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 Cardano trying to say in this passage of his Ars Magna Arithmeticæ?
It is well known that Cardano considered the problem of "dividing 10
into two parts the product of which is 40" in his Ars Magna. This problems leads to the complex solutions $5+ \sqrt{-15}$ ...
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How did someone discover LCM?
How did someone came up with an idea that if we do prime factorization of two numbers and then multiply all the prime factors but including common ones only once, we will get a number that is the ...
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Origin of exact and closed differential expressions
In differential geometry and other fields, an expression involving differentials can be closed or exact. In $\mathbb R^2\setminus\{0\}$ for example, $dr$ is exact whereas $d\theta$ is closed but not ...
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Seeking Comprehensive References on the History of Scientific Notation
I am on a quest to uncover the rich tapestry of history surrounding scientific notation as a way of expressing numbers. Specifically, I'm interested in scholarly books, peer-reviewed articles, and ...
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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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Whence Whitehead's essence?
In the article Quine’s New Foundations of The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Thomas Forster writes:
In [1944] Hailperin gave the first of a number of finite ...
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Was "potency set" used for power set?
Cross posted at Math Overflow
For historical reasons, the English term "power set" in set theory is a translation of the German "Potenzmenge", which is still in use in German ...
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Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved?
I asked this question on MSE and comments suggested I should ask it here
I am currently reading Baby Rudin as my second analysis book (after Introduction to Real Analysis by Robert G. Bartle and ...
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Did anyone apply for a patent based on sphere packing?
Some while ago we had a question about mathematicians patenting their work Examples of mathematicians who applied to patent their work I was about to answer when I realised I needed to find a ...
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Did Fibonacci not grasp the idea of zero?
Indian mathematicians (e.g., Brahmagupta in the 6th century) developed the idea of 0 as more than a placeholder.
In 1202, Fibonacci wrote "These are the nine figures of the Indians: 9 8 7 6 5 4 3 ...
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Group theory in non-European/subaltern cultures?
I'm doing undergraduate research on the history of abstract algebra (specifically permutation groups) and the notion of symmetric groups in indigenous artwork has come up several times. Is anyone ...
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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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Did J. W. Gibbs “invent” Hilbert spaces before Hilbert formulated the notion of such spaces?
I was surprised to see a reply to a comment on his answer to a Quora question by a research mathematician claiming that Hilbert spaces were actually due to J. W. Gibbs rather than to D. Hilbert. The ...
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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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Source of a Poincaré quote: "Logic sometimes makes monsters..."
There's a quote by Poincare on the "new functions", such as continuous functions without derivatives, that were appearing during the second half of the 19th century. The fullest version I've ...
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Etymology of "discrete" in mathematics
People sometimes make a distinction between continuous mathematics and discrete mathematics.
Continuous mathematics study objects that abstract the notion of a continuum and typical examples are the ...
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Who introduced the terminology “nondecreasing” for weakly increasing (i.e. x≤y ⇒ f(x)≤f(y)), and when/why?
Arguably one of the most hated parts of English mathematical terminology is the word “nondecreasing”, referring to a function such that $x\leq y \;\Rightarrow\; f(x)\leq f(y)$ (what other conventions ...
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Can I find the number e in the tables of Napier?
As Napier calculated the logarithms of numbers in the base $\left(1-\frac{1}{10^7}\right)^{10^7}$, I expected to find the number $e$ in the tables of his Mirifici Logarithmorum Canonis Descriptio. ...
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When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?
A nice property of algebraically closed fields is that the theory that describes them ($ACF$) admits quantifier elimination: any statement can be shown equivalent (in the theory) to another statement ...
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History of right hand rule
I am curious to know when the right-hand-rule for vector product was established and used consistently in mathematics.
I read here
Who gave right hand thumb rule for circular loop of current ...
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List of textbooks on Abstract Algebra in the order of time
I am knowing Abstract Algebra things; I am searching aims of Abstract Algebra and origins of parts of Abstract Algebra. I thought original initial textbooks have explicit links to aims and origins of ...
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Who first referred to the number of nonzero entries of a vector as its $\ell_0$ norm?
It is common in the compressed sensing literature to refer to the number of nonzero entries of a vector as its $\ell_0$ "norm." The scare quotes are there because strictly speaking, the $\...
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How to find the first appearance of a theorem?
I often have questions of the "who predicted and proved this theorem when and in what context?" kind.
There are two ways I can think of.
Read books on the history of mathematics.
Find ...
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Did the Romans really use the binomial formula to calculate products?
I'm not quite sure if this is the right place to ask this question (in fact, I was redirected to this SE from the Math Stackexchange), but it's probably more fitting than the original posting place.
I ...
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Any notable large proof that took a long time before anyone checked it?
Have there been any examples in the past where a large proof is claimed but nobody takes the time to check and that has been proved/disproved after a long time? I am not interested in proofs that were ...
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Who was the first to use bijections?
I know that Bourbaki were the first who used the word 'bijection', but one-to one functions were for sure used before them. So do you aware of the earliest examples of one-to-one correspondences?
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Where can I find the Royal Society report on the controversy over the invention of differential calculus?
Where can I find the report on the Leibniz–Newton calculus controversy mentioned in this article?
