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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51 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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Aristotle's and Plato's even and odd numbers, sets and actual infinity

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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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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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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The originator of the sum rule in calculus

One of the most essential theorems in differentiation is the sum rule. I cannot seem to find the originator of this theorem? And, I have the same problems for many of the other basic differentiation ...
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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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108 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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44 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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63 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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201 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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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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91 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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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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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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215 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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144 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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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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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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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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89 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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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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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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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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How did Gauss able to come up with complex formulas at very early age? [duplicate]

This question marked as a duplicate for no clear reason. It's not a duplicate. I was not asking whether Gauss came up with this specific formula. I was asking how he able to invent such a formulas at ...
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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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How did Gaussian and Eisenstein integers get their names?

I can separate this into two questions at some point if necessary, but it's possible that sources for the answer to one will provide the answer to the other at the same time. I learned about ...
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84 views

How were number symbols derived/shaped up?

This question was sitting on my to do list for sometime. So, as I was reading a book on history of science, I came across of a paragraph where the author attempted to give a historical development ...
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Why was the Vietnam Day Committee, begun by Stephen Smale and Jerry Rubin, named as it was?

Stephen Smale, an American mathematician and Jerry Rubin, who was at Berkeley before dropping out to organise around left wing causes, set up the Vietnam Day Committee in 1965 during a 35 hour anti-...
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284 views

How did Von Neumann come up with his Merge Sort algorithm?

Since merge sort is the first $O(n\log n)$ time general purpose sorting algorithm I find it rather surprising that it was discovered without having any obvious conceptual predecessors. Are there any ...
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1answer
119 views

Why did the existential and universal quantifiers in logic took so long to become formalized into symbols after the invention of boolean algebra?

Was there a specific reason that prevented researchers in boolean algebra to invent such quantifiers in the flexible format that are known today earlier? Since the compact symbols for multiplication ...
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Finding sources for “computers will become so powerful that special functions will become obsolete” as a zeitgeist

In Why are special functions special [Physics Today 54, 11 (2007); eprint], Michael Berry makes the following observations: This continuing and indeed increasing reliance on special functions is a ...
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Advantages of numbers growing to the left (ordered Arabic numbers)? [closed]

I was just thinking how the BOM in files makes a lot of sense. I mean the fact that by default numbers grow to the right. And then I started wondering why our numbers grow to the left, if columns of ...
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54 views

History of Mathematical Biology - Resource Recommendations

Biology nowadays is filled with mathematics. Indeed, the field of mathematical biology is huge, and shows no sign of decay. But the mathematisation of biology is, to my knowledge, a recent phenomenon -...
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Is there any book or site where Gauss' collected philosophical writings are presented?

From the questions on some of Gauss' philosophical ideas here at HSM stackexchange it's clear that Gauss had some major philosophical ideas that despite their profundity don't seem to have had much ...

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