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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4
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81 views

Equations in right-to-left languages

Is there an historical tradition in languages read right-to-left (Arabic, Hebrew, Urdu, etc.) to display mathematical equations in some right-to-left form? So, instead of $$x = \frac{-b \pm \sqrt{b^2 -...
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1answer
103 views

In what sense was it thought (at one point) that every lattice was distributive?

Blyth writes in his 2006 book Lattices (p. 65): It is a curious historical fact that it was originally thought that every lattice was distributive! That this is not so is shown by the following ...
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1answer
110 views

Stories about the consequences of statistical simplification?

I am currently preparing a presentation about the value of more complex (specically: non-Gaussian) statistical inference. I thought it might be interesting to start the presentation with a small real-...
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1answer
183 views

A quote attributed to Gauss

A version of this question first appeared in MathOverflow but did not receive satisfactory answers. This quote is often attributed to Gauss: Die Mathematik ist die Königin der Wissenschaften, und ...
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1answer
173 views

($\varepsilon$, $\delta$)-definition of limit by Weierstrass

I am looking for the original ($\varepsilon$, $\delta$)-definition of limit by Weierstrass, but I cannot find an exact quote or a reference. I saw that somewhere it was claimed that this definition ...
3
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1answer
118 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 ...
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1answer
80 views

History of primality testing

Consider a uniform random variable $n$ which is an integer in the interval $2^{1023} < n < 2^{1024}$. What is the oldest algorithm capable of determining whether or not $p$ is a probable prime ...
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1answer
89 views

How did the idea of a formal derivation emerge?

Infinitesimal calculus and the introduction of derivatives is often linked to Newton and Leibniz. I was wondering, when and why the idea of studying formal derivatives (e.g., of a formal polynomial) ...
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100 views

Fourth powers and quartic equations before Descartes

How did mathematicians interpret quartic equations and fourth powers before Descartes propose to perform elementary arithmetic on line segments? I ask this because it seems strange to me that ...
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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. ...
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What is the source of Hopf's (boundary) lemma?

In an introductory course on PDE's I got as a project to prove and present a version of Hopf's (boundary) lemma. Namely: Let $\Omega \subset R^{d}$ be an non-empty open connected set with a twice ...
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1answer
156 views

Who came up with the name "Manhattan distance"?

Who came up with the name "Manhattan distance" (for the distance between two points as measured by the sum of the horizontal and vertical distances, as opposed to the length of the straight ...
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1answer
195 views

Who was Paul Gerwien?

The famed Wallace–Bolyai–Gerwien theorem has got its name from three mathematicians who proved it independently. More precisely speaking Farkas Bolyai first formulated the question. Gerwien proved ...
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72 views

How did Fourier determine the coefficients of Fourier series?

I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the ...
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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-...
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1answer
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Is the story about Fermat's writing on a margin true?

Is there any evidence that Fermat wrote on the margin of a book "I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." Everyone ...
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62 views

What was the first automated theorem prover?

From a lot of googling, it seems like the answer might be "Mizar", but I am not completely sure. What was (or is?) the first automated theorem prover (i.e. not necessarily active right now)?
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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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1answer
342 views

What is the origin of the "Japanese bracket"?

In discussions of Sobolev spaces one often sees the Japanese bracket, $$\langle x \rangle = (1+|x|^2)^{1/2},$$ as useful shorthand. I was not easily able to find information about this term. (1) What ...
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1answer
157 views

What are the earliest known proofs that planimeters 'work'?

The dates of various physical implementations of planimeters are pretty well known. I'm interested in discovering when formal mathematical proofs were published that any given design does calculate ...
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3answers
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Why are quaternions more popular than tessarines despite being non-commutative?

Is this simply because of marketing, hype, etc? The bicomplex numbers (especially tessarines) look just great being commutative and all. Images source:https://citeseerx.ist.psu.edu/viewdoc/download?...
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How did Dyck originally state and prove his theorem in topology about the connected sum of a torus and projective plane?

Dyck's theorem in topology is sometimes stated as follows: the connected sum of a torus and projective plane is homeomorphic to the connected sum of three projective planes. Certainly, this is the ...
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1answer
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What is the difference between Einstein's and Boltzmann's definitions of probability?

In the Einstein's 1905 paper on light quanta, he wrote the following about Boltzmann's notion of probability: The word probability is used in a sense that does not confirm to its definition as it ...
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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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Why are linear forms called "forms"?

My question is about linear forms, quadratic forms, n-linear forms, differential forms and so on. The first term of these names seem clear to me, but I cannot make a link between these mathematical ...
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How did Euler obtain this formula from a paper/work in 1748?

