**11**

votes

**3**answers

700 views

### The role of the Automatic Groups in the history of Geometric Group Theory

What is the role of the theory of Automatic Groups in the history of Geometric Group Theory?
Motivation:
When I read through the "Word Processing in Groups" I was amazed by the supreme beauty and ...

**9**

votes

**0**answers

418 views

### What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...

**60**

votes

**24**answers

6k views

### Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...

**2**

votes

**0**answers

317 views

### What is the oldest known evidence of application of mathematics?

According to Wikipedia the Lebombo bone (age 35 KY) and the Ishango bone (age at least 20 KY) presently are believed to show the first evidence for application of mathematics by humans. (Possibly ...

**9**

votes

**1**answer

320 views

### Discrete Morse theory and chess

There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. ...

**21**

votes

**2**answers

1k views

### Hahn's Embedding Theorem and the oldest open question in set theory

Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper Über die nichtarchimedischen Grössensysteme (Sitzungsberichte der ...

**10**

votes

**4**answers

1k views

### History of the high-dimensional volume paradox

Inscribe an $n$-ball in an $n$-dimensional hypercube of side equal to 1, and let $n \rightarrow \infty$. The hypercube will always have volume 1, while it is a fun folk fact (FFF) that the volume of ...

**11**

votes

**1**answer

203 views

### What is flexible about flexible algebras?

A possibly non-associative algebra is flexible if it satisfies the identity $$(xy)x=x(yx).$$ This is clearly a very weak form of associativity —and obviously an associative algebra is flexible— but it ...

**7**

votes

**1**answer

137 views

### Who first observed that Conjugate Gradient for Symmetric Positive Definite linear systems is a Krylov method?

Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived ...

**28**

votes

**13**answers

2k views

### Great mathematics books by pre-modern authors

Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...

**5**

votes

**1**answer

652 views

### Why do mathematicians prefer one definition over the other when they both define the same concept?

Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...

**1**

vote

**1**answer

308 views

### Newton integration without integration

Consider a function f continuous on a compact interval.
Approximate it by a sequence of polygonal functions (you can).
Then consider a sequence of primitives of the polygonal functions (you can).
...

**6**

votes

**1**answer

217 views

### Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define
...

**4**

votes

**2**answers

237 views

### Was Desargues more an Euclid or an Eudoxos?

In the course of preparing lessons on projective geometry I want to give an account on the historical development. It is easy to obtain an overview of the history starting with G. Desargues. And with ...

**13**

votes

**1**answer

1k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**3**

votes

**0**answers

121 views

### Citation for subset complement result

Let $S=\lbrace s_1,\ldots,s_n \rbrace \subset \lbrace1,\ldots,2n\rbrace$. Consider two operations on $S$: the complement $C(S)=\lbrace 1,\ldots,2n \rbrace \setminus S$ and a reflection* ...

**0**

votes

**1**answer

378 views

### Has the controversy about *fiducial distribution* been settled? [closed]

Has the controversy about the correct meaning of Fisher's notion fiducial distribution meanwhile been settled? And are there newer applications than quoted in the following literature?
G.P. Klimov: ...

**3**

votes

**3**answers

696 views

### Lapses of “the early proponents of the doctrine of limits”

I have a question that I have been wondering about for a long time without finding any answer. Concerning the period around 1900, Robinson commented in his 1966 book that "there is in the writings of ...

**8**

votes

**3**answers

805 views

### who invented projective space $\mathbb{P}^n$?

Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$?
Who was the first person to consider projective closure of
plane affine algebraic curves (curves ...

**18**

votes

**9**answers

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**

votes

**3**answers

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

**0**answers

196 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

**0**answers

205 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

**2**answers

1k 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

**2**answers

168 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

**4**answers

933 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

303 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

570 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 ...

**38**

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

389 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

586 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 ...

**32**

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 ...

**22**

votes

**6**answers

917 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

242 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

693 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

778 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?

**3**

votes

**1**answer

379 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

256 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 ...

**15**

votes

**0**answers

415 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 ...

**52**

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

2k 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 ...