**8**

votes

**1**answer

453 views

### “'Category' was defined in order to define 'functor', which was defined in order to define 'natural transformation'”

I am looking for the source (and original version) of the above oft-repeated quotation. Mac Lane mentions it in Categories for the Working Mathematician, attributing it to Eilenberg-Mac Lane; however, ...

**0**

votes

**1**answer

399 views

### What was the original/historical motivation for introducing Grothendieck (pre-)topologies

The title essentially explains it, but I'll give some background:
I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck ...

**9**

votes

**2**answers

520 views

### Origin of the Socle of a module

Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?

**6**

votes

**1**answer

458 views

### Where does the notation $\pi_1(X,x)$ for the fundamental group first appear?

I've spent the last half hour browsing Stillwell's translation of Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this ...

**22**

votes

**3**answers

1k views

### What exactly does this diagram of Omar Khayyam represent?

Evidently Omar Khayyam (1048-1131) was quite the mathematician. He did groundbreaking work on finding geometric solutions to the cubic equation, which is all the more notable since he did not have a ...

**11**

votes

**2**answers

663 views

### Le Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides

Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in Roger Godement's Analyse mathématique, IV, p.313 (perhaps the only treatise on ...

**19**

votes

**14**answers

2k views

### Insightful books about elementary mathematics

What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful ...

**13**

votes

**6**answers

1k views

### The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...

**8**

votes

**0**answers

523 views

### Has anyone pursued Frege's idea of numbers as second-order concepts?

Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" ...

**18**

votes

**1**answer

1k views

### Arnold on Newton's anagram

Arnold, in his paper
The underestimated Poincaré, in Russian Math. Surveys 61 (2006), no. 1, 1–18
wrote the following:
``...Puiseux series, the theory which Newton, hundreds of years before ...

**7**

votes

**4**answers

2k views

### Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$

In the book, "Pi and the AGM" by Borwein and Borwein, it is mentioned that Gauss computed the following integral to the eleventh decimal palce.
$\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How did he do it? ...

**42**

votes

**18**answers

6k views

### What are some deep theorems, and why are they considered deep?

All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...

**4**

votes

**1**answer

404 views

### Where does the name “Reynolds operator” come from?

I always found it strange that, in the context of invariant and representation theory,
averaging over the group is called the "Reynolds operator". As far as I know the work of Reynolds was in fluid ...

**5**

votes

**2**answers

474 views

### When did the meaning of the term “metabelian” change?

I just realised that the meaning of the term "metabelian", when applied to groups, or Lie algebras, seems to have changed over years. (These days, it means that $[[G,G],[G,G]]$ is trivial, while in ...

**2**

votes

**3**answers

362 views

### Good Books on the history of Zero

I am looking for books that discuss the origins of the zero, specifically the differences in the use and concept of the zero number among different civilizations (considering also the Mesoamerican ...

**6**

votes

**1**answer

662 views

### What are current trends/questions in algebraic logic?

What are current trends/questions in algebraic logic? I mean the research developed by Paul Halmos.
Could anyone give some references for the overview of its history? Any overview of its application ...

**8**

votes

**1**answer

323 views

### historical antecedents of mathematical talks

Is there a general reference of how mathematical talks, say academic talks, evolve in history? Before the International Congress of mthematics, is there any antecedent of todays talks?

**1**

vote

**1**answer

249 views

### What is the name of the following theorem: dimension of complex irreducible representation divides order of group

Who proved it? When?
See also:
Irreducible Degrees and the Order of a Finite Group
http://planetmath.org/proofthatdimensionofcomplexirreduciblerepresentationdividesorderofgroup
Why would dim ...

**5**

votes

**1**answer

274 views

### Origins of Axiomatic Reasoning

Is there any evidence that axiomatic reasoning has been used prior to Thales of Milet (624-547BC), who is generally credited for the "invention" of axioms.
In this context I understand axioms in the ...

**25**

votes

**2**answers

783 views

### Origin and first uses of $\ell_p$ norms?

When exactly were $\ell_p$ norms first defined and used?
(Here is what I know, or think I know: Lebesgue and/or Riesz had something to do with them, but in some sense they go back to Minkowski, since ...

**12**

votes

**2**answers

618 views

### To what extent can fields be classified?

The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and ...

