# Questions tagged [ho.history-overview]

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

1,411
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### Diagrammatic representation of sets as irregular plane figures

I am trying to find the earliest use of plane diagrams of various shapes for representing sets. For example, this is snapshot is from the book called Some Modern Mathematics for Physicists and Other ...

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### Etymology of “real numbers"

I would like to know why the real numbers are called “the real numbers.” I would also like to know the meaning of “real” in the phrase “real number.”
Further questions and clarifications:
I’d like to ...

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### Hilbert's and Gödel's expanded definition of "Recursive Function"

There is a very interesting comment in this post:
I must also make one terminological caveat: Hilbert, and later Godel, used the phrase "recursive function" in a way very different from the ...

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### Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?

$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...

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### Curiosity about "conditional trig identities"

Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...

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### Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?

Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise.
In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in ...

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### What's the earliest result (outside of logic) that cannot be proven constructively?

Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law ...

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### History of knot enumeration tables

There is much arbitraryness in the Rolfsen (and later) tables.
Of course anyone would name $7_1$ to be the first knot with
$n=7$ crossings, but already my own "natural" ordering attempt
(...

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### About the exact origin of a binomial congruence

Given a prime $p$ and an integer $0 \leq k \leq p-1$, a famous congruence on binomial coefficients states:
$$\binom{p-1}{k} \equiv (-1)^k \pmod{p}$$
It is generally taught as a consequence of Pascal’s ...

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### Connection of principal fiber bundles — history

I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the ...

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### What was the "stormy discussion" about differential Galois theory at IHES?

In Kazuo Okamoto and Yousuke Ohyama's paper "Mathematical works of Hiroshi Umemura", Annales de la faculté des sciences de Toulouse Mathématiques, XXIX, no. 5 (2020) pp. 1053-1062, there is ...

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### What is the status of the theory of motives?

It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.
But what is the ...

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### Minkowski problem for polytopes: the origin of necessary condition

Minkowski's uniqueness theorem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets.
Theorem (Minkowski). Let $A_i$ be positive faces areas ...

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### How are Lie groups and polynomial resolvents related?

I came across the following sentence in Stevenhagen and Lenstra's wonderful little article Chebotarëv and his density theorem:
Nikolai's interest in [polynomial] resolvents led him to study Lie ...

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### Video abstracts for mathematical papers

I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...

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### Geometric construction of real root of quintic using marked ruler and compass

My question is motivated by a geometry problem about special folded rectangle:
'A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order ...

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### Meaning of a result of Gauss on "Mensura" of cyclotomic numbers

(This question was asked before on mathstackexchange. I received a few useful comments there, which helped me answer it for a special case, but I did not succeed in proving the general case.)
In an ...

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### ID needed for one mathematician in group photo

The photo below was taken at MSRI in 1984 and MSRI has asked me to try to find out (on behalf of Lou Kauffman, Sofia Lambropoulou and Martha Jones) the identity of the mathematician farthest left, in ...

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### Explicit character tables of non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...

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### Is GCH useful in proving theorems?

By GCH I mean the Generalized Continuum Hypothesis. Let me give some context before presenting my question.
When the axiom of choice was introduced by Zermelo in his 1904 proof of Well-Ordering ...

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### When has the scaffolding been more important than the completed building?

Niels Abel once said(1) of Gauss, "He is like the fox, who effaces his tracks in the sand with his tail." to which Gauss replied, "No self-respecting architect leaves the scaffolding in ...

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### "On models of elementary elliptic geometry"

While perusing p. 237 of the 3rd ed. of Marvin Greenberg's book on Euclidean and non-Euclidean geometries, I learned that it can actually be proven that "all possible models of hyperbolic ...

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### History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi

In 1774 Laplace understood that $I≔∫\textrm{d}x \exp kf(x)$ for $k≫0$ can be estimated if one knows 2-jet of $f$ at its point of maximum (as $I₀ ≔ ∫\textrm{d}x \exp kf₀(x)$ with $f₀$ quadratic with ...

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### Who introduced the concept of beyond planar graphs?

The concept of planar graphs seems to be standard (I'm also not sure who first used this term), and recently, beyond planar graphs attract a lot of interest in the field of graph drawing. I know that ...

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### History of right hand rule

I am not sure if this is the right place to ask, but many mathematicians are knowledgeable and interested also in history of math, so here I am.
I am curious to know when the right-hand-rule for ...

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### How did the term "space" in mathematics started to be understood as a set with a structure?

In mathematical literature, the term 'space' is often used to describe a set endowed with additional structure, such as a metric space or a vector space. What is the historical evolution of the ...

