Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations.
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2answers
1k views
How did the square root get its shape?
I was wondering when in history did people start use the $\sqrt{}$ sign for square root, what did they use before, and why this curious nomenclature is adopted.
10
votes
2answers
699 views
Curious about math and Soviet Union
Why so many very good books were written by authors with Russian surnames?
10
votes
1answer
111 views
Any branch of math can be expressed within set theory, is the reverse true?
Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property?
I am asking ...
10
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2answers
213 views
A quote from Arnold
Arnold said the following in a talk on teaching:
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as ...
10
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3answers
459 views
What was the notation for functions before Euler?
According to the Wikipedia article,
[Euler] introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical ...
10
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2answers
907 views
History of dot product and cosine
The fact that the dot product and the cosine of the angle between two vectors are mutually computable is easy to show (see the two sides in the two answers at Dot product in coordinates).
But looking ...
10
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1answer
274 views
Is Hilbert's second problem about the real numbers or the natural numbers?
In his famous "23 problems" speech, Hilbert gave his second problem as follows:
The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the ...
10
votes
1answer
315 views
Fibonacci, compositions, history
There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets):
a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
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votes
1answer
253 views
Old versus New enunciation of Taylor's Theorem.
I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows:
THEOREM
Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by
...
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1answer
140 views
Who is responsible for the analytical/topological proof of FTA?
The fundamental theorem of algebra asserts:
Theorem Let $P$ be a polynomial of degree $\geq 1$ in $\Bbb C$. Then there exists a $z_1\in\Bbb C$ such that $P(z_1)=0$.
The proof sketch goes as ...
10
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1answer
277 views
Who was the first to use dual space?
Who was the first person who used the dual space? In which paper / book did he or she use the dual space?
Who was the first who called it dual space and in which paper / book?
9
votes
2answers
850 views
Which symbol should be used for an empty set?
Currently, a discussion started on the German Wikipedia article for Empty Set (the German discussion), whether $\emptyset$ or $\varnothing$ should be used or is more common as a symbol for an empty ...
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votes
2answers
904 views
Hardy / Wright's intro to number theory is highly praised but has no exercises
"An introduction to the theory of numbers, G.H Hardy, E.M. Wright, revised by D.R. Heath-Brown, J.H. Silverman. Originally published 1938. Sixth edition 2008 with a foreword by Andrew Wiles" is AFAIK ...
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10answers
500 views
Good (Auto)Biographies of von Neumann and other physicists/mathematicians
Which is the "best" biography of von Neumann available to the casual reader (math undergrad)? Also, other than the Ulam book, which other good biographies of physicists/mathematicians can be ...
9
votes
3answers
529 views
Who introduced the notation $x^2$?
In the book 'Problem Solving and Number Theory' I read
The law of quadratic reciprocity was discovered for the first
time, in a complex form, by L. Euler who published it in his paper
...
9
votes
3answers
269 views
Fibonacci numbers modulo $p$
If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol.
Who first proved this? Is there ...
9
votes
4answers
277 views
Why the terminology “monoid”?
As I am not a native English speaker, I sometimes am bothered a little with the word "monoid", which is by definition a semigroup with identity. But why this terminology?
I searched some ...
9
votes
2answers
49 views
Origin of well-ordering proof of uniqueness in the FToArithmetic
In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it ...
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1answer
670 views
Proof of Euler's Theorem without abstract algebra?
Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ...
9
votes
1answer
140 views
Serge Lang and categories
I was told that (Serge) Lang has never fallen in love with categories, to use a polite euphemism. Other people claim that, in some occasion, he has even declared his lack of interest in the subject in ...
9
votes
3answers
370 views
History of elliptic curves
In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
9
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2answers
519 views
Original source for a quote by Lobachevsky
Lobachevsky is quoted in many places to have once written (said?) "There is no branch of mathematics, no matter how abstract, which may not someday be applied to phenomena of the real world." (In the ...
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votes
1answer
230 views
Origin of the name 'test functions'
This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
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1answer
294 views
What is the “etymology” of the notation “:=”?
I've noticed that sometimes people use ":=" to set variables, like "With $f(x):=x^{2}$, we have $f(1) = 1$." This is also the variable definition operation in Mathematica. My question is, did ...
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votes
3answers
837 views
Why has the Perfect cuboid problem not been solved yet?
Why hasn't Perfect Cuboid Problem been solved yet, whereas (possibly) more nontrivial ones such as FLT and Sphere packing have been solved?
I understand that calling some problems more nontrivial ...
9
votes
1answer
251 views
Sperner's theorem on antichains - where does it come from?
Sperner proved in 1927 (the paper was published in 1928) his theorem stating that the maximal size of an antichain of subsets of $[n]$ is $\binom{n}{n/2}$. In the introduction to his paper, he ...
