# Questions tagged [mathematics]

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

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### 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 ...
726 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 ...
508 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 (...
311 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 ...
735 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 ...
138 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 ...
135 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 ...
297 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:...
1k 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 ...
156 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 ...
186 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 ...
62 views

### What was the first automated theorem prover?

From a lot of googling, it seems like the answer might be "Mizar", but I am not completely sure. What was (or is?) the first automated theorem prover (i.e. not necessarily active right now)?
199 views

### Who coined the Hawaiian Earrings?

I hope to know who first used the name "Hawaiian Earrings." Barratt, Milnor(1962) says "This example was suggested by Steenrod" in its Introduction: https://www.ams.org/journals/...
248 views

### What is the history on the term 'co-domain'?

I am wondering if anyone knows any more on the history of the term 'co-domain' as it relates to functions. Two sources I found: Russell and Whitehead, Principia Mathematica, 1915, page 34 : the class ...
149 views

### Books on elliptic functions

In the end of his address to Annual Meeting of the Mathematical Association in 1933 titled "The marquis and the land agent: a tale of the 18th century", Association president G. N. Watson ...
249 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 ...
100 views

### What is the origin in the discrepancy between engineers' and physicists' notation of waves?

my question is very simple. Physicists use this notation in order to write a (for example) plane wave: $$\xi(z) = \xi^+ \mathrm{e}^{+\mathrm{i}kz} + \xi^- \mathrm{e}^{-\mathrm{i}kz},$$ where $\xi^+$ ...
281 views

### Origin of Tensor Product

When and why did Mathematicians saw a need to define Tensor Products? I want to know the historical development of the idea "Tensor Product"?
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### Did Hardy and Ramanujan miscalculate these values?

When I read Dickson's History Of The Theory Of Numbers Vol-2, I found that there seems to be a mistake in the approximation of partition numbers p(200). For this reason, I found the original text ...
148 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: ...
385 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 ...
135 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 ...
249 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 ...
131 views

### Photo of Wilhelm Ackermann

I am writing a text on the Theory of Computation. I am looking for a photo of the mathematician Wilhelm Ackermann. He is well-known in the field, was a student of one of the most famous ...
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### Were pictorial notations like Feynman diagrams for integrals used before Feynman?

In the book Mathews, Walker: Mathematical Methods of Physics, Addison-Wesley(1969), there is a pictorial notation of the solution found by Fredholm about an integral equation.p.304, p.305This circle ...
128 views

### Who introduced the comma notation for partial derivatives?

In general relativity, it is common to use the comma notation for partial derivatives $$\frac{\partial g_{\mu\nu}}{\partial x_\rho} = g_{\mu\nu_,\rho}$$ Where did this notation first appear? Was it ...
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### 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 ...
183 views

### Origin of the Fourier transform (1878)

I located Joseph Fourier's book, Analytical 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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### 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, ...
108 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 ...
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### 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'...
94 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 ...
228 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 ...
277 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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### 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 ...
130 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?.
116 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 ...
123 views

### Who bet against the usefulness of matrix inversion – or is it a myth?

In my linear-algebra and numerics courses, I frequently heard an anecdote about some professor betting – literally, with money – that there would never be any application where computing the actual ...
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### Equations in right-to-left languages

Is there an historical tradition in languages read right-to-left (Arabic, Hebrew, Urdu, etc.) to display mathematical equations in some right-to-left form? So, instead of x = \frac{-b \pm \sqrt{b^2 -...
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### How did Dyck originally state and prove his theorem in topology about the connected sum of a torus and projective plane?

Dyck's theorem in topology is sometimes stated as follows: the connected sum of a torus and projective plane is homeomorphic to the connected sum of three projective planes. Certainly, this is the ...
108 views

### What role has Whitehead's Conjecture played in the thinking on the plurality of set theories?

I am curious about the history of the Whitehead's Conjecture, as this was the first natural mathematical statement, in the sense that mathematicians were actually interested in the answer, that was ...
94 views

### Lagrange's Theorem as he stated it

In Wikipedia, I found that Lagrange did not state Lagrange's Theorem in its general form. He stated "If a Polynomial in $n$ variables has its variables permuted in all $n!$ ways, the number of ...
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### How did Peano come up with his space filling curve?

Wikipedia cites an earlier result of Cantor as an inspiration but I wonder if there are any previous results of some kind of recursive curve constructions that may have also "inspired" him ...
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I want to understand why, according to several sources, Gauss's discovery of Gaussian quadrature in his 1814 article was "the most significant event of the 19th century in the field of numerical ...
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### What paragraph was written by Emanouil Atanassov to solve problem 6 on the International Mathematical Olympiad in 1988?

From the Wikipedia page about Vieta jumping: Emanouil Atanassov, Bulgaria, solved the problem [assumed to be the most difficult one on the 1988 International Mathematics Olympiad] in a paragraph and ...
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### 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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### 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 ...
220 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/...