Questions tagged [mathematics]

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

127 questions with no upvoted or accepted answers
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18
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1answer
465 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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651 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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21k 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 \...
12
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1answer
359 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 (...
8
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0answers
256 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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655 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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131 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 ...
7
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166 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:...
7
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0answers
696 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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135 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 ...
7
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176 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 ...
6
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123 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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215 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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191 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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158 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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233 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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75 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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72 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: ...
5
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168 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 ...
5
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237 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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99 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 ...
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60 views

Did Gosper or the Borweins first prove Ramanujans formula?

This is a copy of my question on MSE (https://math.stackexchange.com/questions/3372432) because this forum seems better suited for historical questions: In 1985, Gosper used the not-yet-proven ...
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114 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 ...
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196 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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188 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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96 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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55 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'...
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109 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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121 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 ...
4
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122 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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230 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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192 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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118 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 ...
3
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119 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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77 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,\...
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205 views

Were ancients really so bad at computations before Arab numerals?

It is often said that Romans (see below) had a terrible number system, which made a computations a mess. I do believe this, but I'm really suspicious of the claim that nobody had better ways to do ...
3
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2answers
169 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 ...
3
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2answers
205 views

Why is the representation of the direction of the x and y axes in two dimensions different than in three dimensions?

So I apologize if this question seems a bit nit-picky, but it has bothered me for a while. Usually when a coordinate system is represented in two-dimensions, the x-axis is pointing towards what might ...
3
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0answers
165 views

Discovery of the Power Series Form of the Exponential Function

How was the power series form of the exponential function disovered? Was it just observed? By the exponential function, I mean the solution to the differential equation $\frac{df}{dx} = f$ with the ...
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0answers
95 views

Earliest drawings of the plots of trigonometric functions

[Even though this question may seem as a duplicate of this question about the History of sine function, I'd like to ask it again - with a more specific title and a more specific focus (on specific ...
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74 views

When was the nine point conic discovered?

I wonder when was discovered the nine point conic. English Wikipedia article about it https://en.wikipedia.org/wiki/Nine-point_conic is misleading. The nine point conic wasn't discovered in 1892. In ...
3
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270 views

Strange pattern in Math Genealogy

Math Genealogy, https://www.genealogy.math.ndsu.nodak.edu/search.php is a funny site which aims at listing all PhD's in mathematics, with years, place, titles and advisers. Of course it cannot be ...
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181 views

Priority on lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). Historically who has priority ...
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64 views

Alligation - when and why did it disappear?

I have a book from 1795 where the mixing of quantities is done by Alligation. Depending on the supplied data this can be Partial, Alternate, Medial or Total. When I asked several teachers 'how would ...
3
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153 views

What is the name given to the principle that guides mathematical conventions like the product of two negative numbers is positive

I recall that I read---in a book by Constance Reid---of a named principle that guided the arithmetic conventions that applied to operations on newly discovered mathematical objects. For example, when ...
3
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0answers
149 views

Why are rings called rings?

I copied the question from https://math.stackexchange.com/q/61497/378968 because I think it is more suitable for this site and I think an answer to this question here could do better than: Hilbert ...
3
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0answers
125 views

In the scholastic challenges of renaissance Italy, what restrictions were considered appropriate regarding the incumbent's choice of subject?

EDIT Following Mauro's comment, I have altered my question to ask only about any restrictions that may have been considered concerning the suitability of the incumbent's choice of questions for the ...