**0**

votes

**0**answers

37 views

### Shuffle multiplication and generalized Leibniz rule in tensor calculus

The headline already says it: Is anybody (except me) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the commutative shuffle product ...

**1**

vote

**0**answers

71 views

### Why are they called 'pernicious' numbers?

The definition of a pernicious number:
In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.
The meaning of ...

**1**

vote

**1**answer

173 views

### When was the “arrow notation” for functions first introduced?

When was the "arrow notation" $f: X \to Y$ for functions first introduced? Who introduced it and with which motivation?
I ask this question in order to understand whether it was, in part, this ...

**5**

votes

**2**answers

218 views

### Convention about “long” roots for simple Lie algebras of types ADE?

The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in ...

**4**

votes

**1**answer

388 views

### Did Brouwer evade uncountability?

I have the distinct memory of having often heard and read that intuitionism was inter alia geared to avoid Cantor's uncountable sets, and it may be that this was Brouwer's plan. But are there accounts ...

**5**

votes

**0**answers

233 views

### Motivation behind Euler Theorem in differential geometry [migrated]

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...

**10**

votes

**1**answer

420 views

### Metric $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$

I'm wondering where the relative probabilistic distance was first studied:
$$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$
where $\overline A$ is the complement of $A$.
A web search ...

**49**

votes

**4**answers

4k views

### Are there any serious investigations of whether “mathematicians do their best work when they're young”?

There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious ...

**7**

votes

**1**answer

325 views

### Discovery and Study of Conic Sections in Ancient Greece

Is there anything known about what drew the attention of ancient greek mathematicians to conic sections and, what were the models they used to study conic sections?
What I would like to know, is ...

**8**

votes

**0**answers

154 views

### Who first proved the fundamental theorem of projective geometry?

The following theorem is often called the fundamental theorem of projective geometry:
Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of ...

**6**

votes

**0**answers

234 views

### What is the reason that $\sigma$-algebra replaced $\sigma$-ring in introductory measure theory?

May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of $\sigma$-rings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis ...

**4**

votes

**0**answers

247 views

### Reference request : Grothendieck's topological space valued integral

As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...

**6**

votes

**2**answers

505 views

### Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...

**3**

votes

**1**answer

164 views

### Longevity of “random” conjectures

The "random" sample is obviously very, very skewed: If you would be asked to name a random conjecture, it probably will be a "famous" conjecture, and the longer a conjecture stands, the more famous it ...

**9**

votes

**1**answer

585 views

### What is a totient?

In addition to the Euler totient function, there are a great many generalizations and related functions which go by the "totient", usually with some name: Jordan, Lehmer*, Schemmel, Nagell, Alder, ...

**12**

votes

**4**answers

891 views

### Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...

**11**

votes

**2**answers

332 views

### A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ?
I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...

**5**

votes

**1**answer

461 views

### Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of
Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...

**4**

votes

**1**answer

328 views

### Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...

**28**

votes

**1**answer

703 views

### Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...

**1**

vote

**0**answers

45 views

### Reason for the Choice of Line Parameters in the Radon Transform

Why are the lines, over which the integrals in a Radon Transform are calculated, apparently always parameterized as $L(t,\phi,\alpha) := ...

**5**

votes

**0**answers

133 views

### Link between abelian groups and endomorphisms

When teaching Algebra, I try to share my fascination about two apparently unrelated questions, which turn out to involve the same theory:
classifying the finitely generated abelian groups,
...

**7**

votes

**1**answer

256 views

### Euler's Triangular Number closure properties

Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:
If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.
R. F. Jordan in the J. ...

**11**

votes

**1**answer

729 views

### Von Neumann's consistency proof

In the paper Zur Hilbertschen Beweistheorie, John Von Neumann has proposed a consistency proof for
a fragment of first-order arithmetic (the fragment without induction and with
the successor axioms ...

**9**

votes

**2**answers

261 views

### When was Bounded Zermelo set theory first formulated?

Bounded Zermelo set theory, and many variants named for MacLane in some way, are used in equiconsistency proofs for Simple Theory of Types plus infinity, and for the Elementary Theory of the Category ...

