Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations.

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Theorem that von Neumann proved in five minutes.

In "How To Solve It", George Pólya writes: "There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it ...
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461 views

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

There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
15
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361 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
13
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116 views

How and why did Weierstrass $\wp$ get its special symbol?

I kind of always hated drawing the Weierstrass $\wp$ symbol by hand, and it struck me as odd how and why it achieved its special status in the first place. After all, there are tons of other important ...
13
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699 views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
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How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
9
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120 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
8
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107 views

Did Field's Medalist Klaus Roth suffer from test anxiety?

I remember hearing the story that Fields Medalist Klaus Roth was convinced that he could not pass a qualifying exam when he was a graduate student. He was then given a so called practice exam for him ...
8
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178 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...
7
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98 views

To whom is the proof that $A_n$ is simple for $n\geqslant 5$ due, in Rotman's book?

The proof in Rotman's book, Introduction to the Theory of Groups, that $A_n$ is simple consists of the observation that $A_n$ is generated by the $3$-cycles, and hence that if a normal subgroup $H\lhd ...
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189 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
7
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355 views

notation for ramification index and inertial degree

For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$. What is the origin of the ...
6
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51 views

Decoding Gauss' Easter Algorithm

In 1800, Gauss published this algorithm for computing the date of Easter in a given year $year$: $a = year \mod 19$ $b = year \mod 4$ $c = year \mod 7$ $k = \lfloor year/100 \rfloor$ $p ...
6
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133 views

Origin of $\mapsto$ notation

Who invented the brilliant $\mapsto$ notation for describing a function's action on a point, as in $x \mapsto x^2$? This is in some sense a counterpart to Who came up with the arrow notation $x ...
6
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183 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
6
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174 views

Priority of the content of a note by Lebesgue from 1905

I refer to a note by Lebesgue Remarques sur la définition de l'intégrale, Bull.Sci.Math. 29 (1905) 272-275 not very known (see pdf for an exposition in English). It is a pedagogical note containing a ...
5
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127 views

How do mathematicians know what is known?

How do mathematicians know that what they are researching has not been already know for 200 years? Obviously if they are researching something that is cutting edge it is not a problem, but if one is ...
5
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188 views

Ramanujan and sum of four cubes

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of ...
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244 views

History of line integral.

I'm looking for some information about how the line integral was discovered, since I've been looking for a long time for this. I found that Riemann could integer discontinuity functions, then Poisson ...
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233 views

Bloom of Thymaridas

I'm interested in learning more about the Bloom of Thymaridas, a description of which can be found here. Obviously the mathematics behind the identity is not particularly deep from a modern ...
4
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114 views

What is the origin of “how the Japanese multiply” / line multiplication?

A few months ago I made a video about a way to multiply numbers using lines (here) and it got really popular. I had heard about this trick before and I wanted to trace its origins. It seems to me to ...
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36 views

C. Neumann passage in Latin from *Annali di Matematica Pura ed Applicata*

Neumann, Carl. “Theoria nova phaenomenis electricis applicanda.” Annali di Matematica Pura ed Applicata 2, no. 1 (August 1868): 120–128. doi:10.1007/BF02419606. p. 121: Nova introducitur ...
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73 views

Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
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97 views

Earliest precursor to category theory

In the Historical notes section of the Wikipedia article on category theory, it is mentioned that in 1942-1945, Samuel Eilenberg and Saunders Mac Lane, in the course of their work in algebraic ...
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43 views

Original proof of the Invariance of Domain Theorem (in English)?

Does anyone know where I can find a translation of the original proof of the Invariance of Domain Theorem in English? Wikipedia cites the original proof to be in: Beweis der Invarianz des ...
4
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67 views

Mathematical journals (maybe in the past) with regular competitions?

I just finished reading Yōko Ogawa's "The Housekeeper and the Professor". One of the main characters - "the professor" - is a retired mathematician who regularly takes part in contests published in a ...
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70 views

The Leibniz rule in Euler's works

Does anyone know if the Leibniz rule (the method of differentiation under the integral sign), or a variation thereof, has ever appeared in any of Euler's papers? Any references would be appreciated. ...
4
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216 views

Mathematics felt by Srinivasa Ramanujan

At the moment I am reading the book Ramanujan's Papers by B.J. Venkatachala, V. Vinay and C.S. Yogananda; when clarifying some doubt with a professor, he told me that Srinivasa Ramanujan used Galois ...
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42 views

Reference? filler: IRS, Rhind Papyrus, High-school algebra

I believe something like this was included as a filler in one of the MAA journals many years ago. I am searching for the exact reference (for the filler, or an earlier source). Someone dies, and ...
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96 views

Generic Points to the Italians

When I first learned algebraic geometry, I naturally wiki-ed the subject and there was a line there that said the old school Italians used the notion "generic points without any precise definition." ...
4
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Why are 'Groups' and 'Rings' in Algebra called so ? and other nomenclature questions

I have always wondered why are the following concepts/objects in Mathematics named so : 1.) Group 2.) Ring 3.) Exterior derivative (in differential geometry) 4.) Interior multiplication (in ...
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213 views

How much are mathematics driven by applications?

