# Questions tagged [ho.history-overview]

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

1,125
questions

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### history of geometric mechanics

I was thinking about the foundations of geometric mechanics and its precursors.
I wondered who was the first to realized the equivalence between Riemannian geometry and Lagrangian mechanics. In ...

**4**

votes

**1**answer

116 views

### Is this elementary formula for the parabolic segment new?

Recently (May 2020) a formula for the area of the parabolic segment (i.e. the region enclosed by a parabola and a line), in terms of the coefficients of the Cartesian equations, has been published by ...

**19**

votes

**1**answer

571 views

### Digitalized version of “Cours de topologie algébrique professé en captivité”

It is historically known that Jean Leray gave a course on algebraic topology while captive in the Officer's detention camp XVI in Edelbach, Austria during WW2. (References to this topic ...

**28**

votes

**3**answers

4k views

### Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation
$$
y^2 + y = x^3 - x^2.
$$
My guess is that there is some problem ...

**6**

votes

**2**answers

316 views

### Unknown work of Nöbeling on topological/Hausdorff dimension

Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$.
A well known result of
Szpilrajn (He changed his name to ...

**11**

votes

**0**answers

149 views

### Galois group of polynomials related to Fibonacci and Catalan numbers

Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers.
Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$.
For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$.
And another ...

**6**

votes

**1**answer

402 views

### Origin of the term 'index of a subgroup'

The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$.
Why did someone decide to call this an 'index'? What's the rationale for this?

**31**

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3k views

### The origin(s) of the word “elliptic” [migrated]

The word elliptic appears quite often in mathematics; I will list a few occurrences below. For some of these, it is clear to me how they are related; for instance, elliptic functions (named after ...

**18**

votes

**1**answer

594 views

### Why 'excedances' of permutations? [closed]

For a permutation $\pi=\pi_1\pi_2\cdots\pi_n$ written in one-line notation, an index $i$ for which $\pi_i > i$ is usually called an 'excedance.' To me, this seems like a mispelling of what should ...

**6**

votes

**0**answers

134 views

### Two questions on history of symplectic geometry in the 80's

I have a question about the history of two important results from the eighties in symplectic geometry. In both cases it seems that important results were developed (almost) simultaneously by ...

**2**

votes

**0**answers

112 views

### Why is faithful actions called faithful and who first called it faithful?

Sorry for this question. I asked this on MSE and hsm but no one answered and I decided to post it here that is full of experts.
I want to know why is faithful actions called faithful and who first ...

**3**

votes

**0**answers

91 views

### Pyramids whose volume can be computed by simple cutting and glueing

Since this question remained without answers even after a bounty, I thought it might be time to ask it here.
For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn ...

**14**

votes

**1**answer

506 views

### History of the notion of irreducible representation

I am looking for the earliest references where the study of irreducible representations appears. There has been many articles and books on the history of representation theory. A fundamental feature ...

**6**

votes

**1**answer

220 views

### Why are orthogonal matrices so often denoted $Q$?

I apologize for the stupid question in the title. Of course, we can baptize a particular given matrix as we want but, for example, the QR-decomposition has a fixed meaning.
My humble guess is that ...

**25**

votes

**3**answers

1k views

### What aspects of math olympiads do you find still useful in your math research?

I was rereading the book Littlewood's Miscellany and this passage struck me:
It used to be said that the discipline in 'manipulative skill' bore
later fruit in original work. I should deny this ...

**28**

votes

**2**answers

2k views

### Who introduced direct limits?

The general notion of a direct limit of a commuting system of embeddings, indexed by pairs in a directed set, has seen heavy use in set theory. It is the same notion as in category theory. I was ...

**4**

votes

**2**answers

136 views

### Function of moderate growth: history, motivation, and uses

I recently came across functions of moderate growth via this post and I was wondering, what are some concrete uses or applications of this space? Where does it appear and why was it introduced ...

**54**

votes

**4**answers

5k views

### Bourbaki's definition of the number 1

According to a polemical article by Adrian Mathias, Robert Solovay showed that Bourbaki's definition of the number 1, written out using the formalism in the 1970 edition of Théorie des Ensembles, ...

**13**

votes

**1**answer

350 views

### History of well founded relations

I have changed the title in the hope of attracting the attention of someone who knows about the history of set theory as well as intuitionistic logic:
Who was the first to state the definition of ...

**203**

votes

**34**answers

27k views

### Conway's lesser-known results

John Horton Conway is known for many achievements:
Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-...

**4**

votes

**2**answers

380 views

### Earliest use of deconvolution by Fourier transforms

From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...

**6**

votes

**2**answers

599 views

### How did Hilbert prove the Nullstellensatz?

All of the many proofs of the Nullstellensatz I have seen use results from long after Hilbert’s time: Zariski’s lemma, Noether normalization, the Rabinowitch trick, model theory, etc. How did Hilbert’...

**12**

votes

**1**answer

496 views

### Source of a quote by Ferdinand Rudio

I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...

**11**

votes

**1**answer

330 views

### Aleksandrov's proof of the second order differentiability of convex functions

Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...

**1**

vote

**0**answers

175 views

### Why is the exterior algebra symbol a Lambda? [closed]

My guess is that this is because the capital lambda looks like a wedge. In this case why is the symbol a wedge.
Does anyone have any sources for this?

