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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When and why was the concept of "having a least upper bound" dubbed "completeness", as in Axiom of Completeness?

The Axiom of Completeness states that any non-empty set with an upper bound has a least upper bound. When and why was this concept of least upper bound dubbed "completeness"? It's true, of ...
SRobertJames's user avatar
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What does Dedekind mean by "laws characteristic for the concepts"?

I’m slightly confused by what Dedekind means by “characteristic for the concepts they designate” in the quote below: "But [. . . ] these extensions of definitions no longer allow scope for ...
Jerry's user avatar
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Notation for Propositional values in Church's "Simple Theory of Types"

In Alanzo Church's "A Formulation of the Simple Theory of Types" (The Journal of Symbolic Logic 5 no.2 (1940) 56--68, DOI:10.2307/2266170), he adopts the ...
Alex Nelson's user avatar
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Did the principle of permanence have an influence on mathematicians like Dedekind and Cauchy?

Around the time when mathematics was becoming formal, the notion of detaching from attaching "contextual interpretation" to symbols in algebra, up to the point of avoiding inconsistency (...
Demon's user avatar
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What were Auguste Comte's contributions to mathematics (if any)?

Auguste Comte is often described (e.g., on Wikipedia) as a “mathematician” besides being a philosopher of science. I am aware that he taught mathematics (he was at various times a répétiteur and/or ...
Gro-Tsen's user avatar
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0 answers
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I would like to read about Euler's view on negative numbers

So, I've been over fixated on negative numbers lately. I'm coming to the conclusion that, mathematics is usually progressed if it is "useful". The more "useful" a mathematical ...
Demon's user avatar
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1 answer
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Why and how did the study of complex numbers progress despite the denial of negative numbers?

I am going over some history of the complex numbers, and two things baffle me (and they are not mathematics). From Cardano's time to around the 18th century, negative numbers were not accepted by all ...
James's user avatar
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When was it the negative numbers were accepted as “negatives” instead of “subtractions” in European Mathematics?

After Cordano and Bombelli had ways of dealing with “subtractions” and “imaginary” numbers. John wallis published a book where he examined “negatives” motivated by physical and scientific applications ...
Fraser's user avatar
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How did negative numbers “force themselves” onto Cardano, and was it analogous to how imaginary numbers were forced upon him?

I was reading “A brief history of numbers” by Corry, but I came across a part that confused me. Cardano accepted the law of signs for “subtractions” proposed by an older group of Italian ...
Fraser's user avatar
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2 votes
1 answer
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Did Rafael Bombelli write any commentary about his rules for arithmetic involving negative numbers?

Rafael Bombelli was the first European mathematician to write about the laws of arithmetic for negative numbers. On Wikipedia I read that he wrote: “Minus 5 times minus 6 makes plus 30”. I also read ...
Fraser's user avatar
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3 votes
1 answer
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History of cohomology theory

I saw this post. And I already posted it on Math stack exchange, but since someone recommended this site, I'm refining it and posting it again. And I understand that the mathematical object called ...
user1274233's user avatar
1 vote
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66 views

How exactly did Auguste Bravais come up with the regression line?

I am new to statistics and linear regression and I came across the face that auguste bravais discovered regression line but didn't realize it. Auguste Bravais (1811-1863), professor of astronomy and ...
Alexander's user avatar
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3 votes
1 answer
111 views

Mathematization of natural sciences

When was a mathematical formula (instead of just words) used for the 1st time in natural sciences to describe a natural phenomenon?
Sedat Olcer's user avatar
1 vote
1 answer
103 views

David Hilbert's paper: Substitution of the group of cyclotomic field

A question about a notation in David Hilberts's "Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper" (here a german online available source, not sure if there ...
user267839's user avatar
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0 answers
88 views

History of Bernoulli numbers

I have been trying to understand what is the meaning of Bernoulli numbers, but to my mind it has been obscured behind complicated formulas without much explaination. I presume finding the history ...
Gustamons's user avatar
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1 answer
345 views

Ancient drawing board in mathematics

According to Van Der Waerden's "Science Awakening", it was common for Ancient Greek mathematicians to use a board filled with sand to draw their figures, ie : But the ancients made their ...
Slereah's user avatar
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1 answer
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How good was Newton at definite integration?

On Math Stack Exchange, I am impressed by users' skill at finding closed form expressions for definite integrals. For example: Example 1: $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^...
Dan's user avatar
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1 answer
774 views

What were the obstacles that made the discovery of calculus very late?

I wonder What were the obstacles that made the discovery of calculus very late ? Why the discovery of calculus took so long. I know that some of the ideas and techniques of calculus appeared in ...
pie's user avatar
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1 vote
1 answer
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Is there any meaningful history behind harmonic mean?

I am trying to understand the origin of harmonic mean and get an intuitively feel for why it was invented in the first place. I have surfed the web but I keep seeing things like harmonic series / ...
Alexander's user avatar
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6 votes
5 answers
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Why was the development of mathematics very slow between Ancient Greece and Descartes?

I asked this question on MSE here In my studies of mathematics (I am not very good at mathematics, I only studied real analysis, some linear algebra, geometry and calculus ), I noticed that ...
pie's user avatar
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8 votes
2 answers
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When did "neighbourhood of a point" first appear in the history of Taylor series?

I am trying to track down at what point mathematicians started to use the terminology of expanding a function "around a point" or in the "neighbourhood of a point". Neither Taylor ...
StormyTeacup's user avatar
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2 answers
268 views

How was addition and multiplication of natural numbers defined before 1870 (Cantor and modern set theory)?

