers of $\mathbb{Q}(\sqrt{-d})$. Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$. Let $\mathcal{H}^3$ be the upper half-space ...
2
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6answers
2k views

Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
12
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1answer
553 views

Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...
11
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1answer
1k views

What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
30
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1answer
952 views

Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid)

Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction ...
16
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0answers
400 views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
2
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2answers
212 views

Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta u_{t}+\...
3
votes
1answer
377 views

Who is the original author of this simple paradoxical decomposition?

Paradoxical decompositions of sets usually require the axiom of choice; Hausdorff or Banach-Tarski are well-known examples. A paradoxical decomposition of a point set without the axiom of choice has ...
0
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1answer
474 views

The $\zeta-$word [closed]

I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder ...
10
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2answers
560 views

Who was the first to discover that the curvature of an embedded surface is the product of the principal curvatures?

The invention of intrinsic differential geometry is usually attributed to Gauss in the context of his theorema egregium but the notion of the curvature of an embedded surface existed before. Who was ...
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1answer
185 views

When do Theorems (or Algorithms or Methods) Become Celebrated? [closed]

I recently noticed that certain theorems (e.g. Tutte's 1-factor theorem or, Edmond's Blossom algorithm) are attributed celebrated. A quick search on the internet yields further examples: Carleson'...
10
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1answer
402 views

What are “Artin fractions”?

The German Wikipedia entry for Ernst Witt https://de.wikipedia.org/wiki/Ernst_Witt has a photo of his grave in Hamburg. The bottom part has a visible text "Artin Brueche" (Artin fractions) but the ...
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14answers
6k views

Do mathematical objects disappear?

I am asking this question starting from two orders of considerations. Firstly, we can witness, considering the historical development of several sciences, that certain physical entities "disappeared"...
5
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1answer
131 views

Historical refererences for Castelnuovo-Mumford regularity

Does anyone know a good reference to understand the historical background of Castelnuovo-Mumford regularity? I know the backgound for the modern commutative-algebra approach (using free graded ...
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0answers
167 views

Why the Castelnuovo exact sequence is named after Castelnuovo

I have seen variations of the following exact sequence referred throughout the literature as the Castelnuovo sequence: $$0\longrightarrow \mathscr I_{X:H}(-d)\longrightarrow \mathscr I_X\...