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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36 views

Did anyone ever propose the distinction between “divergent to infinity” as opposed to “divergent but with finite average”?

There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...
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181 views

Did Euler produce any Russian text?

Wikipedia says that Euler (1707 - 1783) "mastered Russian and settled into life in Saint Petersburg" in 1727. Did he produce any Russian text, mathematical or personal? I can only find Latin,...
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How did Roger Cotes come up with logarithm form of Euler formula?

I have been trying to get my head around how Roger Cotes first discovered Euler Formula. I knew how Euler did it, but I wanted a new perspective, especially from someone who discovered it earlier. ...
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2answers
107 views

Is there any English translation of this Gergonne paper?

This is the paper: “Variétés. Essai de dialectique rationnelle”. Annales de Mathématiques pures et appliquées, tome 7 (1816-1817), p. 189-228 (“Varieties. Essay about rational dialectic”, By J.D. ...
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73 views

Did anyone ever propose a hypercomplex numbers system with more than one anisotropic axis?

The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, ...
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When did it become a norm for mathematicians not to read proofs of all the results they use? [closed]

Please include evidence in your answer. This question is not opinion-based. I've heard that mathematicians sometimes don't read proofs. The following is a quote from mathoverflow. ... I think a vast ...
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1answer
61 views

Mathematical research institutes similar to Banff and Oberwolfach [closed]

What other institutes such as these two exist for a visit by a scientist for an undisturbed period of short research? Ideally with a good landscape. Dagstuhl is another one I found.
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59 views

Beltrami's Essay on the Interpretation of non-Euclidean Geometry

I am reading the Essay of the title written by Beltrami in Italian and I found a specific point of the essay which in my opinion could be fully clarified only if compared with its translations. At the ...
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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 ...
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2answers
306 views

A branch of mathematics which refused to be rigorous?

I'm currently in a class on formal mathematics/formal logic/axiomatic set theory. Someone asked, "At the end of the day, as mathematicians, why do we care about rigor?" My professor gave an ...
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1answer
111 views

First time the real numbers were axiomatized as the “unique complete ordered field”

(originally asked at M.SE: https://math.stackexchange.com/questions/4094361/first-time-the-reals-were-axiomatized-as-the-unique-complete-ordered-field) I'm looking for historical references on the ...
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What is the middle name of George A. Grätzer?

What is the middle name of George A. Grätzer?
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Appearance of the Dirac delta operator in Laplace's work

I found a reference to the following article ; O B Sheynin, The appearance of Dirac's delta functions in the works of P S Laplace (Russian), Istor.-Mat. Issled. Vyp. 20 (1975), 303-308, 381. I don't ...
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118 views

What was the influence of Gauss on the understanding of the importance of non-Euclidean geometry?

As is known, Gauss came to similar conclusions as Lobachevskii in the problem of fifth postulate, but I don't know whether Gauss had any impact to the contemporaries in that story. Can anybody ...
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72 views

Origin of “Inverse Pythagorean Theorem”

There is a lot of information on the history of the Pythagorean theorem, but not much on its closely related cousin; the Inverse Pythagorean Theorem. Would appreciate any resources on the the history ...
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Why do we still use degrees and cycles if we all implicitly use radians in trigonometric functions?

For various reasons already discussed in other stackexchange posts, we implicitly use radians in trigonometric functions by convention. For example, one period of $sin(x)$ lies in $0 \leq x < 2\pi$,...
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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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91 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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75 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
147 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 ...
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1answer
97 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
62 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
74 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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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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60 views

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
131 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
115 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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52 views

Aristotle's and Plato's even and odd numbers, sets and actual infinity [closed]

Plato and Aristotle both use the terms even and odd about numbers (and have a separate discussion of the number 1). From this point, it seems, there would be no great distance to the sets of even and ...
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64 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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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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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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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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111 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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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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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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60 views

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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118 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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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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2k views

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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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 ...
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211 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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