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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Who first proved that there is no five-digit perfect number?

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The first five perfect numbers are 6, 28, 496, 8128, and 33550336. Nicomachus ...
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Source of a Quote by M. Stone on Poincaré and Bourbaki

The quote in question is the following: For Bourbaki, Poincaré was the devil incarnate. For students of chaos and fractals, Poincaré is of course God on Earth. The common reference for this quote ...
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What is the historical significance of Riemann’s discovery of non-flat metrics?

From a perspective of math history, what discoveries resulted directly from Riemann’s discovery and codification of non-flat metrics on Riemannian manifolds? After Albert Einstein modeled the Universe ...
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Reference for "A manifold is a topological space which satisfies a long series of axioms."

In On teaching mathematics, Vladimir Igorevich Arnold states "What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion ...
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Were 3-dimensional split-complex numbers ever described in literature?

Basically, if you add two complex dimensions to reals, say $i$ and $j$, you automatically get a fourth dimension $ij$ because this number cannot be expressed using only the three dimensions. The ...
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Origin of Catalan numbers and original intuition of the first interpretation and proofs

I was reading about Catalan numbers and how they provide the number count in a variety of situations. I found it very interesting how the main proof is about the underlying common patterns and ...
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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 book did Maria Gaetana Agnesi write which contained both differential and integral calculus?

Wikipedia says the following about Maria Gaetana Agnesi: She is credited with writing the first book discussing both differential and integral calculus and was a member of the faculty at the ...
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Intuitions for Frobenius' generalization of characters to nonabelian finite group given the historical context

I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius. According to most of the related papers (e.g. Pioneers of ...
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Where did we find Ramanujan's series for the exponential integral?

According to Wikipedia, the following series for the exponential integral $$\operatorname{Ei}(x) = \gamma \ln x + \exp{\frac{x}{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^n}{n! 2^{n-1}} \sum_{k=0}^{\...
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Who first used exponential function as a solution of second order differential equations? [duplicate]

You can refer this question from math SE which explains why exponential function is used as a solution of second order differential equations. However, I am interested to know who was the ...
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2 votes
1 answer
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A randomly started branch of math

In the past a lot of math was motivated by practical applications from real life; that's how geometry started for example. Some other areas were developed when trying to solve problems that already ...
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Context of the discovery of ultraproducts

Łoś's theorem is a fundamental theorem in model theory (a branch of mathematical logic). Historical question: What was Łoś's original motivation to define ultraproducts and prove Łoś's theorem? Which ...
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4 votes
1 answer
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Who, between Cayley and Hamilton, first worked on the theorem that bears their name?

I know that Frobenius is the one who proved the Cayley-Hamilton theorem in all its generality. However, between Cayley and Hamilton, who did first work on the subject? In English: Cayley–Hamilton ...
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Frege: Defining a universe of ordered pairs?

Does anyone know how to define the set (or class) of all ordered tuples? In Frege's Begriffsschrift a tuple $(x, ..., z)$ is defined as a primitive object which is provided as an argument to a ...
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Did the logarithm function, with a continuous domain, come before the exponential function? [duplicate]

I was digging about the discovery of logarithms by John Napier. It is clear that he was looking for a function he could compute (at least approximately) that would map products into sums. He came up ...
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11 votes
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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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4 votes
3 answers
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When did linear algebra become the study of vector spaces?

All of the concepts and terminology central to linear algebra were established in the late 19th century. Following recent comments by user KCd, that in the early 20th century determinants were the “...
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Early uses of the word "operator" in mathematics

I'm looking for early uses of the word "operator" in mathematics and how it evolved to become a synonym for the modern "function" in certain areas. Jeff Miller's page only has an ...
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2 votes
0 answers
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Why is the ring of algebraic integers denoted by $\mathcal O_K$?

Why/when was the curly-O notation chosen for the ring of integers of an algebraic number field $K$?
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Research on Pre-Columbian Polynesian geometry?

Has there been any historical/cultural anthropological research into how Polynesian cultures understood geometry before contact with Europeans? In part what I am interested in is how did the dealing ...
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3 votes
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Can not find reference for "uniform convexity implies existence of unique conjugate" mentioned by Pettis

In A proof that every uniformly convex space is reflexive in footnote 3 (available at that link without a paywall), author Billy Pettis mentions that the first half of Lemma 1 in that paper "was ...
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$\pi(x)$ and $\operatorname{li}(x)$ cross infinitely many times -- due to Littlewood 1914 or Schmidt 1903?

In Montgomery-Vaughan's Multiplicative Number Theory I: Classical Theory, they prove in Theorem 15.3 that $\pi(x)-\operatorname{li}(x) = \Omega_{\pm}(x^{\Theta-\epsilon})$ for every $\epsilon>0$ ...
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Looking for math history but keep finding the same old stuff

I've browsed many math history books, but I've never read too deep into any single one. I always find myself reading the about the same facts and same people over and over -- the set of topics doesn't ...
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23 votes
9 answers
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Are there any mathematical objects that got renamed over time?

I'm wondering if there are any mathematical objects that were given a name when first discovered (and wildly used at their time), but then got renamed to match their characteristics later? Counter ...
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What were the standard algebra texts used in elite French universities during the 20th century?

