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

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Euclid vs Eratosthenes

Very little is known about Euclid's life--much less than about other famous ancient Greek mathematicians, which is puzzling. It is also strange to me that Euclid didn't write about the Eratosthenes ...
4
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1answer
239 views

History of the notation for substitution

One of the very common notations for syntactic substitution is $[\ /\ ]$. There seems to be a large inconsistency in the literature about its use. Many write $[t/x]$ for substitute $t$ for $x$ (e.g. ...
4
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2answers
704 views

Unreasonable application of mathematics to the other areas [on hold]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science? I found ...
2
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1answer
159 views

Cambridge Mathematical Tripos papers from late 19th century

Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?
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18answers
8k views

What are some applications of other fields to mathematics?

It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely: What are some ...
18
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6answers
1k views

Mathematics contests before 1800

Aside from well known examples of mathematics contests in 1535 and 1548, what are some other examples before 1800? Background: In The History of Mathematics: an Introduction, 3rd edition (1995), ...
59
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22answers
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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 ...
5
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0answers
158 views
+50

Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
8
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1answer
308 views

Trinity College, Cambridge, circa 1896 maths scholarship papers [on hold]

I've been searching around looking for the (maths component) of the scholarship papers to Trinity College (Cambridge) from around 1890. Can anyone provide a link to a pdf scan of these papers? Was ...
13
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2answers
2k views

A certain mathematical competition in the UK

There is a foreword, written by professor Snow, to the book A mathematician's apology. In the foreword, it is written some thing like the following: "Hardy was opposed to a certain mathematical ...
30
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3answers
4k views

Did Euler prove theorems by example?

In his 2014 book, Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7: Capitolo I Esempi e metodi dimostrativi Introduzione In The Calculus as Algebraic ...
15
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3answers
713 views

History of the analytic class number formula

The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). ...
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0answers
85 views

What are some of the most complete genealogies of mathematical subject areas? [closed]

Cross-posted from HSM: I am interested in the way scientific and mathematical subject areas developed (and are still developing). One of the great visual tools that can help us gain insight in how ...
35
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8answers
4k views

Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
7
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1answer
368 views

What was Gödel's Constitutional Problem? [closed]

It is well known that Kurt Gödel had doubts concerning the US constitution and believed that it somehow was inconsistent and opened up for a dictatorial grab. What was he thinking?
4
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0answers
170 views

Origin of the name ''momentum map''

Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?
79
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23answers
13k views

Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
1
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0answers
595 views

Whether to posthumously honor Grothendieck's request to stop publishing his works? [closed]

Why did Grothendieck say stop publishing his works? https://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter/ Any edition or dissemination of such texts which have been made in the past ...
18
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2answers
1k views

Felix Klein on mean value theorem and infinitesimals

This is a reference request prompted by some intriguing comments made by Felix Klein. In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
9
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2answers
2k views

Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...
5
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5answers
547 views

Important results with one or more than one proof [closed]

Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How ...
3
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1answer
187 views

Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
11
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3answers
2k views

Has Dedekind's proof of existence of infinite sets been analyzed by historians?

This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set. The proof exploits the assumption that there exists a set $S$ of all ...
35
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3answers
2k views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

(This question was posted on math.stackexchange a week ago at http://math.stackexchange.com/questions/187315/definitive-source-about-dirichlet-finally-proving-the-unit-theorem-in-the-sistinbut and ...
144
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136answers
31k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
4
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2answers
228 views

Historical reference request on Nilpotent groups

From Wikipedia: "Abelian groups were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the ...
20
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3answers
801 views

A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$

The connectedness of the moduli space $\mathcal{M}_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the ...
18
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2answers
1k views

Context for “Coronidis Loco” from Weil's Basic Number Theory

In Samuel James Patterson's article titled Gauss Sums in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Patterson says "Hecke [proved] a beautiful theorem on the different ...
2
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2answers
369 views

What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper, Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495. the authors give the following quote of Frege, from his paper "&...
5
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2answers
640 views

Where does the definition of “Tower of Algebras” come from?

A tower of algebras is a sequence of algebras $$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$ with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...
62
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15answers
7k 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 ...
55
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1answer
3k views

Did Bourbaki write a text on algebraic geometry?

Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?
64
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59answers
15k views

Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
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0answers
62 views

Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15): Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the Gromov-...
13
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2answers
2k views

What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
2
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6answers
2k views

Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
20
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2answers
2k 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 ...
8
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3answers
741 views

English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...
24
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1answer
2k views

More on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful '...
-4
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1answer
284 views

Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
1
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0answers
56 views

History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of - $\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...
42
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15answers
6k views

Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...
10
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1answer
503 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?
0
votes
1answer
201 views

Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true. Since posting the question, ...
49
votes
9answers
6k views

When have we lost a body of mathematics because errors were found?

The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, ...
64
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10answers
7k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
3
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1answer
221 views

Have there been any claims of mathematical breakthroughs while in altered states of consciousness?

This certainly is a related question: Have you solved problems in your sleep? Has anyone seriously attempted to make a similar claim for other altered states, besides dreaming? I know the claim has ...
11
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5answers
1k views

Texts on the General History of Contemporary Combinatorics

I am looking for some core texts (books, book chapters, papers) about the general history of contemporary combinatorics, starting, say, from the interwar period up to today. Texts about the history ...
14
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5answers
2k views

What are hypergroups and hyperrings good for?

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...
5
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0answers
103 views

Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of non-...