**2**

votes

**1**answer

173 views

### English translation of Steinitz 1910?

Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042

**12**

votes

**3**answers

975 views

### When was the continuum hypothesis born?

The question Solutions to the Continuum Hypothesis states that the continuum hypothesis was posed by Cantor in 1890. In http://en.wikipedia.org/wiki/Continuum_hypothesis the year 1878 is quoted ...

**11**

votes

**2**answers

985 views

### What might extraterrestrial mathematics look like? [closed]

In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely ...

**14**

votes

**3**answers

1k views

### Who introduced the terms “equivalence relation” and “equivalence class”?

Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence ...

**12**

votes

**2**answers

741 views

### An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...

**18**

votes

**5**answers

899 views

### Historical use of figures in geometry

I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single ...

**12**

votes

**2**answers

634 views

### Why did Voiculescu develop free probability?

I was recently asked why Voiculescu developed free probability theory. I am not very expert in this and the only answer I was able to provide is the classical one: he was challenging the isomorphism ...

**1**

vote

**1**answer

249 views

### Understanding the rationale behind “batch means” estimation

Hello all,
I am implementing an MCMC algorithm for my work, and I've come upon something in the literature which I just can't understand.
Specifically, I am attempting to estimate the amount of ...

**5**

votes

**1**answer

177 views

### How did Hankel determinants get the name Hankel-Hadamard?

My question concerns the name for determinants of Hankel-matrices $H = (s_{i+j})_{i,j = 0}^n$.
In the classical textbook of Shohat and Tamarkin (1943) "The Problem of Moments", these determinants are ...

**15**

votes

**2**answers

1k views

### Where did Sophus Lie write the group commutator for two one parameter groups

If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...

**26**

votes

**1**answer

1k views

### Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...

**7**

votes

**3**answers

339 views

### Meaning of historical fluxion notation

I've noticed that in 18th century English books on calculus writers would say that 'the fluxion of $ax$ is $a\dot{x}$' and 'the fluxion of $x^n$ is $n x^{n-1} \dot{x}$'. What does this extra ...

**37**

votes

**4**answers

3k views

### What is the source of this famous Grothendieck quote?

I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck.
It is better to have a good category with bad objects than a bad category ...

**22**

votes

**2**answers

1k views

### The Erdős-Turán conjecture or the Erdős' conjecture?

This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ ...

**7**

votes

**0**answers

228 views

### History of the characterization of commutative Artin rings

When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as ...

**4**

votes

**1**answer

250 views

### Historical precursor for Peano's axioms of a linear space?

Peano is typically credited with giving the first abstract definition of a vector space (1888):
http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_linear_spaces.html
Apparently, Peano credits ...

**7**

votes

**7**answers

1k views

### famous papers/results by non professional mathematicians [duplicate]

Possible Duplicate:
What recent discoveries have amateur mathematicians made?
Dear overflowers
Out of curiosity: do you know any famous papers and/or results by non professional ...

**12**

votes

**0**answers

425 views

### Unpublished Lecture Notes

Hi, Overflowers
There was a time (not so long ago) where lecture notes were not published, not commonly at least, and their reproduction was expensive. In my case, that was precisely the time when ...

**19**

votes

**1**answer

429 views

### Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...

**11**

votes

**0**answers

677 views

### Why did Bourbaki not use universal algebra?

I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra?
Well, universal algebra is not much older than category ...

**18**

votes

**5**answers

3k views

### Why did Bourbaki ignore the theory of categories? [closed]

QUESTION
They had plenty of time to adopt the theory of categories. They had Eilenberg, then Cartan, then Grothendieck. Did they feel that they have established their approach already, that it's too ...

**6**

votes

**1**answer

470 views

### Generalization of the Lefschetz fixed point theorem

I have encountered a certain generalization of the Lefschetz fixed point theorem as folklore, and I am hoping that someone out there knows its provenance or can otherwise refer me to a source where it ...

**8**

votes

**0**answers

471 views

### Reference/quote request: “All of combinatorics is the representation theory of $S_n$”

I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like
"All of combinatorics is essentially [or can be reduced to?] the representation ...

**6**

votes

**1**answer

237 views

### Did Smith correctly state the mass formula?

Did Smith correctly state the mass formula?
H.J.S. "normal form" Smith was the first, in 1867, to state the mass formula for integral quadratic forms in a genus of 4 or more variables. This was ...

**2**

votes

**0**answers

140 views

### Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...

**11**

votes

**3**answers

906 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

472 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 ...

**66**

votes

**25**answers

8k 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 ...

**1**

vote

**0**answers

382 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 ...