In 1712 the Royal Society in England wrote a report purporting to settle the matter — except, the ...
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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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Motivation to extend sin and cos to angles > 90 degree
What was from a historical point of view the motivation to extend the definition of sin and cos to angles larger than 90 degree?
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Dissemination of Calculus in China
Much has already been written about the dissemination of Euclidean geometry into China: https://www.maa.org/press/periodicals/convergence/mathematical-treasure-euclid-in-china, https://academic.oup....
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Why did Abel choose 6064321219?
In August 1823, Abel wrote a letter to Holmboe with a date:
Copenhague, l’an $ \sqrt[3]{6064321219} $ (en comptant lafraction d´ecimal).
$ 1823 \frac{215}{365} < \sqrt[3]{6064321219} < 1823 \...
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What were the initial physical applications of vector calculus such as curl, div, circulation, and flux?
In what context where vector calculus concepts, such as:
Circulation
Flux
Curl
Divergence
first developed? I had assumed they were developed first in fluid dynamics, since the flow of water is ...
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Unramified étale cohomology groups & unramified Milnor-Witt K-groups: any relation to notion of unramifiedness in number theory
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every point $x \in X^{(1)}$ of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
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History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring
In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (if you don't know what a scheme is, you can read the definition for a commutative Noetherian ring ...
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Lefschetz historical proof of Hyperplane Theorem
I would like to understand the the idea behind the historically original proof by Lefschetz of his Hyperplane theorem sketched roughly Here. The basic setup:
Let $X$ be an $n$ -dimensional complex ...
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Teiji Takagi's Fondation of Class Field Theory in Terms of Norms
I have a question about Teiji Takagi's original motivation for his groundbreaking reformulation of class field theory implementing the ideal norms $N_{\mathfrak{m}}(L/K)$ as new "key players"...
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When did modular forms start to get studied via algebraic geometry?
I'm looking, for instance, as to when people started studying modular curves and modular forms as sections of line bundles on them, as opposed to the point of view of modular forms being holomorphic ...
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Cantor, set theory and foundations
Did Georg Cantor ever think that set theory could serve as a foundational system for all of mathematics?
He died in 1918, but Zermelo set theory (just Z, no ZF or ZFC yet) was described in a paper by ...
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Earliest mention of permutation matrices, or equivalent? More generally, matrices for arbitrary functions between finite sets?
Permutation matrices I assume have a long history, and would be surprised if they were first considered only long after the work of Shur just after 1900, on the representation theory of $S_n$.
...
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Why did Hilbert believe consistency implies existence?
I am reading Sieg's "Hilbert's programs and beyond" and I am having difficulty understanding this quote by Hilbert on page 74:
In the Paris Lecture Hilbert re-emphasized and expanded this ...
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Sylvester's Quote on Determinants
What does the following quote by Sylvester mean?
"A general algebraical determinant in its developed form may be likened to a mixture of liquids seemingly homogeneous, but which, being of ...
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Was the small Desargues Theorem known to ancient Greeks?
My question concerns the classical Desargues Theorem and its simplest version
The small Desargues Theorem: Let $A,B,C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$, be ...
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In historical studies of mathematics, how to chose the corpus of texts to work with?
When studying the evolution of some concept in the history of mathematics, is there some established practice or methodology for selecting the corpus of texts to work with, or at least for selecting a ...
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First use of ~ and ≍ (\sym and \asymp)
The relations ~ and ≍ are frequently used in math and computer science, at least within number theory and analysis of algorithms. What is their origin?
Definitions
Suppose $g(x)$ is an eventually-...
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Did ancient Greeks have a numerical value for the Golden Ratio
Did they calculate a numerical value for the "extreme and mean ratio" or did they just have ways to construct it geometrically? If so, what value did they use and how did they calculate it?
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17th or 18th century use of the continued fraction expansion of $(1 + \sqrt D)/2$ to solve the diophantine equation $x^2 - D y^2 = 4$
Can someone please provide an early reference to the use of the continued fraction expansion of $\frac{1+\sqrt D}2$ to solve the Diophantine equation $x^2 - D y^2 = 4$ for a positive integer $D$ ...
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Why doesn't John Snow's Voronoi diagram look like one? How was the diagram made? (distance to cholera-spreading water pump in 19th century London)
After about 03:03 in Dr. Trefor Bazett's June 2023 video Why this pattern shows up everywhere in nature || Voronoi Cell Pattern there is a discussion of a diagram representing work by John Snow
John ...
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Who was J. M. Gandhi?
I am looking for more information about J. M. Gandhi who is
the creator of Gandhi polynomials. A MathSciNet search finds
only one J. M. Gandhi with 35 publications in Number Theory,
3 in Combinatorics,...
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To what extent were Riemann surfaces a precursor to algebraic geometry?
I read that Riemann started studying the so-called Riemann's surfaces in the second half of the 19th century, introducing tools like meromorphic functions and meromorphic 1-forms. The culmination of ...
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Arrow heads on coordinate axes
Where did arrow heads on coordinate axes appear for the first time?
I checked Descartes' "Discourse on Method, Optics, Geometry, and Meteorology"
but he does not use any arrow heads.
In ...
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