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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Original description of point sources and point spread functions
I already asked this question in the Astronomy community, but there it was recommended to me to also try my luck here.
I would like to know the original description of point sources and point spread ...
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How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?
As described here, we know that:
In the foundations of mathematics, Russell's paradox (also known as
Russell's antinomy), discovered by Bertrand Russell in 1901, showed
that some attempted ...
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Did Russell's paradox affect Peano's construction in the Formulario?
Russell's letter to Frege in 1902 was after the first edition of the Formulario Mathematico but before the fifth one (1908).
So I wondered if the discovery of Russell's paradox had made Peano to edit ...
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Is there a theorem proof whose accuracy is doubted because it is short?
Is there a theorem proof whose accuracy is doubted because it is short?
He told me while chatting with a friend of mine. It's about a mathematician who proves a difficult theorem very briefly and ...
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Cauchy integral formula [duplicate]
There is an integral Cauchy formula. It would be nice to know who exactly (for the first time) added Cauchy's name to a formula that Cauchy was not the author of. Who do you think it could be?
But ...
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Jacobi's product for the discriminant
Did Jacobi prove the product formula $\Delta(\tau) = (2\pi)^{12}q\prod_{n \geq 1} (1 - q^n)^{24}$ for the modular discriminant function, and if so where?
I have tried without success to track down ...
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Origin of Riemann-Stieltjes Integral
What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?
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New mathematics theory vs new mathematical theorem
On the Academia site, there is a recent question that asks about obtaining reviewers for a "new theory". I'm only an amateur mathematician, not a professional, and the question got me ...
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When was the first time/s that sheaves entered algebra and algebraic geometry?
I'm interested in knowing about the first published texts in which sheaf-theoretic methods were used in algebra and/or in algebraic geometry.
The oldest instance I am aware of is J.-P. Serre, ...
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What are some concepts/discoveries in mathematics and science that found practical application years after they had been formulated/discovered?
I'm looking for examples of ideas/discoveries/concepts in Maths or Science that had no practical application at first and were maybe considered as nothing but a theoretical concept but they turned out ...
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Use of the verb "induct" in proofs by mathematical induction
Occasionally, in a proof by mathematical induction, the writer will say something like, "We induct on $n$" or "We induct on the number of vertices." This usage of the verb induct ...
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In which work did Euler invent the Euler Substitutions for a quadratic composed into a radical?
A famous technique in the modification of integrands is the set of “euler substitutions” that provide substitutions for the structure
$$\sqrt{ax^2 +bx+c}$$
That is a fairly common occurence in ...
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Early results on the Fourier transform
Published tables of Fourier transform pairs have been available for many years. One such example is the paper by George Campbell in the Bell Systems Technical Journal in 1928. Most such tables simply ...
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Question about Leibniz's "characteristic numbers" and propositional logic
The Wikipedia article on Gottfried Wilhelm Leibniz mentions, in the chapter on symbolic thought, that:
Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers ...
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Philosophy behind category theory
Category theory represented a huge change in the way the community thought about mathematics, leaving its the set theoretic nature behind and bringing up the importance of arrows between the objects ...
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What is the title of the 1676 Memoir in which Leibniz first used the Chain Rule?
On Wikipedia it says:
"The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative. He first mentioned it in a 1676 memoir [ Chain Rule ...
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Origin of the usage of $\lambda$ to represent eigenvalues
I'm curious whether anyone knows how $\lambda$ came to be used to represent eigenvalues and or who (if anyone) was responsible for the convention. I've looked through a couple of books on the history ...
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Where can I find the early proofs for the simplicity of $\text{PSL}(n,q)$?
I am doing a research project on the history of group theory, and want to know about the early developments on the subject. There are plenty of proofs about why the projective special linear group ...
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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 ...
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The Originator of Cobweb Diagrams
A cobweb diagram is a visualization tool that allows one to qualitatively study the iterates of a self-map of the real line based on the graph of the function; here is an example:
(Here the map is ...
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Markov chains origins and how is Christianity involved
In a book called Advanced Data Analysis from an Elementary Point of View by Cosma Rohilla Shalizi, page 405, the first instance of "Markov process" is accompanied by a footnote which reads
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Where did the index of a subgroup notation $[G:H]$ begin to be used?
In texts of algebra, the cardinality of cosets is written in $[G:H]$ or $|G:H|$. Where did this notation originate?
The history about $G/H$ can be found here. $[G:H]$ is called index of a subgroup. ...
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Continuation on Galois’ lost memoir(s): was Poisson right in his review of Galois’ memoir?
A recent question asked whether Galois’ “lost” memoir has been since recovered (similar to Abel’s lost memoir). The memoir referenced in the question is the one Galois submitted to the French “...
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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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When do we see for the first time the use of the Cartesian coordinates?
I want to see an exact image of the first use of the Cartesian plane. I guess it came the first time with Descartes.
