Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations.

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How was the first log table put together?

Henry Briggs compiled the first table of base-$10$ logarithms in 1617, with the help of John Napier. My question is: how did he calculate these logarithms? How were logarithms calculated back then? ...
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1answer
270 views

history of the contraction mapping technique

If $|f(x)-f(y)| \leq k|x-y|$ for all $x,y$ then $f$ is Lipschitz with constant $k$, if $k<1$ then $f$ is called a contraction mapping. The beautiful result that a fixed point is associated to a ...
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1answer
369 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
96 views

History of the point at infinity?

I'm curious to learn more about the history of the introduction of the concept of the point at infinity into mathematics. The sum of my knowledge of the historical aspect is from this paragraph (which ...
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1answer
333 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
77 views

Cauchy gave 1st example of a Lie algebra in 1847 & exterior product in 1853‽

I read in PDF pg. 5 of this that Cauchy gave the first example of a Lie algebra in 1847: It also claims that he invented the exterior product in 1853. Does anyone have references for this?
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1answer
299 views

Hao Wang's $\mathfrak S$ system/$\Sigma$ system: a “transfinite type” theory that avoids the Goedel's theorems.

Long ago, while I was reading a book ($*$) about the various way to build set theories (Zermelo-Freankel, Von Neumann–Bernays–Gödel, and type theories), I read about a variant of type theory with ...
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5answers
3k views

Why do we consider prime numbers important, and what are their applications other than number theory in pure math?

Why do we consider prime numbers important, and what are their applications other than number theory in pure math? I know that Number theory is devoted to studying prime numbers, but there must be ...
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6answers
4k views

Why do we need vectors and who invented it?

It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what? EDIT: As George Lowther pointed out, the problem is too broad; I added the ...
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3answers
1k 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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3answers
628 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 ...
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347 views
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452 views

Where, specifically, did Principia Mathematica fail?

I'm very fascinated by the book Principia Mathematica. From what I've learned so far, Principia Mathematica set out to be, essentially, the bible of mathematics and logic, from which all mathematical ...
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532 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 ...
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2answers
353 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
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3answers
748 views

Where is the name “coset” in group theory from?

One of the most important application of "coset", I think, is to prove the Lagrange's theorem, which was not originally stated in the group theoretic terms. In some textbooks I have read about ...
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5answers
331 views

Purpose of the Peano Axioms

Wikipedia says the Peano Axioms are a set of axioms for the natural numbers. Is the purpose of the axioms to create a base on which we can build the rest of mathematicas formally? If this is true ...
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1answer
722 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
833 views

Why are even/odd functions called even/odd?

Bit of a silly question, someone told me that the reason even functions are called 'even' and odd functions are called 'odd' is that all (single-variable) monomials with even powers are even functions ...
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2answers
350 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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1answer
131 views

What was the planned topic of Gödel's second paper on incompleteness?

Gödel's incompleteness theorems first appeared together in a paper titled (translated to English) "On formally undecidable propositions of Principia Mathematica and related systems I," with the Roman ...
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2answers
593 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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1answer
464 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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1answer
313 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
341 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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179 views

Newton's “Famous Blunder”?

On page $225$ of Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini (see here for a link), I read In the following demonstration... Newton made a famous blunder... He wrote, ...
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176 views

Ramanujan's personification of small positive integers

I dimly recall reading somewhere (perhaps in "The Man Who Knew Infinity"?) that Ramanujan associated personalities (perhaps it was mystical personalities, e.g. specific gods and goddesses?) with small ...
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1answer
175 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 ...
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1answer
730 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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274 views

Where can I find the 1960s New Math syllabus?

I've been looking everywhere for even a short summary of the content of the 1960s New Mathematics Math education reform in the US but I cannot ;-; Does anyone know?
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1answer
190 views

Articles on ideas in the history of mathematics notation?

I'm teaching a course this term on the history of scripts (writing systems) and rather than talking interminably about Semitic and Chinese and their spawn, I'd like to give students a more varied ...
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1answer
348 views

Origin of Littlewood's idea about sign changes of $Li(x) - \pi(x)$

Background (skip if you like). Skewes and Littlewood are closely identified with the idea that $Li(x)- \pi(x)$ changes sign infinitely often, but Littlewood closed a gap in the work of Schmidt, who ...
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4answers
985 views

Angle brackets for tuples

I've recently noticed that use of angle brackets for writing tuples, e.g. $\langle x, y \rangle$ instead of the usual round brackets in a few books I've been reading — Lawvere's Sets for Mathematics, ...
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146 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
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119 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
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1answer
117 views

The average of the roots of a polynomial equals the average of the roots of its derivative

Background: It's straightforward to check that the average (i.e. the mean) of the roots of a nonlinear polynomial equals the average of the roots of its derivative: if $$f(x) = x^n + a_{n-1} x^{n-1} ...
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8answers
1k views

Why are derivatives specified as d/dx?

Is the purpose of the derivative notation d/dx strictly for symbolic manipulation purposes? I remember being confused when I first saw the notation for derivatives - it looks vaguely like there's ...
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4answers
3k views

Math story: Ten marriage candidates and 'greatest of all time'

I remember a story about a famous mathematician who was offered ten marriage candidates and had to pick one of them, with the condition he had to meet them in turn and propose during that meeting, ...
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4answers
506 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
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597 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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230 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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276 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
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4answers
268 views

Interesting Medieval Mathematics Lecture/Activity Ideas?

Recently I have been invited to give a talk about Medieval Mathematics or mathematics in the 500 AD - 1500 AD time frame. I have been researching the time frame for the past week and have found ...
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2answers
176 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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3answers
886 views

When the trig functions moved from the right triangle to the unit circle?

I have to write a paper about the unit circle and I'm trying to uncover some of its origins. Also, when the trig functions were expanded to angles greater than 90° and what was the rationale behind ...
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2answers
441 views

Notation for intervals

I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed intervals. I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall ...
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2answers
701 views

Why is “h” used for entropy?

Why is the letter "h" (or "H") used to denote entropy in information theory, ergodic theory, and physics (and possibly other places)? Edit: I'm looking for an explanation of the original use of "H". ...
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1answer
359 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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253 views

Etymology of the name “deck transformation”

What does the word "deck" mean in "deck transformation"? What's the idea behind this name?
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490 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 ...