**5**

votes

**1**answer

338 views

### Oldest photographed mathematician [closed]

Who is the most ancient mathematician of which we have a photograph?
(or, in the same vein, what is the oldest photograph of a mathematician)
A quick search on MacTutor History of Mathematics gives ...

**5**

votes

**0**answers

202 views

### When did “Betti cohomology” come to be used the way it is today? (and how is it used)

This is sort of a mixture of a math and history question.
First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I ...

**10**

votes

**3**answers

359 views

### Riemann's formula for the metric in a normal neighborhood

I would love to understand the famous formula $g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(||x||^3)$, which is valid in Riemannian normal coordinates and possibly more general situations.
...

**13**

votes

**2**answers

472 views

### Who first noticed that Stirling numbers of the second kind count partitions?

When the Stirling numbers of the second kind were introduced by James Stirling in 1730, it was not combinatorially; rather, the numbers ${n \brace k}$ were defined via the polynomial identity
$$
x^n = ...

**2**

votes

**1**answer

269 views

### Disruptive innovations in mathematical notations [closed]

I am wondering whether there are examples of mathematical notations that, once introduced, have drastically changed or simplified the way to address a problem or a mathematical area, or that have ...

**2**

votes

**3**answers

373 views

### How did the summation operation come into use? [closed]

So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...

**5**

votes

**2**answers

470 views

### Who first defined quantum integers?

Who first gave the defintion of quantum integers
$$ [m]_q = \frac{1 - q^m}{1 - q} $$ and addition as
$$ [m]_q \oplus_q [n]_q = [m]_q + q^m [n]_q $$ and multiplication as
$$ [m]_q \otimes_q [n]_q = ...

**30**

votes

**1**answer

2k views

### What did Rolle prove when he proved Rolle's theorem?

Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...

**6**

votes

**2**answers

319 views

### Who first used/gave a coordinate representation of a graph?

In his proof of the Shannon capacity of a graph, Lovasz utilizes a coordinate representation of the pentagon (namely an orthonormal representation). Who first utilized a coordinate representation for ...

**57**

votes

**21**answers

10k views

### Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...

**19**

votes

**8**answers

6k views

### Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such.
Yet, they could have been just ...

**5**

votes

**3**answers

288 views

### A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal:
(1) Извѣстія Физико-математического общества при Казанском университете
I am surprised by the ...

**12**

votes

**0**answers

204 views

### Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...

**3**

votes

**3**answers

203 views

### What do you call a fixed point theorem for a mapping from a subset of a space to the whole space?

There are a number of fixed point theorems in which we have a map from some subset of a (metric, topological, ...) space to the whole space. (Usually, there is some condition regarding the behavior ...

**34**

votes

**1**answer

2k views

### Did Leibniz really get the Leibniz rule wrong?

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical ...

**8**

votes

**1**answer

763 views

### Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...

**3**

votes

**0**answers

91 views

### Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...

**33**

votes

**1**answer

2k views

### Hilbert's Hotel

Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943).
Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?

**10**

votes

**1**answer

265 views

### History of powers beyond squares and cubes

The ancient Babylonians understood squares:
Plimpton 322
The ancient Athenians understood cubes, if we can take
doubling the cube, i.e., the Delian problem, as evidence.
My ...

**7**

votes

**2**answers

380 views

### How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...

**1**

vote

**0**answers

154 views

### Motivating mathematics(particularly algebraic number theory) through historical problems [closed]

Most mathematical textbooks start a subject by going backwards, historically. They will define the terms that were invented to solve a problem in their polished form and then use these definitions and ...

**12**

votes

**2**answers

510 views

### History of the analytic class number formula

The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). ...

**6**

votes

**1**answer

352 views

### Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is:
Question 1. Which different approaches to set theoretic forcing are ...

**6**

votes

**0**answers

231 views

### Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.
At most papers ...

**20**

votes

**2**answers

926 views

### construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...