**19**

votes

**4**answers

973 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

**7**answers

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

**1**answer

304 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

**1**answer

637 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

**19**answers

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

**3**answers

394 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**

votes

**1**answer

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

**1**answer

622 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

**3**answers

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 ...

**36**

votes

**31**answers

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

**6**answers

931 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

**2**answers

253 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**

votes

**1**answer

729 views

### Elements of the history of mathematics

Is it known who actually wrote Bourbaki's Elements of the History of Mathematics?

**7**

votes

**3**answers

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**

votes

**2**answers

1k views

### What is the history of $\sqrt{}$

Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?

**27**

votes

**4**answers

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

**4**answers

812 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**

votes

**10**answers

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

**1**answer

484 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

**0**answers

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

**0**answers

422 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 ...

**53**

votes

**61**answers

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

**3**answers

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

**4**answers

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

**4**answers

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

**1**answer

351 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

**1**answer

866 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

**1**answer

942 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

**3**answers

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

**0**answers

427 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

**1**answer

471 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

**1**answer

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

**4**answers

882 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

**7**answers

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

**9**answers

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

**1**answer

543 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

**0**answers

252 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

**2**answers

552 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

**2**answers

622 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

**2**answers

458 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

**6**answers

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

**5**answers

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**

votes

**1**answer

784 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

**3**answers

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 ...

**16**

votes

**1**answer

1k views

### On a theorem of Galois

I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois :
Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...

**30**

votes

**10**answers

3k views

### Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...

**6**

votes

**3**answers

669 views

### Classification of Platonic solids

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula ...

**17**

votes

**2**answers

2k views

### Location of Archimedes' grave in Syracuse (math/archaelogy trivia)

This is really a question for our archaelogist friends, but I could not find an "archaelogy overflow" and some mathematicians might find it interesting.
In a few weeks I am giving a talk in which I ...

**12**

votes

**1**answer

1k views

### Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below.
In ...

**5**

votes

**1**answer

226 views

### constant averages along orbits

What should one say to describe the situation in which a function $T$ from some set $X$ to itself, and a function $f$ from $X$ to some characteristic-zero field $K$, have the property that the average ...