At some point this provocative question came to my mind: Are mathematics mostly driven by applications? I am taking into account some of the comments made to my original question so I want to ...
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The number of solutions of $ax^4 - by^4 \equiv 1$ (mod $p$) for a prime of the form $p = 4n + 1$

Weil writes in his paper "The number of solutions of equations in finite fields" that Gauss finds the number of solutions of $ax^4 - by^4 \equiv 1$ (mod $p$) for a prime of the form $p = 4n + 1$ in ...
4
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174 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
4
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113 views

The mathematical heritage of Lewis Carroll

Which mathematical results has Lewis Carroll, the author of Alice's Adventures in Wonderland, produced? Wikipedia is very vague with regard to this topic and gives us little more than a matrix ...
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117 views

Using other fields of math to simplify a proof.

One of the first non-trivial results given in most courses on algebraic topology is the proof of the Fundamental Theorem of Algebra using topological methods. This is on page 11 of J.P. May's A ...
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231 views

Divisibility notation history

I'm writing a paper project for school about divisibility, so I'd like to include a bit of history about that subject. I'm mostly interested in notation of $|$ sign used in past, but everything else ...
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190 views

Clarification: intersection of a finite number of subgroups of finite index and Poincaré

From Scott's book Group Theory $1.7.10.$ (Poincaré) The intersection of a finite number of subgroups of finite index is of finite index. My question is: Did Poincaré prove the Theorem as stated ...
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210 views

Where does the 'divides' sign come from?

When $a$ divides $b$ we say $a | b$. Where does the $|$ sign come from? This is not homework, just personal interest in the history of mathematical language.
4
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157 views

History Question - Branch Cut

My professor began discussing branch cuts in class today and mentioned that he did not know the origin of the term. Does anyone know the origin of the term and perhaps a source that talks about it?
3
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0answers
73 views

What's so special about binomial coefficients that someone decided to organize them in a triangle?

I know that binomial coefficients are related to figurate numbers (which were studied by Greeks a loooong time ago, because of its connections to geometry). I also understand how the Pascal's triangle ...
3
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0answers
107 views

History of a combinatoric problem: exchanging numbers by throwing stones

Another user recently asked a question on the Puzzling stack: Two spies throwing stones into a river. Suitably generalised, it becomes: Two spies (Alice and Bob) need to exchange a message. Each ...
3
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67 views

Grothendieck's manuscript on differential manifolds

I have a Japanese book on Grothendieck's life and his mathematical works. The author writes that Grothendieck wrote manuscripts(over 250 pages) on "the category of manifolds" and "differential ...
3
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0answers
30 views

History of Moment Generating Functions

I am beginning to appreciate how important Moment Generating Functions (MGFs) are regarding various common probability distributions and the ways their expectations/variances are calculated. My ...
3
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0answers
113 views

Why do Mathematicians use $u$ and $v$ as variables?

I'm sure this has happened to you as well: you are reading some hand-written work, the variables used are $u$ and $v$, and at some point the handwriting becomes unclear and you cannot distinguish the ...
3
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0answers
30 views

Terminology Regarding Basic Properties of Functions

Is there a cultural difference between saying that a function is 1-to-1 or injective, onto or surjective and a 1-to-1 correspondence or bijective?
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95 views

History behind the choice of letters $h$ and $k$ for the vertex of a parabola?

After failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...
3
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0answers
74 views

History of incenter and Euler line

It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and ...
3
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0answers
69 views

Is there a link between level of abstraction and use of numbers?

One of my friend who stopped studying maths in high school told me once You study maths, can you help me fill my tax forms? In her mind, advancing in maths studies implied manipulating an ...
3
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51 views

Question about the first step in Mann's original proof of the Schnirelmann-Landau Conjecture

I was reading Henry Mann's proof for the Schnirelmann-Landau Conjecture from 1942 which can be found in JSTOR here Today, the Schnirelmann-Landau Conjecture is known as Mann's Theorem: $$d(C) \ge ...