# Questions tagged [ho.history-overview]

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

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### Is spherical trigonometry a dead research area?

When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for ...

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### Relegation of Farkas' fundamental theorem to a lemma

Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of ...

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### British "unrigorous" mathematics work prior to G.H. Hardy

I was looking at a bio-movie of Ramanujan last night. Very poignant.
Also impressed by Jeremy Irons' portrayal of G.H. Hardy.
In G.H. Hardy's wiki page, we read:
. . . "Hardy cited as his most ...

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1
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### History of the characteristic matrix

Let $\mathbb{F}$ be a number field, $A$ and $B$ be two $n\times n$-matrix over $\mathbb{F}$. It is known from some textbook that $A$ is similar to $B$ iff there exists $n\times n$ nonsingular matrix $...

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### Freeman Dyson's approach to string theory [closed]

Context:
In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists [1]:
My dream is that I will live to see the day when our ...

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### Great polyhedra: What does "great" signify?

Great Cubicuboctahedron
Great Icosacronic Hexecontahedron
Great Rhombic Triacontahedron
Great Snub Icosidodecahedron
Great Stellated Dodecahedron
Great Triakis Octahedron
...
There are many polyhedra ...

28
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### 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 ...

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### Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians

I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....

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### Mathematical life of Friedrich Ulmer

In the little galaxy of Category Theory, Friedrich Ulmer is known for being one of the authors of Lokal Präsentierbare Kategorien, a book that laid the foundations for the theory of locally ...

7
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### A lecture by Rudin

Is it available a written version of this lecture by Rudin on the relation between Fourier analysis and the birth of set theory?
https://youtu.be/hBcWRZMP6xs
If not Rudin himself, maybe someone else ...

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### Adrastus, Proclus, and 2+8+50+288+… vs. 1+9+49+289+…

According to the MacTutor essay "D'Arcy Thompson on Greek irrationals" (which I take to be a version of Thompson's original essay whose only liberty with the original text is giving English ...

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### Why is $H$ the standard notation for mean curvature?

I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$.
I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, ...

6
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### What is Euler's method in linear algebra?

I have been interested in the following paper ("On systems of linear indeterminate equations and congruences" by
Henry J. Stephen Smith,
Philosophical Transactions of the Royal Society of ...

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1
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### Who is M. Meyniel?

Does anyone know anything about M. Meyniel? According to zbMath, he published precisely one mathematics paper, in which he gave a sufficient condition for hamiltonicity of digraphs:
"Une ...

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### Motivation behind the Bohr-Mollerup Theorem relating the Gamma function

In Wikipedia, it states about the Bohr-Mollerup Theorem:
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
If anyone knows, ...

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### Who introduced the discrete Fourier transform?

I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of ...

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### Restricted sumsets - the origins?

The sumset of the subsets $A$ and $B$ of an additively written group is defined by $A+B:=\{a+b\colon a\in A,\ b\in B\}$. The basic idea to add sets has been around since Cauchy at least.
Erdős and ...

62
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1
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### Identity of J. L. Rabinowitsch (of Rabinowitsch Trick)

For some time, it seemed widely accepted that G. Y. Rainich was the author of the note Rabinowitsch, J. L., Zum Hilbertschen Nullstellensatz., Math. Ann. 102, 520 (1929). JFM 55.0103.04., which ...

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1
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### Hensel's proof that $e$ is transcendental

When he introduced $p$-adic numbers, Kurt Hensel produced an incorrect local/global proof of the fact that $e$ is transcendental. Apparently, the intended proof goes along the following lines: ...

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### First uses of the term "crackpottery" in mathematics [closed]

I first stumbled over the term crackpottery on MO and as I am not a native English speaker, I had first assumed that the meaning were pots for making crack, i.e. the drug of the name crack.
Question:
...

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### Archiving mathematical correspondence

What are great examples of comprehensively archived mathematical correspondence (including both handwritten and electronic items)?
Context: polished papers usually don't reveal the full process that ...

4
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1
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### Examples of rich theories that started out as an infinite-dimensional inquiry

It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it ...

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### Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?

Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...

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### Modern results that are widely known, yet which at the time were ignored, not accepted or criticized

What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It ...

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2
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### Early successes of Schwartz distribution theory

What are the early successes of Schwartz distributions theory?
What are the hard theorems that became simple and what
open problems were solved with this new tool soon after Laurent
Schwartz released ...

1
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1
answer

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### Church-Turing tests and quasi-computational models [closed]

What came to mind intuitively is what I would call C-T tests that are more or less methods of accepting some model as being a computational model or not. The question is in what amount and how could ...