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1answer
377 views
ancient concepts and modern concepts
Is there an extant published expository account, comprehensible to all mathematicians, of the conceptual differences between ancient Greek mathematical concepts and modern ones?
I have in mind things ...
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0answers
215 views
Who was Hermann Künneth?
Question as in the title:
Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia?
The well-known Künneth formula, for example in the form of ...
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0answers
356 views
How do Greek mathematicians name variables?
I've always wondered how people in Greek name variables that other people use greek letters e.g. $\theta$. They use latin?
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3answers
809 views
Provenance of Hilbert quote on table, chair, beer mug
All over the web one can find statements to the effect that:
"One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs"
There are many ...
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votes
6answers
445 views
Read old articles instead books.
I'd like to know if there is a site, or maybe a collection of books, where I can read old articles in mathematics in order to study topics directly from the source, instead reading books in the field. ...
8
votes
4answers
361 views
When/how did “formal” come to mean “informal” in mathematical contexts?
A question for the mathematical etymologists in the room:
Often when I see formally used in mathematical writing, it seems to be an indicator to put the whole sentence in quotation marks - the writer ...
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votes
2answers
400 views
Motivation for Tom Lehrer's song “Lobachevsky”?
I am trying to understand the motivation for the jingle about plagiarism written by Tom Lehrer. A YouTube version can be found here http://www.youtube.com/watch?v=IL4vWJbwmqM .
Where does history ...
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votes
3answers
254 views
What have been some of the most revolutionary philosophical shifts in perspective in mathematics?
Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from ...
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votes
3answers
307 views
The aim in a course of differential equations?
As I used to understand the primary aim of a student learning differential equations is that given a differential equation he should be able to solve it. However while recently reading a note on the ...
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votes
2answers
321 views
Gödel's ontological proof - How does it work?
Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel. Can someone please explain what those symbols are, and explain the proof? Thanks.
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2answers
147 views
What was Klein working on when he “replaces his Riemann surface by a metallic surface”?
I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting:
Look at Professor Klein: he is studying one of the most abstract questions of ...
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2answers
152 views
integrating the secant function, who figured this out?
I was looking at how the secant function is integrated. The process is not obvious, and I don't expect it to be but I wanted to know if anyone knows who figured this out. Here's what I'm talking ...
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2answers
537 views
Who are some forgotten mathematicians? [closed]
In Thomas' Calculus, he presents ''Nicole Oresme's Theorem'':
$$
\sum_{n=1}^\infty {n\over 2^{n-1}}=4.
$$
My first reaction was "who is this person?''.
As it turns out, he was a Frenchman from the ...
8
votes
1answer
524 views
What is the name of the $\in$ symbol and where does it come from?
It looks like a lower-case epsilon, but the Wikipedia page on epsilon states that they are not the same.
Does this symbol have a typographic identification outside of mathematics? Where did the ...
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2answers
262 views
Varieties as schemes
Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne.
Firstly, my main question. I understood that Grothendiecks introduction of schemes ...
8
votes
2answers
328 views
What's the history of the result that $p_{n+1} < p_n^2$, and how difficult is the proof?
In Edsger Dijkstra's monograph "Notes on Structured Programming", he describes a simple imperative program for generating an array of the first $n$ primes. For each prime $p_n$, it finds the next ...
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2answers
208 views
The history of set-theoretic definitions of $\mathbb N$
What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages?
I read about Frege's definition not long ago, ...
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2answers
227 views
Etymology of the name “deck transformation”
What does the word "deck" mean in "deck transformation"? What's the idea behind this name?
8
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1answer
237 views
When was the term “mathematics” first used?
By the second century, in the Almagest, Ptolemy provides a modern conception of "mathematics" as a "science":
'Mathematics' ... is an attribute of all existing things, without
exception, both ...
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1answer
186 views
Journals of math history?
In a related question to this one, in what journals do math historians publish their article in? Brian M. Scott provided a link to Judy Grabiner's, who is a math historian, home page and it seems that ...
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votes
1answer
272 views
Riemann's thinking on symmetrizing the zeta functional equation
In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as
...
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2answers
362 views
Mathematical Discoveries that were made or supported by savants
I just read something about Rüdiger Gamm, who recited $81^{100}$ (191 digits), which took approximately 2 minutes and 30 seconds. So I asked myself:
Are there any kind of mathematical ...
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1answer
240 views
How do mathematicians think about the existence of numbers?
Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept?
I know that existence of numbers is a ...
8
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1answer
121 views
Who first explicitly noted that second-order logic is unaxiomatizable?
As every student now knows, second-order logical consequence is unaxiomatizable. (At least when we read the second-order quantifiers in the natural way, as running over all possible properties on the ...
