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 well-known are these contra-Bernoulli inequalities?

The standard and extremely useful Bernoulli inequality states that $(1+x)^n \ge 1+nx$ for positive integer $n$ and $x \ge 0$. I have needed an inequality of the form $(1+x)^n \le 1+c(n)x$ where $x$ ...
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What's the intuition behind definition of chaotic function?

I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos" ...
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107 views

Mathematical logic and foundations of mathematics in the 20th century

I would like some references regarding the foundations of mathematics in the 20th century, and mathematical logic, e.g. (1) I want to understand what happened to the foundation, what originated the ...
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1answer
253 views

What is “Bourbaki's style in mathematics”?

I know Nicolas Bourbaki "is the pseudonym of a group of (mainly) French mathematicians who publish an authoritative account of contemporary mathematics." But what characterizes "Bourbaki's style in ...
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62 views

Infinite Product Identity for Hyperbolic Sine

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$. I saw this formula in a book and have no idea ...
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203 views

In Whitehead and Russell's PM, does not identity imply existance?

At the end of ✳96.48, $ \sim(w=\overset{\smile}{R}‘max_R‘J_R‘x)$ is chosen over $ w\neq\overset{\smile}{R}‘max_R‘J_R‘x$, on account of the latter's implication of existence. But ✳13.02 states that ...
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1answer
94 views

Differences in how mathematical results are proved in the time of Euclid and in the twentieth century [closed]

What is the difference in the manner demonstrated in Euclidean time and as demonstrated in the twentieth century?
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762 views

Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
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210 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
103 views

The meaning of Differentials in Integration

This is further to the questions discussed in a previous post Here is an example of what I mean: Suppose that $C$ is a closed path in the plane and consider the line integral of $xy\,dx+x^2\,dy$ over ...
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1answer
104 views

Fibonacci's proof that $x^3+2x^2+10x=20$ has no solution in radicals?

I read on a poster today that Fibonacci showed that $x^3+2x^2+10x=20$ has no solution expressible in radicals, way back when. I couldn't find the proof anywhere. Does anyone know where I can find it? ...
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301 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
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1answer
105 views

Who introduced the finite difference notation using $\Delta$?

We all know that Leibniz introduced the differential notation $dx, dy$, and that in developing his calculus for infinitesimal differences he was inspired by his previous work on finite diffences. ...
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24answers
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Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
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1answer
99 views

History of the Basel problem

Why were the people of the $18$th century interested in the Basel problem? (The Basel problem asks for the value of $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$).
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1answer
97 views

eulers original derivation for the Euler–Maclaurin formula?

Please does someone know a good description of how Euler did derive his summation formula? Thank you!
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1answer
430 views

History of Lagrange Multipliers

How did Lagrange discover Lagrange multipliers? Also, was it related to his work on the calculus of variations? And how did he originally understand/implement the technique?
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165 views

Theory vs problems in modern math

Quick background: I'm a fourth year undergraduate entering graduate school next year. I am trying to identify areas of mathematical research in which there tends to be more emphasis on developing new ...
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History Question re Euler's Constant $\gamma$

What used to be be called "Euler's Constant" (http://en.wikipedia.org/wiki/Eulers_constant) is now frequently called the "Euler-Mascheroni Constant". I have tried to find out what contribution ...
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Did Field's Medalist Klaus Roth suffer from test anxiety?

I remember hearing the story that Fields Medalist Klaus Roth was convinced that he could not pass a qualifying exam when he was a graduate student. He was then given a so called practice exam for him ...
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2answers
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Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
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1answer
117 views

In Whitehead and Russell's PM, is there a typo in ✳72.23?

It seems that where $\gamma$ appears at the end of line 1, 4, 5, there should be a $z$ instead, i.e. $✳72.23\hspace{10pt} \vdash : R,S \in 1 \rightarrow Cls .\supset. ...
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10answers
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Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
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1answer
164 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
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3answers
114 views

History of infinite series representations of $\sin(x)$ and $\cos(x)$

When did the famous infinite series representations for $\sin(x)$ and $\cos(x)$ came about? To be specific when did people realise that the ratio of the two sides of a right triangle with one angle ...
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Euler's errors?

What mathematical errors is Leonhard Euler known to have made? PS: As I wrote in a comment below: "However, I would not consider proof to be an error merely because it's not a proof by present-day ...
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1answer
117 views

First usage of the symbol ∈

Concerning a book [1] I am reading the symbol $\in$ was first used by Giuseppe Peano and is the first letter $\epsilon$ (epsilon) of the word ἐστί (means "is"). Does anyone know in which work of Peano ...
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1answer
312 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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1answer
167 views

What cubic problems did Tartaglia and Fior pose to each other?

