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

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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
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1answer
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 ...
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4answers
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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
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18answers
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
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1answer
398 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
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2answers
469 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
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3answers
349 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
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1answer
657 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
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1answer
322 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
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1answer
247 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
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1answer
272 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 ...
24
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2answers
776 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
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2answers
601 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
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1answer
170 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
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3answers
952 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
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2answers
969 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
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3answers
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
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2answers
740 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
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5answers
886 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
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2answers
605 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 ...
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1answer
228 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
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1answer
173 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
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2answers
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
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1answer
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
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3answers
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
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4answers
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 ...
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2answers
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
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0answers
226 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
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1answer
246 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
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7answers
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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
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0answers
422 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
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1answer
423 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 ...
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658 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
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5answers
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
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1answer
457 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 ...
7
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0answers
461 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
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1answer
231 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
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0answers
138 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
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3answers
860 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
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0answers
467 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
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25answers
7k 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
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0answers
379 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 ...
13
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1answer
449 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. ...
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2answers
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 ...
12
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4answers
2k 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
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1answer
224 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
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1answer
158 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 ...
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13answers
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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
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1answer
704 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 ...
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1answer
319 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). ...