Questions tagged [mathematics]

For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns.

138 questions with no upvoted or accepted answers
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18
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486 views

What is the modern significance of Theaetetus's classification of quadratic irrationals?

Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of ...
16
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659 views

Did Kontsevich start a lecture with “one I will not define, the other nobody knows how to define, and we will be proving that they are equivalent”?

The story was circulating in early 2000s, so presumably it happened in 1990s. Kontsevich, it goes, opened a lecture course on mirror symmetry with:"This course is about two categories. One I will not ...
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24k views

Who first defined the “equal-delta” or “delta over equal” ($\triangleq$) symbol?

The symbol $\triangleq$ is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the Unicode Character 'DELTA EQUAL TO' (U+225C). The notation $t \...
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465 views

Conditionally convergent series

I am looking for the original reference discussing a specific, elementary example of a rearrangement of series converging to a value different from the original series. In what follows, I give some (...
9
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0answers
266 views

Did Kronecker say that set theory is not mathematics?

I have frequently come across Kronecker's statement about set theory: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." It ...
8
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84 views

Apéry’s mysterious recurrence relation

A fairly detailed (14 page) account of Apéry’s original proof of the irrationality of $\zeta(3)$ is given in Julian Havil’s book The Irrationals which states that Apéry’s starting point is the ...
8
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679 views

Ramanujan's Method for solving cubic, quartic, quintic

In Ramanujan's Notebooks Volume IV pg. 31 by Bruce C. Berndt, he describes an easy way to solve the general quartic by starting with the system$$x^2+ay=b\\y^2+cx=d\tag1$$ And solving for $x$; which ...
8
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132 views

Contemporary reactions to the rise of axiomatization in the 19th/20th centuries

Starting somewhere in the 19th century, mathematics turned from the study of concrete objects to the study of objects satisfying enough properties to lead to interesting theorems. For example: From ...
8
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177 views

Mathematical counterintelligence at Bletchley during World War 2

Popular works of fiction claim that after breaking the Enigma in Bletchley, some sophisticated mathematics or statistical techniques were used to hide this fact of breaking (not necessarily by the ...
7
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175 views

Who first defined polynomials as sequences?

Question 1. When did the modern definition of a polynomial (as a sequence of coefficients, with multiplication defined by $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$) emerge? Let me clarify:...
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843 views

\mathbb versus \mathbf

When was the use of \mathbb popularized as an alternative to \mathbf? Of course there are the subjective preferences of certain authors, but when I read older articles, there appears to be an almost ...
7
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138 views

Jacobi's product for the discriminant

When did Jacobi prove the product formula for the discriminant function: $\Delta(\tau) = (2\pi)^{12}q\prod_{n \geq 1} (1 - q^n)^{24}$? I have tried without success to track down references (other ...
6
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124 views

F. Schoblik's announced ''ausführliche Darstellung": a lost wrong proof of the Four Color Theorem?

In (the AMS Chelsea Publishing version of) what is perhaps the first genuine textbook on graph theory ever, Dénes Kőnig on p. 28 gives the illustration and the footnote which when translated says ...
6
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218 views

Who gave you infinitesimal epsilon?

As someone reputed among certain historians to have given you the epsilon Cauchy startled me by using $\varepsilon$ to denote an infinitely small number in his 1826 text on differential geometry; see ...
6
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207 views

history of backpropagation

Has anybody read or have access to Alex Andrew Significance Feedback in Neural Nets Report of Biological Computer Laboratory, University of Illinois, Urbana, IL GM-10718-03 TR No 5 September ...
6
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167 views

Who discovered the singular cup product?

Cohomology is a stronger invariant than homology because it can be equipped with a ring structure. To be precise, if one starts with the singular cohomology groups $H^\bullet(-; R)$ with coefficients ...
6
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234 views

Why is Minkowski's question mark function denoted by a question mark?

There are some real odd names for functions in mathematics, but Minkowski's question mark function, denoted by $?(x)$, may be the oddest one I have ever seen. In Zur Geometrie der Zahlen, Minkowski ...
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64 views

Who first called the Brouwer Fixed Point Theorem “the crumpled paper theorem”?

Wikipedia attributes the remark to Brouwer himself, but I am extremely skeptical. Their citation goes to a webpage of a ? French educational TV show, where the remark appears to be a fictionalized ...
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101 views

Origin of Fourier Transform (1878)

I located Joseph Fourier's book, Analytial Theory of Heat (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?
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66 views

Why do Thai numerals look so different than Arabic numerals?

The Arabic numerals I am referring to are “1234567890”. I have read that Thai numerals, “๑๒๓๔๕๖๗๘๙๐”, are actually distantly related. Both descend from the numeral system invented by the Phoenicians, ...
5
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0answers
84 views

What was Littlewood's quip about Hardy and plagiarism?

I'm searching for a quote by Littlewood about Hardy not giving proper credit. The story (as I remember it) is that Littlewood claimed uncredited authorship of something Hardy wrote, Hardy claimed it ...
5
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79 views

What exactly was Lagrange's “grave mistake” with respect to rotating bodies under hydrostatic equilibrium?

A comment below What would be different about satellite orbits if Earth were prolate? Would we have Sun-synchronous and Molniya orbits? got me reading Wikipedia's Jacobi ellipsoid which begins: ...
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184 views

Was there an intentional purge of all audio recordings of Alan Turing?

The YouTube video Alan Turing's lost radio broadcast rerecorded contains a re-enactment of Alan Turing's lecture broadcast by the BBC. In the introduction, the narrator (James Grimes, also of the ...
5
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92 views

Nature of Fermat's friend Lalouvère's activities as censor?

