# Tagged Questions

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

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### Examples of Geometric Constructions in Higher Dimensions

The classical problem of geometric construction seems to be restricted to planar Euclidean Geometry with straight edge and compass as the only admissible "construction-tools". I would like to know,...
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### Was $\Sigma x$ used as quantifier?

Kurt Gödel in 1931 used $x\Pi a$ where we in contemporary notation would use $(\forall x) A$ or $(x)A$, and $Ex a$ where we would use $(\exists x) A$. I believe that I remember that $\Sigma xA$ has ...
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### History of the orientation of Cartesian coordinates in drawing

Is there any actual historical example in which a Cartesian plane with all four quadrants has been used, but with all axes marked with positive numbers? [Please see Sawyer's paper below for a "made-up"...
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### History of Poincare conjecture in higher dimension [closed]

As far as I know, when Poincare formulated his well known conjecture, the original statement was the follwoing: if a closed manifold has the same homology groups as the sphere it is homoeomorphic to ...
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### Banach-Zarecki theorem - who was Zarecki?

I'm writing a paper for real analysis seminar, a paper about Banach-Zarecki theorem and I need some information about the authors. Stefan Banach - there is no problem to find information about him. ...
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### Filmed lectures by Jürgen Moser

Are there any filmed lectures by outstanding German mathematician Jürgen Moser (July 4, 1928 – December 17, 1999)?
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### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught: 1º that $$\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2}, \tag1$$ 2º that, via the fundamental theorem of calculus, this is ...
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### Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on. I am aware that even Gantmacher 1959 has this terminology however I don't know ...
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### Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...
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### Two questions on substitutability

(1) The condition that a term $a$ be substitutable for another term in an expression can be given a recursive definition. Who first developed such a definition? (2) One sometimes see the phrase "$a$ ...
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### Whether to posthumously honor Grothendieck's request to stop publishing his works? [closed]

Why did Grothendieck say stop publishing his works? https://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter/ Any edition or dissemination of such texts which have been made in the past ...
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### Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus

The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products? It is the simplest application of the ...
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### 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 '...
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### 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 ...
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### 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 Bourbaki'...
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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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 $k^n$....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $L(t,\phi,\alpha) := (t*\sin(\alpha)+s*\cos(\alpha),-t*\cos(\alpha)+s*\sin(\alpha))$,...
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### 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, ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 don't....
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...