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

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12
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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 $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
29
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1answer
857 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 ...
14
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0answers
318 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
129 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 ...
3
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1answer
353 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
396 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
512 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 ...
-4
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1answer
155 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: ...
10
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1answer
376 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 ...
39
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14answers
5k 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 ...
5
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1answer
115 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
161 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 ...
24
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5answers
2k views

History of Mathematical Notation

I would like to see a simple example which shows how mathematical notation were evolve in time and space. Say, consider the formula $$(x+2)^2=x^2+4{\cdot}x+4.$$ If I understand correctly, Franciscus ...
12
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1answer
738 views

Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...
3
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2answers
254 views

Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?
4
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1answer
194 views

Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as: Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially? ...
6
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1answer
147 views

How to divide a square into three similar rectangles

Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles? With a bit of algebra it can ...
9
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1answer
375 views

Examples of abstractions that did *not* turn out to be useful [closed]

I’ve read (but cannot find any reference now) that new abstract mathematical concepts like set theory and – not too long ago – category theory were in their time often considered too abstract to be ...
70
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30answers
10k views

What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here. My concern in this question is slightly ...
53
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3answers
4k views

What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem): ...the end goal [is] to establish as ...
11
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1answer
430 views

Two Vinogradovs? Is one the son of the other? [closed]

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming ...
16
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2answers
408 views

Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...
11
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2answers
559 views

First formulation of the Dedekind and Hasse-Weil conjectures

I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated: ...
35
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4answers
2k views

Hilbert's (cancelled) 24th problem

Hilbert's 23 problems, ten of which were presented at the 1900 ICM in Paris, are too famous for any mathematician to not know. If one reads the descriptions of the problems in Hilbert's paper, one ...
2
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0answers
116 views

When did mathematicians begin to use the letter x to denote unknown values? [duplicate]

When did mathematicians begin to use the letter x to denote unknown values ? Gérard Lang
22
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1answer
724 views

How and why did mathematicians develop spin-manifolds in differential geometry?

First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...
11
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1answer
305 views

'Updated' book in the same spirit as Dieudonné's Panorama des mathématiques pures

Today a colleague of mine asked me if I knew of any "more modern version" of J. Dieudonné's Panorama des mathématiques pures. Le choix bourbachique. The very first thing that instantly came to my ...
6
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1answer
162 views

History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...
18
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2answers
534 views

History of set-class distinction

I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class: Who was the first mathematician to make this ...
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0answers
222 views

Why do some people adamantly insist on 'toposes' instead of 'topoi'? [closed]

I've heard that several category and topos theorists, first and foremost Johnstone (see the comments to this question) adamantly insist on 'toposes' as the plural of 'topos'. I was wondering whether ...
6
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1answer
739 views

Windows into new mathematical worlds [closed]

Yitang Zhang's Annals of Mathematics primes-gap result opened a new window, which Polymath's reduction from $70\times 10^6$ to $246$ attests. Perhaps Harald Helfgott's celebrated proof of the odd ...
11
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2answers
782 views

Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?
8
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0answers
215 views

On an unpublished result of Magidor

In 1970th, Magidor proved the following important results: (1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and ...
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0answers
126 views

Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...
43
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1answer
3k views

What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
7
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0answers
182 views

$\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$ Question 1. Who first introduced the above question, and where can I find ...
0
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2answers
345 views

Linear Algebra classic books [closed]

I'm learning linear algebra at the moment, so I'm looking for some great old classic books. Something like Fermat's or Gauss books of some great mathematians. I don't really like the nowadays books ...
8
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1answer
446 views

Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?

A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already ...
19
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2answers
1k views

History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to ...
22
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0answers
502 views

Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson ...
15
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1answer
1k views

The list of problems for Grothendieck's thesis

Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...
4
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1answer
197 views

Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...
20
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1answer
966 views

What is $\infty^6$?

The title of this question may make you want to close it immediately, but bear with me a moment. In several older mathematics papers (early 20th century) I have seen statements such as The ...
1
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0answers
128 views

First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...
14
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2answers
834 views

Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus ...
44
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4answers
2k views

How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...
16
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1answer
384 views

What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
1
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1answer
169 views

Non-Pythagorean proof for the square root of 2 and solution to YBC7289 [closed]

My name is J. Frederic Teubner I am an independent researcher. I wish to publish a proof for the non-Pythagorean solution to the Babylonian tablet YBC7289 and am currently inquiring as to whether or ...
6
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1answer
413 views

Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?
2
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1answer
192 views

What kind of role has Functional Analysis played in Signal Processing? [closed]

Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?