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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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3 votes
3 answers
199 views

Origin of modern definition of a function as a graph

In the past, I came across a very elegant direct definition (below) of a function, which is based on the fundamental concepts of triples, pairs, and sets. However, I find it difficult to search the ...
6 votes
5 answers
323 views

Math concepts introduced by physicists and made rigorous later

I am looking for mathematical concepts (*) which have been introduced by physicists in a non rigorous way (e.g. without a formal definition, without rigorous proofs of the results, etc.) and used to ...
20 votes
1 answer
4k views

Why is the radical symbol $\sqrt{}$ called "radical"?

This question arose in a conversation with a teacher who was introducing square roots to her students. I know from the website Earliest Uses of Symbols of Operation that the symbol $\sqrt{}$ has its ...
0 votes
0 answers
28 views

The Root of a Geometric Progression

Good people! I'm presently in the process of putting something together on Euler and Gauss and cyclotomy and modular arithmetic, and I noticed that when it comes to the terminology primitive root ...
1 vote
0 answers
81 views

Were Ten and One Hundred Thousand treated more like "special" numbers at one point than today?

Note: Not English specific but about any major system of recording or discussing numbers. I know some systems would not give 100 a special place. People today when discussing largish numbers tend to ...
1 vote
0 answers
78 views

Who "made" Gauss?

I was looking at the math genealogy related to probabilities and there was pretty much a straight line going from Kolmogorov to Laplace. Then I got to Markov's sequence which gets a bit messy: Markov -...
2 votes
0 answers
46 views

Equicardinality of $\mathbb{R}$ and $\mathbb{R}^2$ via interleaving decimal expansions

As Fernando Q. Gouvêa notes in his paper, Was Cantor Surprised? (Amer. Math. Monthly 118 (March 2011), 198–209) Cantor initially tried to prove that $(0,1]$ and $(0,1] \times (0,1]$ have the same ...
0 votes
0 answers
52 views

When did people first thought of a purely symbolic logic?

In Euclid's Elements, the famous five planar geometry axioms are formulated in common language (ancient greek in this case) and use ambiguous terms. On the other hand, modern theories like ZFC or ...
3 votes
1 answer
186 views

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 ...
3 votes
1 answer
168 views

What is the origin of the method of undetermined coefficients?

This MSE post asked about a specific integration technique that appears to be attributed to Charles Hermite, per a comment. The OP's source calls the technique el método alemán, i.e. the German method....
4 votes
0 answers
188 views

How and why was catastrophe theory brought to its knees?

How applications of catastrophe theory outside mathematics stalled the theory, and why? I know that the theory had its fair share of popularity during the 1970s, with many distinguished mathematicians ...
1 vote
0 answers
49 views

Was Euler aware of the general form of the characterization of primes of the form $p=x^2+ny^2$ for arbitrary $n>0$?

If $n$ is a positive integer then there is a monic irreducible polynomial $f(x)$ such that if $p$ is an odd prime not dividing $n$ nor the discriminant of $f$ then $$ p=x^2+ny^2\iff \left(\frac{-n}{p}\...
1 vote
0 answers
72 views

What did Gauss think about V. A. Lebesgue's proof of quadratic reciprocity?

The proof can be found here (pdf). It was published in 1838 and Gauss lived until 1855, so I would guess that he read it. Did Gauss say anything about it?
0 votes
2 answers
397 views

Aziz of Complex Analysis

Does anyone know about Prof. Abdul Aziz on whose name Aziz's theorem is named? Aziz's theorem is a theorem about the location of zeros of polynomials.
8 votes
2 answers
947 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 ...
15 votes
4 answers
625 views

Why were 18th century mathematicians interested in extending the factorial to non-integers?

As far as I understand, the Gamma function was developped as a way of calculating "the" factorial of a non-integer number. Why did this problem interest 18th century mathematicians? Was it just a ...
3 votes
3 answers
233 views

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 ...
9 votes
2 answers
307 views

Origin of $\ll$ notation

Vinogradov introduced the notation $$f(x) \ll g(x)$$ to denote that for some $C>0$, we have $|f(x)|\leqslant C\,g(x)$ for all $x$ under consideration; usually for all $x$ larger than a fixed ...
1 vote
0 answers
122 views

Are there alternatives functions for the gamma function that was used as generalisation for the factorials?

I asked this question on MSE here $$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ x>0. $$ Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $...
7 votes
2 answers
583 views

Historical Instances of Set Theory

Context: I've been reading a lot about Set Theory lately, and how it suddenly sprung onto the mathematical scene in the late 1800's, thanks largely to Cantor. But it seems strange to me that no one ...
6 votes
2 answers
616 views

When did set theory throw off theology?

"The general set theory [...] definitely belongs to metaphysics. You can easily convince yourself when examining the categories of cardinal numbers and the order type, these basic notions of set ...
2 votes
0 answers
73 views

Who discovered that the Lanczos method can only calculate extremal eigenvalues of large matrices?