12
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1
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### Who introduced the notation for $\beth$ numbers and when?

Georg Cantor, when developing the basics of set theory, noted that there are two ways to increase cardinality: power sets and successors (or, in modern terms, the Hartogs operation).1
Eventually the ...

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2
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### M. M. Artyukhov / М. М. Артюхов

Does anybody know any biographical information about М. М. Артюхов (e.g., first name, affiliation)?
It seems he discovered a criterion for primality equivalent to the Solovay–Strassen one in 1966, in ...

55
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10
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### Books/websites which have motivating stories of mathematicians overcoming hardships in life

Edit 1: I have received a lot of great answers. I am not accepting any answer because I think there might be in future that some user want to contribute any new answer, as in my opinion some users ...

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1
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### Who introduced nerves in category theory?

Who was the first to consider that categories were semi-simplicial sets (and in particular groupoids were simplicial sets)?
I think there was a concept of nerve of a covering in algebraic topology ...

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0
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### History of Laplacian comparison theorem

The Laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. ...

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### Karatsuba photo [closed]

Can anyone confirm if the following link displays a photo of A. A. Karatsuba?
https://commons.m.wikimedia.org/wiki/File:A.A.Karatsuba_in_Crimea.jpg

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### Tohoku and cohomology of toposes

In McLarty's The Rising Sea: Grothendieck on simplicity and generality I found the following quote:
The same, Grothendieck knew, would work for cases yet unimagined. He notes that Tohoku [...

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### List of problems that Erdős offered money for?

Is there a list somewhere of all the problems that Erdős offered cash awards for, including both solved and unsolved problems? One would think that the answer is yes, but so far I have had no luck ...

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### The history of suprema involving parameters

I had the following historical question (with implications for logic/reverse math):
Who first used the notion of supremum explicitly involving parameters?
Let me provide a positive example of the ...

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1
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### Gauss-Bonnet Theorem: Neither Gauss nor Bonnet [closed]

Tristan Needham says (p.174),*
"While Gauss and Bonnet certainly paved the road to [the Gauss-Bonnet Theorem],
neither one of them was even aware of this extraordinary result, let alone stated ...

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### Attribution of the quote "a mathematician is someone who is cautious in the presence of the obvious" [closed]

A few years ago I came across a quote attributed to a well-known mathematician: "a mathematician is someone who is cautious in the presence of the obvious". I really like this quote but I ...

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### Are there mistakes in the proof of FLT?

This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to ...

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### When did we stop the challenges between two mathematicians? [closed]

In this video(*) of Veritasium, you can see a challenge(**) between two mathematicians : Tartaglia and Fior, during the Renaissance in Italia.
When did we stop the challenges(**) between two ...

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0
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### On the (Brouwer-Koebe) Continuity Method

The so-called Continuity Method is a simple yet powerful method to show that a given continuous injective map is surjective. Namely, suppose that $f:X \to Y$ is a map between two connected manifolds $...

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### Shing-Tung Yau's doubts about Perelman's proof

[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.]
According to Wikipedia, Shing-Tung Yau expressed some doubts about Perelman's proof of ...

4
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1
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### Hegel's disproof of Newton [closed]

I know it's not a very comprehensive question but I've nowhere else to ask. A friend relayed to me a portion of a book from Hegel where he seemingly disproves Newton's way of finding a differential. I ...

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### An historical mathematical result-paradox

If you feel like this, please post an information about a mathematical result that satisfies the following conditions to its best:
the result's authorship is well documented and free of controversy (...

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### Mathematical fictionalism

Have there been any successful mathematicians that also happen to be mathematical fictionalists? Let's say success is defined by at least one article published in a non-pay journal.
I ask because ...

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### What was supposed to appear in EGA after Chapter IV?

We find a nice table on the Wikipedia page mentioning for instance that abelian schemes were supposed to be discussed in Chapter XII.
Did anyone involved in EGA say anything more detailed, verbally or ...

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### Reference request: a tale of two mathematicians

I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so:
When P. Gabriel presented the theorem in a conference [sometime around ...

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### Taking a theorem as a definition and proving the original definition as a theorem

Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are ...

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### Known and fixed gaps in the proof of the CFSG

As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this ...

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### Can the theory of elliptic functions developed from purely geometric considerations?

I always had this question, but was unable to get a definitive answer to it.
There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...

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### When did Grothendieck join Bourbaki? [closed]

Bourbaki listed Grothendieck as a third-generation member. Nevertheless, it does not provide details on when he joined and when he left.
Concerning his departure, there is a
Letter from October 9, ...