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 where was Legendre's Conjecture first published?

When and where did Legendre first publish or write about his conjecture that there is a prime between consecutive square numbers? $$n^2 < p < (n+1)^2$$ I have looked through edition 1 and 2 of ...
2 votes
1 answer
100 views

Which are the most ancient theorems that negate the existence of some deeply valued entity?

Are there theorems earlier to the works of Abel and Galois showing that a problem that mathematicians had been trying to solve for a long time was actually unsolvable? Or are those works novel in the ...
0 votes
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 ...
0 votes
0 answers
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Do I need to understand Euclid's Elements before reading the Almagest? [closed]

I've been looking at Toomer's translation of the Almagest, and in it, he provides a section on how chords were formed from Euclidean geometry. However, I find that it's difficult to learn Euclidean ...
0 votes
0 answers
312 views

Has Cantor's irregular enumeration of rationals ever been discussed?

Enumeration of all positive fractions recently has gained renewed interest (see the list below). By translation invariance we can be sure that in all intervals (n, n+1] of the real axis, there are the ...
5 votes
1 answer
130 views

Where can I find the complete papers of abstracts published by P. G. Tait in Proc. Roy. Soc. Edinburgh in 1880?

I am interested in looking up P. G. Tait's flawed proof of the four-colour theorem, published in 1880. The citation that I have seen is: P. G. Tait, On the colouring of maps, Proc. Roy. Soc. ...
3 votes
1 answer
145 views

Why is the letter $b$ used to represent the y-intercept in the equation of straight line?

The slope-intercept form of a non-vertical line is $y=mx+b$. I have been told that the slope is called $m$ because it is the first letter of the French word for mountain. But why is there the letter $...
5 votes
1 answer
106 views

Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?

In differential geometry, if $e_1$ and $e_2$ are bases for a tangent space $T_pM$, then the coefficients of the first fundamental form is: $$\begin{align}E&:=\left<e_1,e_1\right>\\F&:=\...
2 votes
1 answer
128 views

What mathematical expressions took a long time for a closed form to be found?

The Basel problem, $\sum_{n=1}^\infty{\frac{1}{n^2}}$, took 90 years for a closed form, $\frac{\pi^2}{6}$, to be found. I'm curious to know what other mathematical expressions, especially those ...
2 votes
0 answers
82 views

When did ordered pairs first appear?

I am interested in the history of the ordered pair. When did ordered pairs as a concrete object, not necessarily defined in terms of sets, appear in a mathematical or logical setting? The definition ...
3 votes
0 answers
80 views

What were the main different homology theories in 1930 and how did they get united?

I am reading The Uses and Abuses of the History of Topos Theory by Colin McLarty. On page 353, the following is said: An homology theory associates groups to topological spaces so that the group ...
0 votes
0 answers
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What does P. G. Tait mean by "double points"?

I am reading the following short paper: P. G. Tait. Some elementary properties of closed plane curves. Messenger (2) 6 (1877), 132–133. JFM 09.0393.01 Here, Tait writes The closed curves ...
1 vote
2 answers
125 views

Discussions of why the Greeks "squared the..." geometrically

The Wikipedia article on squaring the circle has a relatively good history of their efforts on this particular problem, but it fails to mention why the Greeks were interested in this methodology on a ...
6 votes
6 answers
2k views

Has a digit ever been used to represent the number "10"?

Ten is special to humans, as there are 10 fingers on two hands, and fingers are still the basic counting medium for people. So, was there any digit representing the number "10" in a positional system ...
1 vote
0 answers
135 views

Who came up with the idea of variables in mathematics?

Who came up with the idea of using variables in mathematics to represent unknown numbers and other unknown entities? It must have been a big conceptual revolution, the idea of variables. Prior to that ...
3 votes
1 answer
311 views

What mathematics did Isaac Newton learn at school?

Since Sir Issac Newton invented a lot of modern mathematics, what mathematics did he already know? Since he was standing on the shoulders of giants which giants was he speaking of? I presume he knew ...
8 votes
1 answer
370 views

Does Blum's speedup theorem have any conceptual predecessors?

Blum's speedup theorem seems to me that bears at least some superficial resemblance to Godels research on the length of proofs under certain axiomatic systems. Does Blum's speedup theorem have any ...
15 votes
3 answers
1k views

How did Grothendieck encounter and adopt the categorical language?

Apparently the first mathematical publication of Grothendieck where he uses the terms “functor” and “category” in the technical sense is the Kansas report. From where Grothendieck had knowledge of ...
2 votes
0 answers
114 views

Is C.S Peirce really an independent co-discoverer of the first-order logic?

