Questions tagged [mathematics]

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

111 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
18
votes
1answer
436 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 ...
17
votes
0answers
633 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 ...
12
votes
0answers
303 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 (...
11
votes
0answers
17k 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 \...
8
votes
0answers
627 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
votes
0answers
124 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
votes
0answers
153 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
votes
0answers
563 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
votes
0answers
240 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 ...
7
votes
0answers
130 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
votes
0answers
174 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
votes
0answers
117 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
votes
0answers
212 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
votes
0answers
174 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
votes
0answers
150 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
votes
0answers
227 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 ...
5
votes
0answers
187 views

How was this equation for pi derived by Ramanujan?

Have we figured out how this series was derived by Ramanjuan? $$\frac{1}{\pi}=\frac{\sqrt{8}}{9801}\sum_{n=0}^{\infty}\frac{(4n)!}{(n!)^4}\frac{26390n+1103}{396^{4n}}$$
5
votes
0answers
147 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
votes
0answers
91 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
votes
0answers
221 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?
5
votes
0answers
97 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
votes
0answers
172 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/...
4
votes
0answers
93 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
votes
0answers
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'...
4
votes
0answers
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, ...
4
votes
0answers
121 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?
4
votes
0answers
226 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);...
4
votes
0answers
184 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 ...
4
votes
0answers
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 ...
4
votes
0answers
113 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 ...
4
votes
0answers
124 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?.
3
votes
0answers
96 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 ...
3
votes
0answers
190 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
votes
0answers
110 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
votes
0answers
116 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 ...
3
votes
0answers
91 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 ...
3
votes
0answers
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
votes
0answers
258 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 ...
3
votes
0answers
132 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 ...
3
votes
0answers
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
votes
0answers
115 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 ...
3
votes
0answers
152 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
votes
0answers
148 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
votes
0answers
121 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 ...
3
votes
0answers
276 views

Are there primitive Pythagoras triplets (in integers), being with all the terms as powerful numbers?

I'm searching a trusted historical sources about primitive Pythagoras triplets as being powerful integers (numerical examples), or a notable work of impossibility of such a triples, but couldn't find ...
3
votes
0answers
3k views

History of the geometric series

I'm interested in understanding the history of the geometric series, especially how it was discovered and whether there exists continued fraction representations for the geometric series, just as ...
3
votes
0answers
61 views

Analytic and holomorphic functions, definitions and foundations?

If you search for the definition of analytic and holomorphic functions in books and online, you get crosses between two definitions (one involving taylor series and one differential) I can find no ...
3
votes
0answers
66 views

Origin of alternate base annotation

In modern arithmetic textbooks, students are taught about alternate numeric bases. The notation for indicating the base of a number is to attach the base as a subscript. The subscript is itself a ...
3
votes
0answers
81 views

First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...
3
votes
0answers
130 views

History of the Wreath product

Why is the wreath product so named? If possible, please provide a citation.