**3**

votes

**1**answer

52 views

### Uniqueness constraints for Delaunay triangulation

Commonly the assumption that is made on point sets that shall be Delaunay-triangulated is that no three are collinear and no four are cocircular.
Those assumptions are however too restrictive: if ...

**2**

votes

**1**answer

104 views

### History- calculating convolution by tabular method

I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1
Basically, ...

**3**

votes

**0**answers

125 views

### Nascent formal group law

The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal compositional inverse, perhaps ...

**-2**

votes

**2**answers

210 views

### Early examples of proof appraisals [closed]

What are the earliest known examples for attributing proofs as 'deep', 'elegant' or 'beautiful' (or their equivalents in other languages)?
Gauß for example called one of his results 'remarkable' ...

**4**

votes

**1**answer

198 views

### Earliest reference on the calculation of derivatives by Fourier transform

I was looking for an earliest reference or the name of the mathematician who showed calculating the derivatives is possible in the Fourier domain?
The Fourier transform of the derivative is (...

**67**

votes

**9**answers

17k views

### Nontrivially fillable gaps in published proofs of major theorems

Prelude: In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem, as presented in Nash's well-known paper "The Imbedding ...

**24**

votes

**5**answers

3k views

### Examples of incorrect arguments being fertilizer for good mathematics? [duplicate]

Sometimes (perhaps often?) vague or even outright incorrect arguments can sometimes be fruitful and eventually lead to important new ideas and correct arguments.
I'm looking for explicit examples ...

**7**

votes

**0**answers

125 views

### Importance of textbooks in health of a sub-discipline

I am interested in published articles, and also more informal writing (blog posts, talk slides etc.) which discuss the importance of textbooks (where this word encompasses research monographs etc.) in ...

**2**

votes

**0**answers

152 views

### What do you call an object constructed as part of a proof? (Terminology)

I find myself wanting to talk about parts of a proof, e.g. the role played by mathematical expressions within a proof.
When proving a theorem it is common to construct some kind of object and then ...

**3**

votes

**1**answer

336 views

### Motivating the coefficient field of $\ell$-adic cohomology

It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures.
It was also ...

**6**

votes

**0**answers

144 views

### Hahn's approach to Hilbert's 17th problem?

The Wikipedia article on Hahn Series mentions that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".
Is this correct? If so, what was this approach, and where can I ...

**53**

votes

**60**answers

10k views

### Mathematicians with both “very abstract” and “very applied” achievements

Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant ...

**0**

votes

**0**answers

53 views

### First appearance of Euclidean algorithm with Euclidean divisions

When was first described the Euclidean algorithm using Euclidean divisions?
In Euclid's original presentation¹ in his Elements, each step consists in repeated subtractions, while the modern ...

**7**

votes

**1**answer

422 views

### Why was the factor $\frac12$ introduced in the Riemann $\xi$ function?

The factor $\frac12$ in the Riemann $\xi$ function:
$$\xi(s)=\frac12 s(s-1)\,\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)$$
was introduced by Riemann, however appears to be redundant. Once he had arrived at:
...

**26**

votes

**6**answers

3k views

### Any real contribution of functional analysis to quantum theory as a branch of physics?

In the last paragraph of this last paper of Klaas Landsman, you can read:
Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...

**4**

votes

**4**answers

450 views

### Reference request: Oldest books on logic with unsolved exercises?

Per the title, what are some of the oldest books on logic out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
Update: Doesn't have to be ...

**72**

votes

**10**answers

10k views

### What are examples of (collections of) papers which “close” a field?

There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
A total characterisation,...

**3**

votes

**0**answers

277 views

### Understanding a part of Friedberg’s Priority Argument Paper

This is Richard Friedberg’s original 1957 proof of the Friedberg-Muchnik Theorem, the origin of the ground-breaking priority argument. (The result proven is that there are two recursively enumerable ...

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votes

**2**answers

1k views

### How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that ...

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**2**answers

1k views

### Origin of the noun “mathematician” [closed]

I have read that Pythagoras's fraternity had two kinds of members, the 'acousmaticians', who were allowed to attend the lectures, and the 'mathematicians', who had been initiated. Is this the origin ...

**3**

votes

**0**answers

83 views

### Terminology for set systems: “trace” or “projection”?

Although the following question is not in itself mathematical, it is the expertise/breadth of the research community in mathematics that I wish to appeal to, beyond the filtered/trained search results ...

**6**

votes

**0**answers

223 views

### Brief history of primality testing theory after 2002?

Its clear that there is about 15 years (2004-2019) after the publication of AKS primality testing in 2002 and its modifications in 2003-2004. AS result, is there any development happened in this ...

**3**

votes

**1**answer

572 views

### Why Bourbaki's epsilon-calculus is not suitable for set theory?

Does anybody shed light on what is A. R. D. Mathias' idea that Bourbaki's $\tau$-calculus (Logically the same as Hilbert's $\varepsilon$-calculus) is not suitable for set theory, especially because of ...

**30**

votes

**2**answers

4k views

### What is the meaning of Text inside of AMS logo [closed]

What is the meaning of the text inside this AMS logo?
The image is from here, and the logo seems to have been frequently used until the 80's. The text is
ΑΓΕΩΜΕΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ
but Google ...

**43**

votes

**4**answers

5k views

### A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?

Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.
Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...