Questions tagged [mathematics]
For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns.
1,062
questions
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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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34 views
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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47 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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0answers
22 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-...
17
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1answer
3k views
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 ...
6
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0answers
43 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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34 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 ...
5
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86 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 ...
3
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1answer
92 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 ...
4
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0answers
47 views
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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36 views
2
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1answer
71 views
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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39 views
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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0answers
22 views
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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0answers
57 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 ...
3
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0answers
128 views
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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3answers
3k views
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?
2
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1answer
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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57 views
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 ...
0
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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 ...
9
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1answer
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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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 ...
6
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1answer
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 ...
2
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0answers
81 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 ...
5
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2answers
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 ...
6
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1answer
127 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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0answers
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 ...
3
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2answers
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.
...
5
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2answers
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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0answers
32 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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0answers
67 views
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 ...
4
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1answer
105 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 ...
2
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0answers
151 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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139 views
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 ...
0
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1answer
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 ...
8
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1answer
149 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 ...
6
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0answers
95 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 $\xi^+$ ...
4
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0answers
87 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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52 views
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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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 ...
1
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1answer
82 views
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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1answer
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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1answer
40 views
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-...
4
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2answers
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 ...
2
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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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0answers
70 views
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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53 views
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 ...
2
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2answers
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 -...
2
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1answer
69 views
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 ...
