# Questions tagged [ho.history-overview]

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

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### What's the unspoken history of compactly generated topological spaces?

Usually, the alleged motivation for the definition of compactly generated topological spaces is Cartesian closedness, which fails for general spaces. Of course, from a contemporary perspective, this ...

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### Mention of Bernoulli principle by Bill Lawvere

In the Author Commentary to the reprint of the paper paper Diagonal Arguments and Cartesian Closed Categories in Theory and Applications of Categories Bill Lawvere wrote:
Although the cartesian-...

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### History of algebraic geometry over finite fields

My question is of historical nature: when did mathematicians start studying algebraic geometry over finite fields in a systematic way, and who were the main driving forces ?
Did it start with Weil (...

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### Glauberman-Thompson normal $p$-complement theorem for $p=2$

I asked this question on Math StackExchange yesterday. As suggested by Professor Derek Holt, this question may be more suitable for this site. So I ask this question here again, but more details and ...

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### Question on a crucial lemma in Euler's approach to Fermat's Last theorem for $n=3$

As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...

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### Who introduced the notion of ringed spaces?

My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...

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### Examples of bad notation and its consequences [closed]

An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary ...

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### Who first proved that algebraic numbers form an algebraically closed field?

I am interested in the history related to algebraic numbers and have two questions:
Who first proved that algebraic numbers form a field?
Who first proved that algebraic numbers form an algebraically ...

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### Proof of an asymptotic formula by Tricomi

Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it.
QUESTION:
Let $P_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ ...

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### A metric characterization of Hilbert spaces

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...

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### The place and year of birth of Henry Maurice Sheffer

I do not know whether this question (in history of math) is proper for MathOverflow, but I know no other places where it can be asked with a hope to obtain an answer.
Reading the biography of Henry ...

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### Phenomena of topos

These days I am wandering on a wild adventure in an incredible but intimidating land. Fortunately, I could find a guide to some animals of this land
Phenomena of gerbes
But someone said to me that ...

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### What is an important mathematical question?

$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste....

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### Notations for open and closed sets

I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.

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### Comparative analysis of history of mathematics

I am a bit scared about writing this question because I am unsure if it is appropriate. However, here it is.
Is there anything written about the history of mathematics from a comparative or (post)...

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### A zoo of derivations

Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...

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### Terminology associated with mathematical induction

In "Number: The Language of Science" (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as "the induction step" (and refers to what we call the ...

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### Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences?

Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences?
Background for the question:
Park, M., Leahey, E. & Funk, R.J. Papers and patents ...

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### Reference request for recurrence relation of division polynomials

The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...

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### Is there another controversial statement by Grothendieck apart from 57 being prime?

There is a well-known story about Grothendieck being asked to explain concretely some result involving prime numbers and of his answering "You mean an actual number? All right, take 57".
...

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### Etymology “Kulkarni–Nomizu product”

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...

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### History of tropical mathematics

This is a follow-up to this question about the origin of tropical mathematics.
Are there any articles, websites or books which deal with the history of tropical mathematics?
I have been trying to find ...

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### Origin of tropical mathematics

On Wikipedia, it is claimed without a source that Imre Simon founded tropical mathematics.
The first work of his I was able to find on the subject is Limited subsets of a free monoid which uses the ...

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### The origin and use of the term "equianharmonic" (elliptic function)

This question has been posted on History of Science and Mathematics stack exchange, but there was no answer or comments there.
In Weierstrass notation, the principal elliptic function $\wp$ is a ...

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### Origin of 'Analytic' Geometry?

My impression is that the name analytic geometry, which I understand roughly to be geometry in Euclidean space using coordinates, is not used that much anymore. We would probably classify the subject ...

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### Landau's century-old problems: Anything comparable?

Landau's four problems
are now over a century old (1912), and each still unsolved.
This seems remarkable, even though he was not the originating author all four
(maybe only the 4th?). Still, he ...

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### Bourbaki-Witt in a textbook, other than in logic?

The Bourbaki-Witt theorem states that, in a chain-complete poset, the subset $X$ generated by an inflationary monotone function $s$ from the least element and joins of chains satisfies
$$ \forall x,y\...

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### What is the origin/history of the following very short definition of the Lebesgue integral?

Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...

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### True origin of the term "Spline"

In mathematical contexts the term spline essentially refers to interpolating or approximating piecewise functions with continuity constraints.
According to the history of mathematical splines
In the ...