I have been researching the history of finding roots to general polynomials and the story of solving for the roots of cubic polynomials ($ax^3+bx^2+cx+d=0$) lead me to find several sources describing ...
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The correct pronunciation of Whaples?

What is the correct the family name of the mathematician George Whaples? Feel free to migrate my question to a another forum if you think here is not the right place.
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1answer
73 views

Enlightening books giving a guided tour of mathematics, in a style that Gian-Carlo Rota would not mind?

I am currently reading Gian-Carlo Rota's Indiscrete Thoughts. What more can I say apart from "the man can write"? (In other words, you should really read it if you are interested in mathematics.) I ...
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3answers
223 views

The Historical Importance of Keynes' A Treatise on Probability

A visiting speaker in Economics recently happened to mention that John Maynard Keynes' A Treatise on Probability revolutionized probability theory. I have not heard any such claim before and it struck ...
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2answers
787 views

How to evaluate trigonometric functions by pen and paper?

How did people determined the values of trigonometric functions before calculators, like e.g. $\sin 37^\circ$ up to five decimal places? Was that possible to find before series were invented?
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In Whitehead and Russell's PM, are overlapping ranges of significance necessarily identical?

In Principia Mathematica summary of ✳63 In virtue of ✳20.8, we have $\vdash : \phi a ∨ \sim\phi a . ⊃ . \hat{x}(\phi x \vee \sim \phi x ) =t‘a$ i.e. if "$\phi a$" is significant, then ...
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3answers
277 views

What is the “Principle of permanence”?

While reading the book "The Number-System of Algebra (2nd edition)." term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students.". I ...
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1answer
88 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
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3answers
66 views

Why the name “umbilic”?

Umbilic points are points on a surface at which the principle curvatures of the surface are equal. "Umbilic(al)" refers to the navel/belly button. But why do we call these points so? What about the ...
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Concept of a function and Idea of a formula as a function; History of

Enderton Elements of Set Theory, p. 43 (1977, Academic Press), writes: There was a reluctance to separate the concept of a function itself from the idea of a written formula defining the function. ...
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How did Newton and Leibniz actually do calculus?

How did Leibniz know to write derivatives as $$\frac{dy}{dx}$$ so that everything would work out? For example, the chain rule: $$\frac{dy}{dz}=\frac{dy}{dx}\frac{dx}{dz}$$ Integration by Parts: ...
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1answer
1k views

Did Albert Einstein contribute to math?

Many great scientists have made important contributations to many related fields. Gauss, Euler and Newton each made many contributions to both math and physic. One of the great scientists of last ...
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1answer
122 views

Definition of the $\sec$ function

I am a postgraduate student of mathematics from Slovenia (central Europe) with quite some experience in mathematics. While answering questions on this site, I often encounter the function $\sec(x)$ ...
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2answers
140 views

Why the $\log$ is so special?

When I first learn about the logarithm function $\log$ or $\ln$. My professor said that $\log x$ is a function that when we derive we get the inverse function $1/x$. This $\log$ becomes very popular ...
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2answers
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Reason behind standard names of coefficients in long Weierstrass equation

A long Weierstrass equation is an equation of the form $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, ...
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4answers
575 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
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4answers
213 views

“important” math concepts to pass on to next generation of creatures at some cataclysm [closed]

This may be somewhat silly to ask, but I couldn't resist the temptation. The idiosyncratic physicist Richard Feynman was once asked If, in some cataclysm, all of scientific knowledge were to be ...
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2answers
164 views

History of Morse theory.

How can I get good references which give many information about history of Morse theory? Now I am interested in how and who found that Hessian have a lot of data. Thank you for your helping!!
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2answers
336 views

Works on Calculus by Newton and Leibniz (primary sources)

I'm trying to find PDFs or hard copies of the following works from the dawn of calculus. Does anyone know where I could find English translations of them? Newton - De analysi per aequationes numero ...
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Lebesgue - differentiation of monotone functions

I was wondering how Lebesgue himself proved the continuous case. Since my French is not good enough to read his own book, I was wondering if someone knows if there exists a translation ? (at least of ...
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1answer
83 views

History of the Enneper Surface

I was just wondering whether anyone could tell me more about the Enneper surface and its history (why it is important historically in the development of mathematics), or where to go in order to learn ...
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1answer
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Soft Question: Algorithms: Will We (One Day) No Longer Need to Study Algorithms? [closed]

I'm just now getting into the study of algorithms and it seems like as computers get faster and faster the need to study algorithms may begin to diminish. How likely is it that in 50 years there ...