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

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26
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Who invented $\vee$ and $\wedge$, $\forall$ and $\exists$?

I can rather easily imagine that some mathematician/logician had the idea to symbolize "it E xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the ...
26
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2answers
1k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
26
votes
1answer
353 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
26
votes
4answers
2k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
25
votes
4answers
6k views

Why are so many of the oldest unsolved problems in mathematics about number theory?

Stillwell mentions in his book, Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... Have ...
25
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11answers
4k views

Good books on Philosophy of Mathematics

Where can I learn more about the implications, meta discussions, history and the foundations of mathematics? Is Russell's Introduction to Mathematical Philosophy a good start?
25
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2answers
1k views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
25
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4answers
1k views

Examples of mathematicians who lost interest in Math and got interested again?

I am looking for some examples, and hopefully some short biographies on mathematicians who lost interest in Math along the way, and somehow got rejuvenated again. (Better still, who managed to do ...
24
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1answer
2k views

What concept does an open set axiomatise?

In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ ...
23
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5answers
2k views

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 ...
23
votes
6answers
838 views

What did Newton and Leibniz actually discover?

Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a ...
23
votes
1answer
480 views

How much math education was typical in the 18th & 19th century?

Was it unusual for people in those days to learn Calculus? Could a grad student take a course in differential equations or multi-variable Calculus, or did they have to learn from journals? I am always ...
23
votes
2answers
975 views

Where can I find the old papers of the Math Tripos?

Is there a repository on the Internet which has the old question papers of the tripos? I am specifically interested in the papers during the 1890-1910 era, which was the era before the reforms, ...
23
votes
3answers
1k views

When did Fubini's name get applied to the theorem without measures?

Fubini's theorem, from 1907, expresses integration with respect to a product measure in terms of iterated integrals. The simpler version of this theorem for multiple Riemann integrals was used long ...
22
votes
5answers
1k views

Importance of rigor

I always have a hard time explaining the importance of rigor to my friends who are not mathematically minded. A lot of past mathematicians develop the foundations of today's mathematics without going ...
22
votes
2answers
2k views

Math symbol in German thesis from 1963

I have the following math symbol in a German thesis written in 1963. Is it anything more than just a function name? It is used in the following context and then goes on to state that "If the ...
22
votes
5answers
827 views

Did Euclid prove that $\pi$ is constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to ...
22
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
22
votes
2answers
991 views

Does there exist a copy of Euclid's Elements with modern notation and no figures?

I am working through Euclid's Elements for fun, but I find the propositions difficult to understand without referencing the provided figures. Unfortunately, the figures usually give away the proofs, ...
22
votes
2answers
798 views

Why is the topological pressure called pressure?

Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy. For ...
22
votes
1answer
345 views

Hilbert's original proof of basis theorem

Does anyone know Hilbert's original proof of his basis theorem--the non-constructive version that caused all the controversy? I know this was circa 1890, and he would have proved it for ...
21
votes
2answers
1k views

Why is lambda calculus named after that specific Greek letter? Why not “rho calculus”, for example?

Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?
21
votes
3answers
683 views

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
21
votes
3answers
777 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
21
votes
2answers
878 views

Did H. Lebesgue claim “1 is prime” in 1899? Source?

John Derbyshire, in his text "Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics" states that The last mathematician of any importance who did [consider the number ...
20
votes
1answer
480 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
19
votes
2answers
529 views

Approximation for $\pi$

I just stumbled upon $$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$ which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I ...
19
votes
3answers
856 views

Who realized $\int \frac 1x dx =\ln(x)+c$?

Who discovered the non-obvious $\int \frac 1x dx=\ln(x)+c$ ? Were power series involved? The series look similar on opposite sides of 1: $$ \frac 1x =\sum_{n=0}^\infty (-1+x)^n \text{ for } ...
19
votes
2answers
633 views

Who first called the Grothendieck's schéma scheme?

Grothendieck called "schemes" schémas in French. I find it strange we call them schemes. In fact, Grothendieck called them (pre-) schemas(this is an English word) in his talk(in English) at Proceeding ...
19
votes
4answers
2k views

Where did mathematicians learn how to do truth tables?