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### Cartier and the continuity of the early history of schemes

If you allow me I would divide the early history of schemes this way
_ Weil, Zariski, Bourbaki, Nagata, Van der Waerden,... up to Chevalley (you can find an interesting blog here)
J P Serre varieties ...

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### Seeking identity of mathematician in photo on the cover of "The Honors Class" by Ben Yandell

All of the photos on the cover of Ben Yandell's book The Honors Class appear in the book, except the one in the upper left. I'd wager a beer that the mathematician in the upper left corner is A.N. ...

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### Papers of the masters translated to English in one location

Surely someone has collected these papers and translations and has them in a single location for download? For music there is musipedia. Surely there is a mathepedia equivalent?
If I search gauss ...

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### Origin of $L$ in $L^1$ and $L^2$ norms

In the German Wikipedia entry for $L^p$-Raum it is stated (Link)
Das $L$ in der Bezeichnung geht auf den französischen Mathematiker Henri Léon Lebesgue zurück, da diese Räume über das Lebesgue-...

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### Examples of mathematical work that gained recognition after it was outlined by journalists

Having a background in both mathematics and journalism, I'm interested in examples of previously barely recognized mathematical achievements that received recognition after having been given attention ...

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### Reference request: Leonardo Da Vinci's supposed math results

Many reputable sources (I can give as many as you want) describe Da Vinci as a mathematician, but they never mention a single theorem, result, or lemma that he proved. There's the golden ratio spiral, ...

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### Did Grothendieck and Hörmander ever meet?

I carelessly announced a seminar talk Grothendieck meets Hörmander in which I try to explain that Grothendieck's early work in functional analysis could have had a bigger influence on Hörmander's ...

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### Examples of "proof by generalising" [duplicate]

In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?).
Are there any compelling examples where it is significantly "easier"/"simpler" to prove ...

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### History of use of "=" symbol to mean "is canonically isomorphic to"

Let $A$ be a commutative ring, and let $f$ and $g$ denote elements of $A$ such that the prime ideals of $A$ containing $f$ are precisely the prime ideals containing $g$ (a not completely trivial ...

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### Best sources for the history of prize money for open mathematical problems

I'm a science/math journalist [ger] and currently I'm working on an article about prize money for open problems (Millennium Prize Problems and such). One section will be about the history of prize ...

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### Why descend a representation (of a finite group) over $K$ to a representation over $k$ with $k$ a subfield of $K$ is useful?

I heard that Schur was trying to answer the following question
Given a representation of a finite group $G \overset{\rho}{\rightarrow} \operatorname{GL}_{n}(K)$, how to find the smallest subfield $k$ ...

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### History of (proposal of) set-theoretic foundations

It is often said that set theory is the de facto foundation of mathematics. Regardless of the truth of this claim, this seems to be the story told to students (and mathematicians) who poke their ...

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### Was failure of choice proved consistent with Zermelo or MacLane or TST before forcing?

It was forcing that first proved that there are models of $\sf ZF$ which in which $\sf AC$ fails.
Was that result known for $\sf Zermelo$ or $\text{MacLane Set Theory}$ or for $\sf TST$ before forcing?...

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### Where did the military money go?

In older papers, one sometimes finds references to sources of funding directly linked to or overseen by military agencies. For example, I have memories of seeing acknowledgments to DARPA funding in a ...

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### Where can I find that Weil suggested a cohomology theory for characteristic $p>0$?

I have seen that in Grothendieck's paper "THE COHOMOLOGY THEORY OF ALGEBRAIC VARIETIES", he says "The need of a theory of cohomology for 'abstract' algebraic varieties was first ...

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### How to prove Gauss's identities on the action of the operator $f' = x\frac{df}{dx}$ on Jacobi theta functions?

(I have previously posted this question on mathstackexchange, but after getting no response there I decided to ask it again here. Here is a link to the same question there, so if this question is not ...

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### Algebraic geometry over the complex numbers, and beyond

My question basically is very simple: when did mathematicians start to do algebraic geometry "outside the complex numbers" ?
In the old days, algebraic geometry was solely done over the ...

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### Motivating unpublished statements of Gauss about congruences and quaternions

Background
Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this ...

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### History of points of view on Eisenstein series

What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in ...

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### Why does Arnold put Hardy on the same level as Bourbakists?

In the preface to his book "Lectures On Partial Differential Equations" Arnold writes:
The effort to destroy this unnecessary scholastic pseudoscience is a natural and proper reaction of ...

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### Formalisation of intuitive concepts in the language leading to mathematical progress

In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...