# 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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117 views

### How did Euler stumble on this proof?

Euler proved $n=641$ divides $2^{32}+1$ by noting $n=5^4+2^4=5\times 2^7+1$ so $$2^{32}\equiv-5^4\times 2^{28}=-(5\times 2^7)^4\equiv-1\,(\text{mod}\, n).$$How did he happen upon this realisation? One ...
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### Timeline of mathematical foundation?

As it is globally known that set theory as a foundation of mathematics, although in the beginning we didn't call it "Set" rather group of elements. For example - set of [1(banana) + 2(apple)+1(cow)] =>...
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### 18th and 19th century skeptics of imaginary numbers?

Complex numbers were used in as early as the 16th century to solve cubic equations, but they didn't gain wide acceptance until the late 18th and early 19th century. What is the reason for the 200 year ...
1k views

### origin of the terms “domain” and “range”?

A 1929 paper of Chittenden contains the following sentence (about the derived set operator on a space $P$): “Thus the relation $E' = K(E)$ defines a single-valued set-valued set-function, whose ...
308 views

### History of Arithmetic and Geometric Inequality

I tried to find up the history of (Rogers-)Hölder Inequality. In wikipedia, I found L. J. Rogers's origin paper, saying how to extent the "well known" AM-GM (arithmetic mean-geometric mean) inequality ...
412 views

### History of the arithmetic mean

The arithmetic mean of a set of points $\{x_1, x_2, ..., x_n\}$ is defined by $$\frac{1}{n}\sum_{i=1}^n x_i.$$ It is remarkable for its ubiquitous use and universal understanding. It represents a ...
146 views

### Which mathematician first proved the laws of arithmetic?

Specifically, the associative, distributive, and commutative laws of addition and multiplication. Was it Peano?
54 views

### Earliest presentation of a 3-D permutahedron?

Below is a picture of a 3-D permutahedron sundial by Stefano Buonsignori (16th century) in the Medici collection presented by Museo Galileo. The permutahedra / permutohedra and the closely related ...
134 views

### Was Newton aware of a nascent inverse function theorem?

More specifically, was Newton aware that given an inverse pair of functions $f$ and $h$ such that $$f(h(x)) = x = h(f(x))$$ about the origin that, for $$(x,y)=(h(y),f(x)),$$ the derivatives satisfy ...
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### Taking less than a minute to solve a problem within 10% which was stated in less than 10 seconds?

I believe there is a story about a 20th century sciencist (either Neumann or Feynmann I think) who, at some dinner or some sort of gathering, made the following claim to his fellow colleagues: He ...
95 views

### Why do we often minimize in optimization?

Because of the following relation, \begin{equation*} \inf(S) = -\sup(-S), \end{equation*} minimization and maximization is essentially the same thing. However, take any optimization routine in R for ...
149 views

### How did “one-to-one” come to be used to refer to injective functions?

I have always had a hard time explaining to my students the term one-to-one. After making sure my students understand "in", "sur" and "bi", the Bourbaki terms, injective, ...
183 views

### Why did Lagrange say that Cauchy should learn Classics before mathematics?

I read somewhere that Lagrange said that Cauchy should concentrate his efforts on Classics/litterature before studying maths. Is there any reasons for this ?
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### Sophie Germain and the outlook on science

I have two questions. What is the importance of Sophie Germain's work with Fermat's last theorem, if her only contribution was Sophie Germain prime numbers? Euler was able to calculate on the problem ...
98 views

### Origins of Stone duality

My question is a mix of mathematical and historical, if you consider my question will be better answered in the mathematics community, please tell me. I want to know the historical roots of Stone's ...
53 views

### What is the middle name of George A. Grätzer?

What is the middle name of George A. Grätzer?
25 views

### Who was the first person to prove the invariance of the Euler characteristic under triangulations?

Given a compact orientable surface $S$ and any triangulation where $F$ denotes the amount of triangles, $E$ denotes the amount of edges, and $V$ denotes the amount of vertices, we know that the Euler-...
37 views

### Old Indian and Chinese references

It is been some years since I completed my graduate studies in mathematics at a Spanish university. I remember one of the most pleasant and enriching moments I experienced was when reading Euclid´s ...
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### Which one goes first - Secant or Newton - in Numerical root finding technique?

In Numerical root solving technique, which comes first in history - Newton or Secant - and each one is named after whom?
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### Singularities on null capacity sets are removable — Wiener or Bouligand?

A classical theorem on harmonic functions states that singularities of bounded harmonic functions are removable if the singular set is of null capacity. This theorem is sometimes attributed to ...
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### History: Direct Product became Tensor Product?