I'm trying to find out who invented truth-tables. Here is what I have so far. Leibniz 'invented' binary arithmetic, or at least is the first one recognized to have codified and explained a base 2 ...
19
votes
2answers
266 views

Curious remark of D. Ravenel

In his beautiful (but difficult) book "Complex cobordism and stable homotopy groups of spheres", concerned mostly with methods of computing homotopy groups of spheres, D. Ravenel describes a general ...
19
votes
2answers
575 views

History of the theory of equations: John Colson

This is an EDIT version of my original question: Recently I've been interested in the history of the Theory of Equations. The thing is that I learned about this mathematician named John Colson, he ...
19
votes
2answers
659 views

A place to learn about math etymology?

I was recently wondering where the word `kernel' comes from in mathematics. I am sure the internet must know. I did manage to find http://www.pballew.net/etyindex.html#k which contains the origin ...
19
votes
1answer
340 views

Matsunaga's Method for solving $x^2+y^2=p$

In his history of number theory, Dickson mentions an 18th century algorithm due to Matsunago [Sic --- he means, presumably, Matsunaga Ryohitsu a.k.a. Matsunaga Yoshisuke] for finding two numbers whose ...
19
votes
1answer
376 views

When, and by whom, was “$\mathbb{C}$ is algebraically closed” dubbed the “fundamental theorem of algebra”?

Wikipedia has this enigmatic sentence on the page for the fundamental theorem of algebra: ...its name was given at a time when the study of algebra was mainly concerned with the solutions of ...
18
votes
8answers
3k views

How did the notation “ln” for “log base e” become so pervasive?

Wikipedia sez: The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians ...
18
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6answers
2k views

What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
18
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4answers
3k views

Why is an image called an “image”?

Given a function $f : A \to B$, the image, denoted by $\operatorname{Im}f$ is the set of all $f(x)$ where $x \in A$. Why do we call this set the image? When was it first used, and what motivated its ...
18
votes
1answer
1k views

What was Ramanujan's solution?

The wikipedia entry on Ramanujan contains the following passage: One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a ...
18
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1answer
2k views

Where did the word “logarithm” come from?

Where did the word logarithm come from? Any relation to the word algorithm?
18
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1answer
656 views

Did Leonardo of Pisa prove $n=4$ case of FLT?

Reputable on-line sources agree that Leonard 'Fibonacci' proved the nonexistence of positive-integer solutions to $c^4 - b^4 = a^2$ . Yet my change to Wikipedia to reflect this was reverted. I hope ...
18
votes
1answer
523 views

The most active fields of mathematics?

Which fields of mathematics are the most active at this time -- by number of papers published, grant money, people working in them or by any other measure? Any trends in this regard?
18
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0answers
478 views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
17
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5answers
1k views

Results that were widely believed to be false but were later shown to be true

What are some results which were widely believed to be false, but were later to be shown to be true, or vice-versa?
17
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3answers
1k views

where does the term “integral domain” come from?

Self-explanatory title really! A student today asked me why they were called integral domains -- and I realised that the word "integral" seems to be being used in a way totally unlike any other way I ...
17
votes
1answer
715 views

What was the last mathematical paper published in Latin?

From an answer to a previous question I learned that Peano published in Latin as long as 1889. What was the last mathematical paper/book of recognized importance published in Latin?
17
votes
2answers
660 views

History of Modern Mathematics Available on the Internet

I have been meaning to ask this question for some time, and have been spurred to do so by Georges Elencwajg's fantastic answer to this question and the link contained therein. In my free time I ...
17
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2answers
1k views

De Moivre's Theorem. Motivation and origins.

I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous $$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$ but ...
16
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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 ...
16
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2answers
637 views

Motivation for/history of Jacobi's triple product identity

I'm taking a short number theory course this summer. The first topic we covered was Jacobi's triple product identity. I still have no sense of why this is important, how it arises, how it might have ...