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### Compare with Weber and Hilbert class field

Heinrich Martin Weber and David Hilbert created their own class fields in 1891 and 1897 respectively.
In the past, Weber continued to name $K={Q}(\sqrt{-m}, j(\omega))$, the Kronecker class field of $...

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### History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring

This is a repost. So far, I've received no answers on HSM Stack Exchange; maybe I do in MO.
In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (for a ...

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### Grothendiecks's lectures on Kohärente Garben und verallgemeinerte Riemann-Roch-Hirzebruch Formel

In his biography of Hirzebruch in Jahresber Dtsch Math-Ver (2015) 117:93–132, Zagier says that
[T]he dominating event [of the first Arbeitstagung in 1957] was unquestionably Grothendieck’s lecture ...

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### Who introduced the term hyperparameter?

I am trying to find the earliest use of the term hyperparameter. Currently, it is used in machine learning but it must have had earlier uses in statistics or optimization theory. Even the multivolume ...

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### references about the state of algebraic geometry between 1930 and 1955?

I asked the question a few weeks ago in another form, but did not have any answers nor comments. So I will try again another way: could you recommend bibliography concerning the state of algebraic ...

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### History of Gauss theorems that say "it clearly follows that" but it did not clearly follow

I studied mathematics fifty years ago. I have forgotten much of what I learned. there is an anecdote however that one of my maths lecturers told that I would dearly love to know more about. He stated ...

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### Mumford–Tate 1962 "Algebraic geometry seminar" citation

In FGA 3.V, there is a citation for
Mumford D. and Tate J., Séminaire de géométrie algébrique, Harvard University, Spring term 1962 (à paraître).
This seems to be the same seminar mentioned by ...

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### Who first introduced the term "categorical group", and when?

The term "categorical group" is often used to mean a group object in Cat; these days we also call such a thing a strict 2-group. Who first introduced the term "categorical group", ...

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### Who was the first to propose a formal definition of infinity?

In a recent Quanta article about the 9th Dedekind number, Dedekind is credited with the first formal definition of infinity.
Is this an accurate attribution?
Folklore where I’m from dictates that ...

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### Discovery of Hilbert polynomial

Presumably it was Hilbert who discovered Hilbert polynomials - where did they first appear?
The basic theorem is that for a finitely generated graded module $M = \bigoplus_k M_k$ over the ring of ...

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### 1958 Lang's book on algebraic geometry

Is the famous 1958 Lang's book "introduction to algebraic geometry" a reasonable account of the pre-Grothendieck era of algebraic geometry ?
Or, in the contrary, does this book present a ...

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### On the origin of power semigroups

Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty ...

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### Questions on Gauss's geometric interpretation of spherical functions

(This question was initially posted on HSM stackexchange, but eventually I came to conclusion that it is too mathematical to be answered there.)
In the physics chapter of his biography of Gauss, W.K. ...

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### Hilbert's approach to Riemann hypothesis using Fredholm's work:

I read somewhere that Hilbert wanted to prove Riemann hypothesis using Fredholm's work on integral equations but I can't find anything online.
Can someone provide historical references for it?
What's ...

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### Locating historical sources showing diversity in mathematics

I am helping my colleague, Brigitte Stenhouse, in a piece of work on “contextualizing the curriculum”. We are looking to assemble a collection of mathematical sources which showcase diversity in ...

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### (Obscure) areas of mathematics that are largely inactive or forgotten today? [duplicate]

I am looking examples of a mathematical theory (i.e. a body of knowledge, with its own definitions, results, principles etc., i.e., its own language) that is completely inactive or forgotten by today.
...

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### Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version
The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...

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### Referring to the countability of $\Bbb Q$ as "Cantor's first diagonal argument"

I had a discussion with one of my students, who was convinced that they could prove something was countable using Cantor's diagonal argument. They were referring to (what I know as) Cantor's pairing ...

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### What causes some mathematical journals to discontinue?

This is soft question. I decided to ask my question on MathOverflow rather than on academia StackExchange because I believe that the community here is more equipped to answer the question.
The ...

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### What is the history of the term "faithful functor"?

Is it known who coined this term and what he meant? By comparison, the association between "full" and "surjective on $\mathrm{Hom}$" doesn't sound so cryptic. (I understand, of ...

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### Original references for the Hall - Witt identity

The group identity
$$
[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}],b]^a = 1
$$
is commonly attributed to Hall and Witt (here $x^y:=y^{-1}xy$ and $[x,y]:=x^{-1}y^{-1}xy$). However, ...

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### Nash–Moser–De Giorgi differences

The names of Nash, Moser and De Giorgi are associated to elliptic and parabolic regularity theory.
But what are the differences in the approach between the three contributions? Can you briefly sketch ...

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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-...