I am reading this book on trigonometric series, "Тригонометрические ряды от Эйлера до Лебега" (Trigonometric series from Euler to Lebesgue) , it is in Russian, and my Russian is abysmal. But ...
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Where were mathematical/science works posted before the arxiv website?

Arxiv website is very good at collecting mathematical/science work updates. However, where were mathematical/science work updates posted before arxiv website existed?
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152 views

What did Roger Apery mean by "Oh they grow in my garden"

Roger Apery, a French mathematician most remembered for proving that $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ (irrational) in $1979$. In this video posted by Numberphile: https://www.youtube.com/...
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Why is Hilbert's Seventeenth Problem important?

I'm self-learning about Model Theory and I just got to the proof of Hilbert's 17th Problem via Model Theory of Real Closed Fields. The 17th problem asks to show that a non-negative rational function ...
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64 views

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 ...
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Story of a student who solves an open problem

A number of years ago, I read a (non-fiction) story about a young mathematician whose professor wrote an open problem on the board and the student went ahead and wrote down a solution after a moment ...
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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 ...
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1answer
246 views

History of Fatou's Lemma

This question concerns Fatou's Lemma: Let $(f_n)$ be a sequence of nonnegative measurable functions on a measure space $X$. Define $f: X\to [0,+\infty]$ by $f(x) :=\liminf_{n\to\infty} f_n(x)$, for ...
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144 views

The origins of $\det(I+AB)=\det(I+BA)$

I am looking for the earliest published source that gives and perhaps proves the identity $\det(I+AB)=\det(I+BA)$ where $A$ and $B$ are just rectangular matrices of finite dimensions (as opposed to ...
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2answers
147 views

History of generating functions

I've seen that there is some information in the first volume of Polya's "Mathematics and Plausible Reasoning". Also, the following paper deals with the usage in probability - "The ...
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1answer
142 views

What is the source of Hermann Weyl's remark about "near-sighted policemen" with respect to compact spaces?

Hermann Weyl is widely reported as making the following observation about a compact space: "If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen." ...
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83 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 ...
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2answers
233 views

What is the oldest open question solved in mathematics?

In mathematics, 1760 Plateau's problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó. ...
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2answers
253 views

How did polar coordinates come into existence?

So, I came here from Mathematics StackExchange where I posted this question. So, I want to know why polar coordinates came into existence. Why exactly did the mathematician who introduced them......
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35 views

Historical proofs of the series expression for the Bessel function of the first kind

Introduction The Bessel function of the first kind $J_n(x)$ ($n \in \mathbb{Z},\ x \in \mathbb{R}$) appeared early among other topics, in Celestial Mechanics, in the series expression of the true ...
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How did definitions of a limit vary before the epsilon-delta definition?

My understanding is that before the epsilon-delta definition of a limit, the rigor and soundness of the definition of a limit was not good enough. So, how did the definitions of a limit vary before ...
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1answer
153 views

Who invented multiplying by the conjugate to rationalize denominators and when?

1860 Manual of Algebra describes a method which is now taught in upper secondary schools worldwide: To rationalize the denominators of fractions which consist of binomial quadratic surds, use the ...
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167 views

why do we write `abelian` group instead of `Abelian` group?

Suppose an object (or a concept or ...) is named after the person X, in honor of Mr. or Mrs. X in mathematics: X-ian objects/ <...
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The mathematics of multiple values

tl;dr Why has the array paradigm, which emerged in the 1950s and 60s amongst mathematicians, and which underpins certain programming languages, apparently failed to capture and maintain interest ...
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1answer
106 views

Are the ancient Egyptians and the ancient Mesopotamians really the cradles of civilization? [closed]

Is it true that ancient Greek knowledge of mathematics, science, astronomy, engineering, architecture, civilization came from the ancient Minoan civilization and ancient Mycenaean civilizations and ...
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1answer
194 views

History of irreducible polynomials and motivation for them

I've been thinking about the history of the irreducible polynomials and why they were introduced. I found What is the origin of polynomials and notation for them?, but it's about polynomials in ...
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100 views

What is the origin in the discrepancy between engineers' and physicists' notation of waves?

my question is very simple. Physicists use this notation in order to write a (for example) plane wave: $$ \xi(z) = \xi^+ \mathrm{e}^{+\mathrm{i}kz} + \xi^- \mathrm{e}^{-\mathrm{i}kz}, $$ where $\xi^+$ ...
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0answers
111 views

What is the basic idea behind calculation of area? [closed]

The system of calculating area in terms of square units is pretty philosophical and not very intuitive. It must have taken a great amount of time for humanity to arrive at such a convention and to ...
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40 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 ...

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