Biographies, philosophy of mathematics, mathematics education, recreational mathematics, communication of mathematics

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18
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9answers
3k views

Was the early calculus inconsistent?

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY. George Berkeley wrote in 1734 with reference to the early calculus that ...
27
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3answers
2k views

What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math. The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...
3
votes
0answers
206 views

Where was the arithmetic zeta function of a scheme first defined?

Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the ...
4
votes
0answers
217 views

origin of the notion of “network” in graph theory

In current graph theory, a "network" is a precisely defined object: It is a directed graph associated with a function, the "capacity", which is defined on the edge set and has certain specific ...
21
votes
2answers
2k views

Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...
2
votes
2answers
178 views

local subrings of matrix ring

When is the subring (containing 1) of a matrix ring $M_n(k)$ over a field $k$ is local? I would be grateful for every reference concerning this matter, Thank you!
19
votes
4answers
986 views

The role of ANR in modern topology

Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
20
votes
7answers
2k views

Modern developments in finite-dimensional linear algebra

Are there any major fundamental results in finite-dimensional linear algebra discovered after early XX century? Fundamental in the sense of non-numerical (numerical results, of course, are still ...
5
votes
1answer
308 views

First mention of the fundamental bigroupoid of a space?

The fundamental bigroupoid $\Pi_2(X)$ of a space $X$ was independently described by Hardie, Kamps and Kieboom (paywall) and Stevenson (arXiv) around the year 2000. HKK cite Baez-Dolan's seminal HDA0, ...
15
votes
1answer
644 views

Mendelson's *Mathematical Logic* and the missing Appendix on the consistency of PA

A very soft question, but I hope not out of order here. In the first edition of Elliott Mendelson's classic Introduction to Mathematical Logic (1964) there is an appendix, giving a version of ...
39
votes
19answers
6k views

Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...
4
votes
3answers
395 views

Biographical Information Concerning Henry Sherwin

Henry Sherwin's Mathematical Tables achieved some popularity. The first edition was published ~1706 and the fifth and last in 1771. Some editions were more erroneous than others and the error rate was ...
4
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1answer
1k views

Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer. As we all know, ...
4
votes
1answer
630 views

Ancient method to study Archimedean spiral

It is well-known the properties of Archimedean spiral ($\rho = k\phi$) which is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant ...
1
vote
3answers
1k views

Who invented the expression “pairwise different” and what is its advantage over “different”

There are many applications of "pairwise", for instance different, disjunct, orthogonal, independent, intersecting, connected, and many more. Some of them like "pairwise intersecting" or "pairwise ...
35
votes
31answers
6k views

Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...
23
votes
6answers
936 views

Concise model of modern fiat money and its non-conservation

A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, ...
1
vote
2answers
257 views

What are Normal Sets (Fréchet)?

In 1913, LEJ Brouwer started a new approach to give a topologist's definition of the notion dimension ("Über den natürlichen Dimensionsbegriff", Journal für die reine und angewandte Mathematik, 142, ...
3
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1answer
738 views

Elements of the history of mathematics

Is it known who actually wrote Bourbaki's Elements of the History of Mathematics?
7
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3answers
2k views

Retracted Mathematics Papers [closed]

Can anyone cite an example of a mathematics paper that has been retracted? It is said that on the order of 100,000 new theorems enter the mathematics literature every year. For a number of reasons ...
15
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2answers
1k views

What is the history of $\sqrt{}$

Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?
27
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4answers
2k views

What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...
7
votes
4answers
814 views

Errors, oversights, and misunderstandings in mathematical research [duplicate]

Possible Duplicate: Examples of common false beliefs in mathematics. Hopefully this is not overly controversial, but I thought it would be instructive to compile a list of errors which are ...
25
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10answers
3k views

real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
4
votes
1answer
487 views

What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...
5
votes
0answers
260 views

Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal ...
16
votes
0answers
424 views

Authorship of Grothendieck universes

Universes seem to first appear in Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is ...
54
votes
61answers
8k views

Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...
11
votes
3answers
3k views

Why is a ring called a “ring”?

Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...
35
votes
4answers
2k views

The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
12
votes
4answers
1k views

Statements which were given as axioms, which later turned out to be false.

I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness ...
6
votes
1answer
354 views

Where do the Kähler Identities first appear?

The Kähler identities (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships ...
6
votes
1answer
882 views

Deligne's letter to Looijenga from 1974

Hello, I wonder if anyone has a copy of Deligne's letter to Looijenga from 1974 mentioned as reference [26] in Bessis' paper Finite complex reflection arrangements are $K(\pi,1)$ from 2006, see ...
11
votes
1answer
943 views

At what point does number theory stop playing with finite rings?

Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for ...
44
votes
3answers
2k views

Groups that do not exist

In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and ...
1
vote
0answers
442 views

Area Under Generalized Parabolas and Hyperbolas without Calculus.

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...
15
votes
1answer
481 views

The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as $$ \Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du, $$ where $\Phi(u)$ is defined as $$ ...
8
votes
1answer
2k views

What is “Teichmüller Theory” and its history ?

What is "Teichmüller Theory" ? What part has been worked out / forseen by O. Teichmüller himself and what is further development ? Is there some current work which might be considered as ...
9
votes
4answers
892 views

Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...
11
votes
7answers
2k views

What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far: No-wandering-domain Theorem (Sullivan) Theorems of Thurston ...
7
votes
9answers
1k views

Examples where adding complexity made a problem simpler

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples: $S^n$ is never contractible, but $S^{\infty}$ is. The ...
8
votes
1answer
547 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
3
votes
0answers
255 views

A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...
2
votes
2answers
555 views

Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry", an English version ...
6
votes
2answers
628 views

Why is Gauss credited with his connection?

Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi_* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose ...
8
votes
2answers
460 views

The history of the geometrization of closed surfaces

Who first recognized that the torus supports a flat structure? Who first characterized the moduli space of flat structures on the torus? Who first recognized that the closed, orientable genus 2 ...
24
votes
6answers
4k views

Who wrote up Banach's Thesis?

Some time ago I read somewhere (and I don't remember where it was) that Stefan Banach -- a highly creative and great mathematician -- did not always write down his ideas. Allegedly, he did not write ...
37
votes
5answers
5k views

Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?

According to Steven Krantz's Mathematical Apocrypha (pg. 186): As was custom, Weil often attended tea at [Princeton] University . Graduate student Steven Weintrab one day went about the room ...
5
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1answer
796 views

Historical questions on the term “general abstract nonsense”

Saunders Mac Lane reports that the contents of his 1942 paper (joint with Samuel Eilenberg), that first introduced categories, were then referred to (in the words of prominent representatives of the ...
3
votes
3answers
439 views

Origin of the theorem on the existence of the smallest field of definition of an affine variety

Weil proved the following theorem in his book Foundations of Algebraic Geometry, p.19. The proof is somewhat involved. I wonder if the theorem is his original. Theorem Let $K[X_1,\dots, X_n]$ be the ...