The Lanczos tri-diagonalization process is widely or even routinely used today. It is said that it is useful for obtaining the extremal eigenvalues, but useless for medium eigenvalues. But who ...
3 votes
4 answers
149 views

Translated articles of Fatou and Julia

Is there any English translation of the 1918-1920 Memoirs of Fatou and Julia on the iteration of rational functions?
0 votes
1 answer
82 views

Where does Oliver Heaviside fit in the ranks of physicists/mathematicians? [closed]

It seems to me that he was able to reformulate Maxwell's equations in a more understandable form and in fact come up with vector calculus without finishing high school would arguably cause him to be ...
0 votes
2 answers
179 views

What geometric results were first proven by assuming all real numbers are rational?

Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural ...
6 votes
2 answers
335 views

What exactly did Poincaré mean by 'simply connected'?

I've been reading John Stillwell's translation of the famous Analysis Situs and have become confused about the exact meaning of 'simply connected' in Poincaré's language. On page 7 (in the ...
2 votes
2 answers
429 views

Poincaré quote regarding instability

Poincaré wrote (English translation of the French) in 1903: ``A very small cause that escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect ...
8 votes
1 answer
390 views

Who discovered the singular cup product?

Cohomology is a stronger invariant than homology because it can be equipped with a ring structure. To be precise, if one starts with the singular cohomology groups $H^\bullet(-; R)$ with coefficients ...
1 vote
1 answer
119 views

F = ma -- How was did we come to understand that this compact form expressed what Newton said in words?

My understanding is, Newton in the 17th century did not use this formula but rather said, in words basically that if you apply a force it will cause a mass to accelerate in the direction of that force....
6 votes
1 answer
1k views

How come there is no portrait of Legendre?

Besides the famous cartoon, of course, there seems to be no portrait of Legendre. Legendre is well regarded nowadays and he was also quite influential at his time, for example, Jacobi and Abel praised ...
4 votes
3 answers
233 views

Inscriptions on a 16th century 3-dimensional permutahedron sundial?

Does anyone know what the inscriptions, written presumably in Latin, are on this 3-D permutahedron sundial by Stefano Buonsignori (16th century) in the Medici collection presented by Museo Galileo? ...
0 votes
0 answers
37 views

Why did Kronecker say "the integers are the work of God, the rest is the work of man"? [duplicate]

To me, it seems no number is the work of God, they are all concepts of the mind. However, it seems negative numbers are more artificial than the rest of the numbers out there. So why did he describe ...
0 votes
1 answer
62 views

Is there a resource about integer constructions and motivations?

I have an assignment about the foundations of mathematics. I am trying to compile a list where I get common construction of integers and a small writing about the constructor and their explanation. ...
1 vote
1 answer
150 views

When were negative numbers fully accepted into mathematics?

Dedekind gave a construction and explanation of integers and rational in 1858. This was as ordered pairs of natural numbers. I'm not sure if this was the standard view of these objects after this ...
4 votes
1 answer
439 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 ...
1 vote
0 answers
46 views

Did Dedekind's construction of the integers and rational numbers become standard in mathematics textbooks?

I am referring to the construction using pairs of natural numbers in 1858. Since we use pretty much the same construction today in some analysis courses (Analysis 1, Terence Tao), except without the ...
3 votes
0 answers
53 views

What can I read to learn the history of multivariable calculus?

People have been doing calculus of several variables since well before the concepts of vectors, matrices, and linear algebra were formalized. Where can I learn about the development of multivariable ...
0 votes
0 answers
28 views

Did Heinrich Weber have a structural approach to mathematics similar to Dedekind?

So I was reading a History of Mathematics by Katz, and noticed that the first definition of a field came from Weber, who had previously done extensive joint work with Dedekind. His definition was used ...
1 vote
1 answer
53 views

Did Dedekind's work directly influence the work of Hilbert?

I am wondering if Dedekind's theory about the structure of deductive science influenced the work of Hilbert. Hilbert obviously favored axioms at the beginnings of a deductive science, whereas Dedekind ...
4 votes
2 answers
231 views

How much ground was prepared for Riemann so that he could conjecture Riemann hypothesis?

Although I do not doubt in Riemann˙s originality, I would like to know how much complex analysis was developed up to the day when Riemann conjectured what is today called Riemann hypothesis and how ...
1 vote
0 answers
57 views

Where did the Delannoy numbers make their first appearance?

I am writing a short exposition on the central Delannoy numbers and would like to find the year which Henri Delannoy first introduced them in a formal setting. I believe Delannoy's initial ...
4 votes
1 answer
248 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 ...
18 votes
2 answers
10k views

Who calculated for the first time the volume (and surface area) of the sphere exactly?

As we know, even Archimedes did soon some experimental calculations. My question were, who calculated first time the exact formulas ($V=\frac{4\pi}{3}r^3$, $A=4\pi r^2$)? As I know, these formulas ...
4 votes
2 answers
768 views

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 ...
0 votes
0 answers
31 views

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 ...
0 votes
1 answer
112 views

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 ...
1 vote
0 answers
70 views

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 ...
4 votes
3 answers
928 views

Is Gauss the first who introduced congruences?

I have read Disquisitiones Arithmeticae and I have read somewhere that he introduced the concept of congruence with this book. Wasn't this concept already used by other Mathematicians in the past?
3 votes
1 answer
340 views

Why did systems theory never gain popularity?

Briefly from wikipedia, Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has ...
1 vote
0 answers
72 views

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 (...

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