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

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18
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Where are Georg Cantor's Original Manuscripts?

Georg Cantor is famous for introducing transfinite numbers and set theory. A main part of his mathematical point of view about this new type of "numbers" and this new "realm of mathematics" cannot be ...
40
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4answers
3k views

The Arnold – Serre debate

I have read (but I cannot now find where) that Arnold & Serre had a public debate on the value of Bourbaki. Does anyone have more details, or remember or know what was said?
2
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1answer
160 views

why the difference between terms and propositional variables?

Reading some old logic texts (written around 1930) I noticed that these texts make no difference between propositional variables and terms. They do make difference between identity and truthvalue ...
55
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14answers
6k views

Mathematical research published in the form of poems

The article Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen, Math. Z. 127 (1972), no. 1, 10-16 is written in the form of a lengthy poem, in a style similar to that of the ...
7
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0answers
191 views

Reference Request: Topological h-cobordism theorem in higher dimensions

I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so. The h-cobordism theorem is true in the topological and in the smooth category in ...
2
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0answers
178 views

Does anyone know what is the right reference for the following simple lemma from harmonic analysis?

The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds $$\|\sum_j a_j\chi_{\lambda B_j}\|_p\leq ...
-2
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2answers
677 views

Accidental, unplanned breakthroughs in Mathematics [closed]

In math/physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts. Moreover, there are some cases that originally having ...
45
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18answers
7k views

Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time. I am looking for a list of major theorems in mathematics whose proofs are ...
8
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1answer
315 views

Why did Alonzo Church choose the letter $\lambda$ as the “binding operator”?

Is there any known reason why Alonzo Church chose Greek $\lambda$ as the "binding operator" for the Lambda Calculus?
7
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1answer
452 views

History of Jordan Canonical Form?

Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in: When and how was it first stated? (I understand it was independently stated ...
4
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2answers
303 views

Who first used the cross-ratio to describe shapes in hyperbolic geometry?

I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...
26
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1answer
899 views

Institutional response to “Esquisse d'un programme”

It is well-known that Grothendieck's "esquisse d'un programme" was submitted in 1984 as part as the author's application for a permanent position of "Directeur de Recherche" at the C.N.R.S. (the main ...
20
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7answers
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Was lattice theory central to mid-20th century mathematics?

Four years ago, I read a book on the history of mathematics up to 1970 or so. It was very interesting up until the end. The last few chapters, though, were on lattices. The author claimed that ...
25
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9answers
2k views

Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background: Gauss invented "Gauss curvature" to measure how surface curves. Riemann gives an ingenious generalization ...
9
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1answer
515 views

Well founded induction attributed to Noether

What I know as well founded induction, namely the rule $$ \big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big), $$ whose validity is the ...
14
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1answer
2k views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
8
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1answer
270 views

History question: Roth's theorem on approximating algebraic numbers…before Roth

Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality ...
17
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1answer
530 views

What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
12
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2answers
843 views

How did Gauss and contemporaries think of modular forms?

Accounts of modular forms say that they were studied in the early 19th century, but then define modular forms using terminology that didn't exist until the 20th century. How did the earliest ...
5
votes
1answer
328 views

Influence of Yau's solution to the Calabi Conjecture on the field of PDEs

I remember reading a long time ago(I can't recall where, unfortunately) that Yau's solution of the Calabi-Yau conjecture introduced new techniques that were very important for the field of partial ...
14
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1answer
490 views

History of Koszul complex

This is a question about history of commutative algebra. I'm curios why Koszul complex from commutative algebra is called Koszul complex? All Koszul's early papers are about Lie algebras and Lie ...
8
votes
1answer
2k views

What is the source of this E̶r̶d̶ő̶s̶ quote?

Namely, the following one "All problems appeared once in the [American Mathematical] Monthly." I remember reading it several years ago... When I first posed the question, I believed that I had ...
7
votes
1answer
254 views

Analogy between Lagrange's Theorem and Rank-Nullity Theorem?

One can view view Lagrange's Theorem $$|G/H|=|G|/|H|$$ and the Rank-Nullity Theorem $$\dim(V/U)=\dim(V)-\dim(U)$$ as directly analogous. Does anyone know a high-level explanation of this analogy? I ...
14
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2answers
971 views

Deligne Weil II

Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period ...
10
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5answers
754 views

What are good English-language sources for reading about the Luzin affair?

What are good English-language sources for reading about the Luzin affair? I'm interested in the subject and am wondering about good historical sources.
10
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1answer
366 views

Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs. Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...
8
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1answer
437 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
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1answer
391 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
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2answers
492 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
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1answer
438 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
3answers
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
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2answers
655 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 ...
18
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14answers
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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 ...
12
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6answers
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
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0answers
486 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
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1answer
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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 ...
6
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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? ...
38
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18answers
5k 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
1answer
382 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
452 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
329 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
643 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
314 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
242 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
266 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
votes
2answers
774 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
554 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
166 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
927 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
932 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 ...