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The history and origin of the Argument Principle ( or Cauchy's argument principle)
I am looking for a book that discusses The history and origin of the Argument Principle ( or Cauchy's argument principle) Thanks!
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Was not chess notation an earlier sort of Cartesian Coordinates?
I realize modern algebraic notation is fairly new but for as long as there have been recorded chess games, every square I think was referred to using two coordinates, in old descriptive notation a ...
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How did Gaussian and Eisenstein integers get their names?
I can separate this into two questions at some point if necessary, but it's possible that sources for the answer to one will provide the answer to the other at the same time.
I learned about ...
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How were irrational numbers accepted by mathematicians?
What was behind accepting the existence of irrational numbers historically? Especially numbers that are not constructible on the real number line, say for example $\sqrt[3]{2}$. Was it a (somewhat) ...
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Upon which incorrect equation of Euler did Sophie Germain rely in her work that won a prix extraordinaire from the Paris Academy of Sciences?
Wikipedia's Sophie Germain; Work in elasticity; Subsequent attempts for the Prize says:
Germain had derived the correct differential equation (a special case of the Kirchhoff–Love equation),31 but ...
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What is the history behind the concept of "schemes" in algebraic geometry?
Let me start by saying that I've just started studying algebraic geometry. I have a habit of trying to find out something about the motivation behind new concepts that I'm studying, in this case that ...
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Where can I find Lagrange’s original equilateral triangular solution for arbitrary masses?
This answer to What kind of triangle is formed by three unequal masses in a circular restricted three body orbit? explains that
In the Newtonian limit, an equilateral 3-body solution exists for any ...
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When did bounties and prize money for open mathematical problems start being a thing?
I'm a science/math journalist [ger] and currently I'm working on an article about the culture of prize money/bounties for solving open mathematical problems (Millennium Prize Problems and such). One ...
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Katz's symbol 兄 for Gauss-Manin connections
In his famous 1970 paper [1], Nicholas Katz used the character 兄 for the Gauss-Manin connection. I have always been curious about the history behind this symbol.
Question: What motivated Katz to use ...
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Pefsu problem explanation
Problem no. 12 from Moscow Mathematical Papyrus:
Example of calculation of $13$ heqats of grain
If someone says to you: Take $13$ heqats of grain to make them into $18$ jugs of beer
Note that the ...
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How and when did the dedicated study of locally compact groups begin?
How and when did the dedicated study of locally compact groups begin?
Specific instances from literature, recorded stories, etc., may help supplement the answers. There seems to be no reason why I ...
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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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When did mathematicians first use the contrapositive form to prove a conditional statement?
I am reading parts of Euclid’s Elements and I am surprised, rightly or wrongly, to see that Euclid did not recognize that a conditional is logically equivalent to its contrapositive form. Indeed, one ...
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Why did Clairaut's theorem take so long to prove?
I was reading the Wikipedia on Clairaut's Theorem (Symmetry of second derivatives) and the article accounts the significant amount of time and failed proofs occurred before the theorem was made fully ...
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History of generating functions
I've seen that there is some information in the first volume of Polya's "Mathematics and Plausible Reasoning". Also, the following paper deals with the usage in probability - "The ...
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What was known about Chebyshev polynomials in 1900?
Around 1900, was it widely known that the Chebyshev polynomial $T_n(X)$ satisfies the identity
$$ T_n(X) \circ \frac{X+X^{-1}}{2} = \frac{X^n+X^{-n}}2?$$
And also, would one expect top-notch ...
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Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...
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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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Leonhard Euler's Mathematical Proof of God
There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis ...
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Is Kline right that Cauchy believed that continuous functions must be differentiable?
Morris Kline, in Mathematical Thought from Ancient to Modern Time, writes in chapter 40 (The Installation of Rigor in Analysis), "Though Bolzano and Cauchy had rigorized (somewhat) the notions of ...
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Did Riemann invent the Riemann curvature tensor?
I'm pretty sure they weren't using tensors in the modern sense at that point, but to what extent did Riemann lay out the structure or significance underlying his eponymous tensor? In his habilitation ...
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Why is $Ax^2+Bx+C=0$ called standard form of Quadratic Equation? And who declared it the standard form?
$Ax^2+Bx+C=0 $ is mentioned as standard form of quadratic equation in every textbook or encyclopaedia, but what's so special about it that its called standard form of quadratic equation.
Also, I am ...
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The origin and use of the term "equianharmonic" (elliptic function)
In Weierstrass notation, the principal elliptic function $\wp$ is a solution of the differential equation
$$ (\wp')^2= 4\wp^3 -g_2\wp -g_3. $$
The case when $g_3=0$ is called lemniscatic (it ...
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Poincaré's definition of mathematics (?)
Poincaré is said to have given this definition of mathematics:
It is the art of giving the same name to different objects. I can not believe this quote is from Poincaré.
Did Poincaré really say this? ...
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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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