I'm reading a 1939 paper by the great and famous J. von Neumann, "On infinite direct products" (of vector spaces), available here http://www.numdam.org/item/?id=CM_1939__6__1_0, legally I ...
64 views

### Luca Pacioli's incorrect solution

In the Summa of Luca Pacioli, he gives a solution for a quadratic equation involving a variable to the third power. The solution he gives is wrong; that is easily verifiable by substituting it into ...
78 views

### How well did Cardano understand elementary probabilities, as shown from his writings?

I was recently reading a paper on the historical development of probability theory where Cardano is presented as having discovered some elementary laws of probability in the 1550s. But he is described ...
35 views

### Who was the first one to rigorously show that quantum fields are operator-valued distributions?

Any Wightman-based approach to Axiomatical Quantum Field Theory states that quantum fields are (operator-valued) distributions. Is there a first rigorous proof of this fact which became trivial as ...
33 views

### What's the history of the heuristics on doing science efficiently and effectively?

I recently read a paper by Taleb about doing science efficiently and effectively and I was wondering if there other predecessors to his work which more or less express the same things(since Taleb ...
44 views

### Was Morse code the first practical application of binary encoding of information?

As I understand schemes for sending different symbols over different wires were implemented but the simplicity of Morse's dot or dash made it the easiest to read and/or implement. It seems similar to ...
76 views

### Was Lagrange the first to have used generalized coordinates?

I was wondering if Lagrange was the first to use generalized coordinates as defined by their wikipedia article.
182 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 ...
48 views

### How did percolation theory come to be established in network science, and who first studied it?

According to the textbook "Network Science" by Albert-László Barabási, percolation theory is a specialized branch of both mathematics and physics [1]. It involves node clustering in a ...
50 views

### Is there an English translation of Kronecker's proof of Infinitude of primes?

Is any English translation of the following paper available? H. Hasse, Vorlesungen ¨uber Zahlentheorie, Second edition, Springer-Verlag, New York, $1964$ (L. Kronecker, $269–273; 440–442;$ K. Hensel, ...
70 views

### How was mathematics used in World War II to “act on the right amount of intelligence”?

In the movie "The Imitation Game", Alan Turing along with his team crack the German encryption machine Enigma but advises his superiors to not act on all decrypted intelligence, as that might lead to ...
64 views

### How did the definition of each “ordered set” come about?

I could get a little intuition about preset, poset, and toset. e.g. A toset is, in effect, a linearly ordered set, and a poset is a set in which no more than one element in the same order exists, ...
60 views

### Did the mathematician Garrett Birkhoff ever work with or mention Feynman's path integrals?

Did Garrett Birkhoff ever work with Feynman's path integral? Did he ever work in his Many-Histories interpretation? Or at least, did he mention it in any of his articles?
106 views

### Why didn't Euclid use equations or numerals in his proofs?

I think the Elements would have been a lot more concise if he did.
31 views

### History of Path algebras

I want some references that point the inventor of Path algebras and history/evolution of these algebras from the first idea. If possible. I tried to search in many different places, but all times, ...
44 views

### On the origin of the concept of aliasing & the Discrete Fourier Transform frequency axis

The development of the fast Fourier transform (FFT) is attributed to Cooley & Tukey, both of whom have written a lot about its historical development. However, I am searching for early ...
66 views

### Mathematical analysis vs. Practical genius

Concerning the role of mathematics in technological inventions: which books would you suggest that examine the historical relation between mathematical analysis & practical wisdom? For example, ...
56 views

### The Hahn-Kolmogorov Extension Theorem

How did Hahn and Kolmogorov prove the Hahn-Kolmogorov Extension Theorem?
39 views

### Rocket & drag equation?

i'm writing an assignment on firework rockets and their trajectory. Now of course im doing this with a lot of limitation as a realistic rocket calculation would be impossible to execute, at least for ...
38 views

### Are there any undergraduate research/internship opportunities in math/science education/history?

(Apologies if this isn't the right place for such a post. I see lots of information out there for math/science students wanting to do research, but haven't seen anything about doing things on the "...
676 views

### Who first proved Rota' conjecture

In Wikipedia in article about Rota's conjecture it is written: “ A proof of the conjecture has been announced by Geelen, Gerards, and Whittle” . In Quanta magazine in article about mathematician June ...
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### Why did Nikolai Luzin almost commit suicide?

I tried accesing the original article without much success. Can someone fill in the details ?
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### Why 1 was source of numbers even though ancient Greeks knew about irrational number?

In Ancient Greek, most people like Pythagoras thought 1 (monad, unity) is no number, but it is ruler and beginning of all other numbers. And Pythagoras thought everything is number. But they found ...
86 views

### Related concept for this equation?

One friend of mine has a cup with a lot of equations and the other day I saw it and I do not know what equation is. Which is the history behind this equation and what is the related concept? Thanks