According to this article copies of Frege's Begriffsschrift were both present during the early 1880s (before Peirce published his works on first-order logic) at the Johns Hopkins University, where ...
4 votes
0 answers
94 views

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 ...
0 votes
1 answer
106 views

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 ...
4 votes
2 answers
278 views

Yuktibhāṣā, 16th century, first modern proof of $\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$

It is an Indian (Kerala) text, it would be the first modern proof (based on earlier knowledge) of $$\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$$ The integral would be a ...
8 votes
3 answers
1k views

What was Richard Courant's saying about mathematical research apart from applications?

I remember reading somewhere (perhaps in The Mathematical Experience) that Richard Courant wrote something to the effect that, without applications to guide the river of mathematical discovery, ...
3 votes
1 answer
91 views

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 ...
0 votes
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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 ...
2 votes
1 answer
150 views

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 ...
5 votes
4 answers
549 views

What contributions to mathematics did Napoleon make?

I have watched a video about Napoleon's theorem — maybe it was contributed by Napoleon, maybe not. I also know that Laplace himself said Napoleon was good at mathematics. However, did Napoleon make ...
0 votes
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154 views

Related concept for this equation?

One friend of mine has a cup with a lot of mathematical expressions printed on it. I would like to know more about one of these. What is the history behind the following inequality, and what is the ...
35 votes
6 answers
10k views

Who introduced the Principle of Mathematical Induction for the first time?

Can you tell me the name of the mathematician, who introduced the Principle of Mathematical Induction for the first time? (with reliable source). Please don't say De Morgan because I have read the ...
0 votes
0 answers
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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 ...
0 votes
0 answers
58 views

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 ...
5 votes
4 answers
1k views

Kronecker vs Cantor — who won?

Now set theory is taught even to kids and it is the foundation of mathematics. Can we say that Cantor won?
4 votes
1 answer
345 views

Why did Kronecker develop the "adjoining a root" construction?

Kronecker is generally credited with the formalization of "adjoining a root to $f(x)=0$". Nowadays it is interpreted as the quotient $K[x]/(f)$, where $K$ is some appropriate algebraic structure in a ...
13 votes
0 answers
361 views

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 ...
3 votes
4 answers
684 views

Did Kronecker attribute immutable origin to the integers?

The familiar quote is often incorrectly attributed to Kronecker directly. Actually a colleague of his named Weber claimed after Kronecker's death that Kronecker said this. I have doubts about this ...
5 votes
1 answer
305 views

Why was Kronecker dissatisfied with Cantor's submitted paper?

It is said here that In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set ...
5 votes
1 answer
238 views

Could a "field" have non-commutative multiplication originally?

Today, when the term "field" is defined in algebra, it is almost always stipulated that all fields are commutative. However, the author of these lectures says that this has not always been ...
3 votes
1 answer
331 views

How many mathematical publications containing non-reprinted, original mathematics did John Nash publish?

American Mathematical Society lists 25 according to one of the comments in the thread below: https://mathoverflow.net/questions/207477/john-nashs-mathematical-legacy. However, as indicated in the ...
10 votes
1 answer
10k views

How much did John Nash contribute to proving the Riemann hypothesis?

At quite the end of the movie A Beautiful Mind, John Nash tells a student "I am making progress" (towards proving the Riemann hypothesis (RH)). Actually, how much did Nash contribute to the ...
8 votes
2 answers
503 views

Why is Sophie-Germain's Identity popular?

Sophie Germain's identity is only about factorising $a^4+4b^4$ as product of two squares.It's not quite difficult. So,why is it so popular?
1 vote
0 answers
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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 ...
13 votes
5 answers
765 views

Are there any anonymous contributions to mathematics that had a great impact?

Are there any examples of mathematical ideas being communicated in an anonymous manner that had a substantial impact on mathematics at the time. It seems to me that it is very rare for an author to ...
-1 votes
5 answers
315 views

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 ...
2 votes
1 answer
158 views

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}^{\...
4 votes
1 answer
547 views

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 ...
4 votes
1 answer
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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 ...
4 votes
1 answer
106 views

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 ...
0 votes
0 answers
64 views

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 ...
2 votes
1 answer
167 views

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 ...
4 votes
1 answer
191 views

Reference - Schwarz's Proof of Clairaut's Theorem

Where can I find a copy (online) of Schwarz's paper that proved Clairaut's theorem for mixed partial derivatives? His paper is: Schwarz, H. A., "Communication", Archives des Sciences ...

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