I know how to define addition and multiplication of natural numbers using set theory, but I think that before Cantor, mathematicians did not try to use set theory as a foundation for mathematics (I ...
pie's user avatar
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1 vote
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How does Math look on Different Planets? [closed]

I have had this theory for a while - basic principles in math were likely known by early humans such as caveman (and possibly some level of understanding might even be possessed by animals). I think ...
stats_noob's user avatar
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What is the earliest documentation of counting and what was it used for?

Counting has been in observed in almost every culture across the world. I think it developed because it proved to be so useful for a range of tasks. I am wondering what the earliest use cases for it ...
Fraser's user avatar
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1 answer
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The history of motivations [closed]

Most histories, that I've encountered, of mathematics about the 18th century and onward focus on a chronology of publications, results, definitions, and similar "pure" interests. However, I ...
Addem's user avatar
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What were people looking for when they started to study bounded linear functionals?

Very briefly, my understanding of the initial motivations for studying $L^p$ spaces included interest in the $\ell_2$ space, due to relationships with quadratic forms that arose from searching for ...
Addem's user avatar
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1 vote
1 answer
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How do I obtain the first column of Plimpton 322 from the other columns?

How is the first column of plimpton 322 derived from the other columns? Consider the first row of the tablet. The second, $b,$ and third, $ c,$ column are pythagorean triples $(a,b,c)= (120, 119, 169)$...
Chris's user avatar
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3 votes
1 answer
107 views

Which geometer first compared a length (one dimensional) to an area (two dimensional)?

What are sources placing a length (one dimensional) in proportion to an area (two dimensional)? The Greek geometers compared quantities of the same dimension: e.g. the area of a circle is in ...
SRobertJames's user avatar
3 votes
1 answer
236 views

How to "prove" recent analysis theorems without rigor?

I asked this question on MSE here but I was told it would do better here I always wonder how mathematicians proved theorems before Cauchy’s epsilon-delta proof. Since many "recent" theorems,...
pie's user avatar
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Where can I find Bourbaki's Latin telegram?

In the Bourbaki archives in Lorainne, France, there appears to be a Latin telegram from Nicolas Bourbaki to Marcel Brelot (see reference 486 at https://iecl.univ-lorraine.fr/files/2021/04/...
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Etymology or genesis of the concept of proof

I'm wondering about the origin of the concept of proof in human thinking. Nowadays the concept is a cornerstone of mathematics, but also in law. Did early mathematicians maybe adopt it from the ...
user313032's user avatar
1 vote
0 answers
40 views

Usage of the word "autonomous" to refer to time-invariant differential equations

At what point in time did the term "autonomous" begin to signify time-invariance within the context of dynamical systems, and what prompted this association?
shamisen's user avatar
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2 votes
0 answers
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History of logarithmic potential

I have some historical questions in connections for the notes to a book I am writing. Who the first person to discover that the Coulomb potential in two dimension is $\log(|x|^{-1})$, equivalently ...
Barry Simon's user avatar
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50 views

Why is the standard deviation bias correction factor called c₄?

The term to remove bias from an estimate of standard deviation for a normal distribution is referred to as $c_4$. What is the origin or reason for using that notation for the correction factor?
feetwet's user avatar
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8 votes
1 answer
1k views

DeMorgan's commentary on Euclid's Elements

Augustus DeMorgan wrote comments on Euclid's Elements, which capture many of the most important points. Heath quotes them extensively. I cannot find any source for the original: Where can I see ...
SRobertJames's user avatar
6 votes
1 answer
1k views

What was the role of Schmidt in derivation of the Gram-Schmidt process?

When reading the section related to Gram-Schmidt process in the book Linear Algebra and Its Applications by Gilbert Strang, I found a foot note that says: If Gram thought of it first, what was left ...
Tran Khanh's user avatar
5 votes
1 answer
62 views

Are adjoint operators historically related to integrating factors?

Birkhoff and Rota, in their book Ordinary Differential Equations (4e), claim on p.55 that: The concept of the adjoint of a linear operator, which originated historically in the search for integrating ...
Alp Uzman's user avatar
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10 votes
2 answers
2k views

How do we explain the lack of activity in the study of Latin mathematics?

A full professor teaching the history of mathematics at Masters level recently told a friend of mine that there was nothing of interest left to explore in the mathematics written in Latin over the ...
user19422's user avatar
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7 votes
2 answers
250 views

Who first stated the "uncertainty principle" for Fourier transforms?

My question is clearly related to this one, but my interest is not specifically in Heisenberg's result. To quote from Wikipedia. A nonzero function and its Fourier transform cannot both be sharply ...
CrimsonDark's user avatar
3 votes
0 answers
103 views

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 $5+ \sqrt{-15}$ ...
Charles Bukowski's user avatar
1 vote
0 answers
137 views

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 ...
Steve's user avatar
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6 votes
2 answers
670 views

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 ...
Mikhail Katz's user avatar
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3 votes
2 answers
186 views

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 ...
Humberto José Bortolossi's user avatar
4 votes
0 answers
128 views

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 ...
James Hanson's user avatar
1 vote
0 answers
83 views

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 ...
Frode Alfson Bjørdal's user avatar
1 vote
1 answer
172 views

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 ...
Frode Alfson Bjørdal's user avatar
4 votes
2 answers
437 views

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 ...
pie's user avatar
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4 votes
1 answer
156 views

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 ...
mdewey's user avatar
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2 votes
1 answer
192 views

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 ...
user19226's user avatar
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0 answers
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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 ...
zomzoms's user avatar

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