Questions tagged [mathematics]
For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns
301
questions with no upvoted or accepted answers
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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 ...
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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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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 ...
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How were contour plots of complex functions produced in the days of mechanical differential analyzers?
I was reading an old paper (specifically, the first appearance of the Pearcey function, here) and I was struck by the beauty of the plots it contains, particularly for a paper from 1945-46:
Pearcey ...
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Origin of the special Finnish notation for difference of antiderivative
Apologies for a question that is specific to one country (but perhaps others find it a curious example of how mathematical notation can vary between countries).
In Finnish calculus texts, if $F$ is an ...
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Did John von Neumann hate pure mathematics that became too abstract?
John von Neumann wrote the following in his essay The Mathematician:
As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only ...
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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 ...
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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 ...
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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 ...
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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", the Association president G. N. ...
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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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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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\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 ...
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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 ...
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History of group actions as their own structures
I'm interested in when (and how) the modern idea of a group action developed and how group actions became their own algebraic structures.
As far as I can tell in the 19th century group actions were ...
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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)?
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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/...
user12897
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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
...
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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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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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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 ...
user13915
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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 $\xi^+$ ...
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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 ...
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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 ...
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History of Algebraic Geometry: Morphisms and Birational Geometry
Good people, I'm trying to get my head around the history of algebraic geometry, and while Dieudonné's tome is a very good source (very often the only source), it can from time to time be very ...
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When were arrows first used to visualise vectors?
I guess the use of arrows to visualise vectors came before the general notion of vectors, so a more precise question is: when where arrows first used to visualise physical (or mathematical) quantities ...
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Does "Metatron's cube" have a history and a serious name in geometry?
This is a figure that I saw while going down the rabbit hole of "Sacred Geometry" back when conspiracy theories and related nonsense were relatively harmless and fun to laugh at. A book ...
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Origin of Problem 6 on the 1988 International Mathematical Olympiad
The recent Numberphile video on the famous Problem 6 of the 1988 IMO (mentioned in a recent answer on this site) got me wondering:
Who came up with this problem in the first place, and how did they ...
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Who proved Rank Nullity Theorem?
I have been learning about the Rank Nullity theorem and was trying to understand Who came up with the rank nullity theorem? While i did look up on the internet i came up with almost no answers. Some ...
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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
$$x = \frac{-b \pm \sqrt{b^2 -...
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Photo of Wilhelm Ackermann
I am writing a text on the Theory of Computation. I am looking for a photo of the mathematician Wilhelm Ackermann. He is well-known in the field, was a student of one of the most famous ...
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Were pictorial notations like Feynman diagrams for integrals used before Feynman?
In the book Mathews, Walker: Mathematical Methods of Physics, Addison-Wesley(1969),
there is a pictorial notation of the solution found by Fredholm about an integral equation.p.304, p.305This circle ...
user12897
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Who introduced the comma notation for partial derivatives?
In general relativity, it is common to use the comma notation for partial derivatives
$$\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 ...
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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, ...
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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'...
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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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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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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 ...
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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?
user2084
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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 ...
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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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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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Is there existing footage of Stanislaw Mazur giving Per Enflo a live goose for solving the approximation problem?
There is a famous incident in the history of mathematics involving the mathematician Per Enflo being awarded a live goose by Stanislaw Mazur for solving problem 153 in the Scottish Book by ...
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Who proposed terminating decimals as a major set and why are them important in France?
After looking at some school sources in French, it is common to provide the various number sets in the following order
$$\mathbb{N}\subset \mathbb{Z}\subset\mathbb{D}\subset\mathbb{Q}\subset\mathbb{R}\...
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When was the calculus first part of college curriculum in USA?
Or I guess the Colonies, if it happened before 1776?
I know that mathematics tended to be both applied and emphasized things like taking fifth roots. Also, I think long ago high school and college may ...
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Did the ancient Greeks know that "most" cube roots are irrational?
It is common knowledge that the Pythagoreans discovered irrational numbers (or incommensurability), for example if the hypotenuse of an isosceless right triangle is compared with one of the legs or ...
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Did Rayleigh or Ritz prove the Rayleigh–Ritz theorem?
The maximum eigenvalue of a real symmetric (or complex Hermitian) matrix is given as the maximum of the associated the quadratic form:
$$
\lambda_{\rm max}(A) = \max_{\|x\| = 1} x^*Ax. \tag{1}
$$
This ...
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