Fermat had a friend at Toulouse named Lalouvère. Lalouvère was censor, jesuit, and mathematician (in alphabetical order). Antonella Romano writes on page 512 of her book La Contre-Réforme ...
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251 views

What is Hensel's lemma a lemma for?

Was Hensel's lemma originally used for proving some other theorem? Or is it meant to be a standalone result? Why is it a "lemma" and not a theorem?
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129 views

Origin of the Hankel contour?

Who was the first to publish a Hankel contour integral? See notes in my answer to the MO-Q How does one motivate the analytic continuation of the Riemann zeta function?.
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101 views

How were the phenomena relating to symmetric polynomials discovered?

The "fundamental theorem of symmetric polynomials" states that any symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials. This, or at least variants on it or ...
4
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1answer
150 views

What is the basis of the claim that $F_5$ was fully factored in 1732?

The Wikipedia Page on Fermat numbers states that $F_5$ was "fully factored" in 1732. This appears to be the same time that Euler found that any factor of a Fermat number $F_n$ was of the form $$2^{n+...
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65 views

Books on elliptic functions

In the end of his address to the Mathematical Association in 1933 titled "The marquis and the land agent: a tale of the 18th century", G. N. Watson says: My final task is to express my gratitude to ...
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81 views

Why did Jordan introduce his canonical form?

Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear ...
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118 views

Reference Request: Comment about Contradictions Proof Method Related to John G. Thompson

I read in a PDF document where the author made a comment that it is “dangerous” to use indirect proof method/contradiction proof method (as far as I can remember, and of course I am paraphrasing) as ...
4
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206 views

How was this equation for pi derived by Ramanujan?

Have we figured out how this series was derived by Ramanjuan? $$\frac{1}{\pi}=\frac{\sqrt{8}}{9801}\sum_{n=0}^{\infty}\frac{(4n)!}{(n!)^4}\frac{26390n+1103}{396^{4n}}$$
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197 views

Entry in Gauss' Mathematisches Tagebuch (Mathematical Diary)

Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his mathematical diary (Mathematisches Tagebuch, full scan available via https://gdz.sub.uni-goettingen.de/id/...
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98 views

Courant (1943) and History of Finite Element Method

I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal ...
4
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59 views

Where is the first reference to the “Z combinator”, a call-by-value fix-point combinator?

I'd like to know the earliest reference to the Z-combinator. This could be either where the name was first coined, or even the first discussion of a need for an applicative-order Y combinator. I didn'...
4
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0answers
111 views

The Principia Mathematica's missing chapters

The numbering of the chapters of Bertrand Russell and Alfred Whitehead's Principia Mathematica is the following : 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 30, etc overall, ...
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126 views

Who was the first known mathematician to graph an equation?

A friend of mine pointed out that there were no graphs in Adam Smith's The Wealth of Nations, which was published in 1776. This surprised me because René Descartes (1596-1650) is well known as being ...
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124 views

How did people calculate the zeros of the Bessel functions before the electronic computer?

The Bessel function appears in many mathematics and physics problems. Their zeros are solutions of many problems. So how did people calculate them at 1900? Did they have some good series?
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94 views

Cryptography in Japan before Meiji

I have a question related to Japan History and cryptography record. As you may know, Meiji is the period in Japan between 1868 and 1912, in which occidental reforming was performed. After (and ...
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0answers
234 views

Did Gauss know Jacobi's four squares theorem?

In p. 283-285 of volume 2 of Dickson's “history of the theory of numbers” appear several formulas of striking similarity: some of them are stated by Gauss (p.283) and some are stated by Jacobi (p.285);...
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209 views

When was the British Flag Theorem discovered or proven?

The British Flag Theorem is a fancy name for a law relating distances from the corners of a rectangle to an arbitrary point. The wikipedia article is small and has no history section. Could not find ...
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1k views

Why Doesn't Einstein Get More Credit for Being the Father of Quantum Mechanics?

I'm not simply referring to the notion that Einstein treated the discrete emission and transference of energy (and matter) as "real" physical phenomena, but rather his major continuous role in the ...
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122 views

A Lecture by Polya on Symmetric Algebraic Equations with an Unexpected Conclusion

Sometime in 1980 George Polya gave a lecture at the University of Minnesota about solutions of algebraic equations that have symmetry in the appearance of the variables in the equation (any ...
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70 views

Literature on Mayan mathematics

I asked this question on math.se and they sent me here. It is well known that Mayan people used vigesimal (base 20) numeral system, and had had an advanced calendar system. Except for these facts, I'...
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65 views

First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
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71 views

What did Dieudonne mean by “Theories in a state of dilution”?

In "A Panorama of Pure Mathematics" by Dieudonne, he said The history of mathematics shows that a theory almost always originates in efforts to solve a specific problem (for example, the ...
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128 views

Did Newton invent convex hulls?

The convex hull of a set of points appears recognizably in a 1676 letter from Newton to Henry Oldenburg describing Newton polygons. Is there an earlier precedent for convex hulls or is this their ...
3
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80 views

Who first proved Fubini's theorem $n$th order integrals?

Who first proved a generalized Fubini theorem for integrals of order $≥3$? An $n$th order integral is $$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\...
3
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0answers
154 views

What was Havil's source for the statement that G.H. Hardy would offer his Savillian chair to whoever could prove $\gamma$ irrational?

In Havil's 2003 book Gamma he states that Hardy offered up his chair in Oxford to whoever could prove that the Euler-Mascheroni constant $\gamma$ is irrational. I'm almost positive I had heard a ...
3
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2answers
195 views

I am searching for a book of this form and content, is there any?

I would like to know is there a book that is both a history of mathematics and a collection of open problems? I know that there exist many books that cover either larger or smaller periods of the ...