France was home to some of the greatest mathematicians of the 20th century such as Weil, Serre, Grothendieck, and Deligne, just to name a few. What algebra texts did they study from during their ...
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What is the grounding of commensurability?

I understand that before Hippasus of Metapontum proved that the square root of 2 is an irrational number, it was commonly assumed that, given two line segments, it would be possible to find a third ...
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2 votes
1 answer
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Separation between Physics and Mathematics

When and under which circumstances did Physics and Mathematics take separate routes? Even though connections between Mathematics and Physics have been strong and prosperous at all times the methods, ...
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Are there examples of mathematical systems that reflect the worldview of the culture they are from?

By this I mean, are there systems of numbers, counting, sorting.. or even higher level mathematical concepts that reflect a culture's worldview? Did some cultures not develop certain math concepts ...
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9 votes
0 answers
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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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2 votes
1 answer
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Grothendieck and Fields medal 1962

We can read as a mathunion excerpt that Grothendieck won the Fields medal in 1966 Built on work of Weil and Zariski and effected fundamental advances in algebraic geometry. He introduced the idea of ...
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Questions related to the theorem von Neumann proved in five minutes

I have recently seen this post about the theorem von Neumann proved in five minutes. Searching for more information about that, it is certain that I can't find any references for what the theorem was. ...
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1 answer
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Physical theories and Mathematics [closed]

I study pure mathematics. In pure mathematics, we begin from some axioms and obtain theorems. I am also interested in studying physics. I have some questions about the relationship between physical ...
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0 votes
1 answer
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Scientific Collaborations during World War

Nowadays, scientific progress is often based on very big collaborations, like the discovery of gravitational waves by the Ligo and Virgo collaborations. But also in many other branches of science, ...
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3 votes
1 answer
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Why is this notation used to define points in (elementary) analytic geometry?

I have always found strange that in elementary analytic geometry points are defined by their names followed by their coordinates, for example: "Find the distance between $A(5, -3)$ and $B(2, 1)$....
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4 votes
1 answer
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Relation between Bourbaki group and Vienna Circle

Background: I'm an undergraduate student in Mathematics and I study Mathematical Logic and Philosophy of Science in an undergraduate research project (here in Brazil, where I'm from, we call this kind ...
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0 answers
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Origin of other names of the Chinese remainder theorem [duplicate]

I went to Wikipedia and decided to look at the official Chinese article of the Chinese remainder theorem. Google translating back to English it is roughly called "Han Xin counting soldiers". ...
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1 vote
4 answers
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Is there any theorem/physical law with different names in more than two different languages/regions?

Mathematical statements and physics equations often are named after a person (like Pythagoras theorem or Newton's second law). Reading from different authors with different origins one may sometimes ...
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6 votes
1 answer
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Why did Borel reject countable additivity of probability?

On page 26 of the 2003 article - The origins and legacy of Kolmogorov's Grundbegriffe by Shafer and Vovk, it is stated that Borel rejected the countable additivity axiom of probability theory as he ...
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Publication information on "Recreational Mathematics Magazine"

I'm currently studying Martin Gardner's 1968 repackaging of two of Henry Ernest Dudeney's collections: "536 Puzzles & Curious Problems", and trying to track down all the references in ...
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3 votes
0 answers
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Origin of the "teakettle principle" joke?

There's a fairly widely known joke about boiling water (one version is below) that pokes fun at how mathematicians like to reduce new problems to known solutions. I've traced it back to a footnote on ...
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4 votes
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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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3 votes
1 answer
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Historical origin of commas and periods in numbers

I have heard that in English when we write a decimal number, then we use a period, like 1.111. When we write large numbers, then we use a comma, like 1,111. Sometimes we use spaces for large numbers, ...
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4 votes
0 answers
109 views

Explanation request for the terminology and notation employed by Gauss in his major 1843/6 treatise on Geodesy

Background: In his 1827 treatise on differential geometry, Gauss in his "theorema egregium" proved that the curvature of a surface is an intrinsic invariant; it doesn't change under ...
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5 votes
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What is the mysterious expression mentioned by Hadamard that Riemann derived properties of zeta function from?

In the Chapter 7 epigraph in Stein and Shakarchi's Complex Analysis, there is a quote, apparently from Hadamard in 1945, saying He proved some important properties of that function, but enunciated ...
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Classification of "Epitaph of Diophantus" problem

The well-known riddle of the Epitaph of Diophantus, attributed to Metrodorus, is one of the style of simple problem in algebra whose pattern when expressed in contemporary algebraic notation is: $$x = ...
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3 votes
1 answer
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Is there a translation of Kronecker's original work on adjoining a root of a polynomial to a field?

I would be interested in reading how Kronecker formally approached this construction, using the mathematical ideas of his time, and possibly some insight as to what he considered its philosophical ...
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1 vote
1 answer
106 views

History of difference equations

Could you please give me some references about the history of difference equations. Thank you for your reply
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Cantor's Art of proposing a question

There is a rather famous quotation of Cantor: In mathematics the art of proposing a question must be held of higher value than solving it. It was in a thesis he defended for his doctoral examination ...
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1 answer
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Was Kolmorogov really inspired to publish Foundations of the theory of probability because he needed money to repair his dacha?

Background It is well known that Kolmogorov published in 1933 his ground-breaking work on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung usually known to English speakers as ...
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