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

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What are some theorems that currently only have computer-assisted proofs?

What are some theorems that currently only have computer-assisted proofs? For example, there's the four colour theorem. I am very curious about this and would like to generate a list.
37
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7answers
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Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
37
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1answer
910 views

What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?

I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract ...
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2answers
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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 ...
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5answers
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Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
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4answers
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Why weren't continuous functions defined as Darboux functions?

When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like "Continuous functions are those you can draw without lifting your pen" With this in ...
32
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6answers
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Why are integrals called integrals?

What is the historical background for this term? I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems a strange choice of ...
32
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3answers
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Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
32
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5answers
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“Stick it to the man!” Mathematical discoveries that resulted in persecution.

As the old story goes, Pythagoras and his followers were adamant that all numbers were rational, until Hippasus came along and proved that $\sqrt{2}$ (the length of the diagonal of the unit square) is ...
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1answer
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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 $F$ ...
32
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1answer
568 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 ...
32
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3answers
544 views

To what extent were mathematicians in previous centuries aware of the lack of rigour in their methods?

By modern standards, much of pre-modern mathematics isn't rigorous. Famous examples include Euler's solution to the Basel problem or literally anything involving sets before Cantor and Russel came ...
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20answers
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Good math bed-time stories for children?

What are some good references/books/articles from which to derive some good bed-time math stories to pique a child's interest in math? I am fascinated by math (used to hate it as a kid) and want my ...
31
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4answers
1k views

Could G. H. Hardy make a product of two primes so big he couldn't find out which?

This question of course began as a slightly irreverent play on the riddle, "Can God make a stone so big He can't lift it?" Then I wondered about the answer. Is it possible to exhibit a number that is ...
31
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1answer
1k views

History of “Show that $44\dots 88 \dots 9$ is a perfect square”

The problem Show that the sequence, $49, 4489, 444889, \dots$, gotten by inserting the digits $48$ in the middle of the previous number (all in base $10$), consists only of perfect squares. ...
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4answers
7k views

Are We Teaching Pre-Calc Wrong?

It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial. When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that ...
30
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3answers
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Where does the word “torsion” in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
30
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2answers
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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 : ...
30
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2answers
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Image of a math problem that was stated in Cuneiform, Arabic, Latin and Finally in modern math notation

Many years ago a lecturer of mine had a photocopy of a page from a book containing a math problem ( I think it was a simple quadradic equation ) that was stated/solved in Cuneiform, Arabic, Latin ...
29
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7answers
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Genius mathematicians who never published anything

Amongst philosophers, Socrates is an example of a genius with a great influence on human history who never wrote anything. Almost all facts which are known about his revolutionary ideas are written by ...
29
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4answers
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Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
29
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0answers
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Irrationality of $e$

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
28
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11answers
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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?
28
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3answers
756 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question ...
28
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3answers
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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 ...
27
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4answers
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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 ...
27
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5answers
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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 ...
27
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1answer
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Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
27
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4answers
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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 ...
27
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4answers
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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 ...
27
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3answers
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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, ...
26
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3answers
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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”?
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4answers
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Why is “mathematical induction” called “mathematical”?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the ...
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5answers
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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 ...
26
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0answers
668 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 ...
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5answers
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Why are turns not used as the default angle measure?

Why is $2\pi$ radians not replaced by $1$ turn in formulas? The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?
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5answers
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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 ...
25
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5answers
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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 ...
25
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5answers
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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 ...
24
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3answers
963 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 ...
24
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5answers
478 views

Examples of Mathematics in Court

In court trials, natural sciences such as physics and biology routinely make an appearance, e.g. when estimating the speed of a vehicle based on impact damage or trying to deduce from the condition of ...
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6answers
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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 ...
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5answers
953 views

Who are some blind or otherwise disabled mathematicians who have made important contributions to mathematics?

Two prominent mathematicians who were disabled in ways which would have made it difficult to work were Lev Pontryagin and Solomon Lefschetz. Pontryagin was blind as a result of a stove explosion at ...
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4answers
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Were “real numbers” used before things like Dedekind cuts, Cauchy sequences, etc. appeared?

Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers. I'm also very interested, if the answer is ...
24
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3answers
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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 ...
23
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11answers
4k views

Why is Lebesgue so often spelled “Lebesque”?

Henri Lebesgue (1875-1941) was a French mathematician, best known for inventing the theory of measure and integration that bears his name. As far as I know, "Lebesgue" is the correct spelling of his ...
23
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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?
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3answers
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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 } ...
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2answers
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How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
23
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1answer
542 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 ...