28
votes
5answers
2k views

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 ...
13
votes
7answers
497 views
2
votes
1answer
105 views

Reference Request - Early Calculus Papers

Question: I am looking for good references on the early calculus papers. Optimally, I want emphasis on the mathematics itself and I want that mathematics to be translated into modern terminology and ...
58
votes
23answers
10k views

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 ...
5
votes
2answers
137 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 ...
83
votes
30answers
16k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
1
vote
1answer
143 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
18
votes
1answer
496 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?
28
votes
20answers
2k views

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 ...
76
votes
24answers
6k views

What are some examples of notation that really improved mathematics?

I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational ...
34
votes
13answers
3k views

Examples of famous problems resolved easily

Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example ...
64
votes
21answers
3k views

What are some examples of mathematics that had unintended useful applications much later?

I would like to know some examples of interesting (to the layman or young student), easy-to-describe examples of mathematics that has had profound unanticipated useful applications in the real world. ...
4
votes
1answer
197 views

List of proofs of non-trivial theorems which were unnoticed to be wrong for at least a few years

For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.
41
votes
19answers
4k views

Theorems' names that don't credit the right people

The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) ...
51
votes
20answers
4k views

Is there a great mathematical example for a 12-year-old?

I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets. Why? Because the maths department at her school is outrageously good, and set ...
51
votes
9answers
3k views

Surprisingly elementary and direct proofs

What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, ...
45
votes
2answers
1k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
4
votes
0answers
112 views

The mathematical heritage of Lewis Carroll

Which mathematical results has Lewis Carroll, the author of Alice's Adventures in Wonderland, produced? Wikipedia is very vague with regard to this topic and gives us little more than a matrix ...
15
votes
1answer
424 views

Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.

In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of ...
5
votes
1answer
118 views

Differences in worlds with and without $\aleph_0<|S|<2^{\aleph_0}$

Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
95
votes
33answers
9k views

Can you provide me historical examples of pure mathematics becoming “useful”?

I'm trying to think/know about something but I don't know if my basis premise is plausible, here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...
62
votes
10answers
10k views

Results that came out of nowhere.

Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. ...
14
votes
7answers
1k views

Films about math: a question about math education and motivation for learning math

I'm interested in movies about or related with mathematics or physics, I mean not documentaries which I also consider movies, but artistic or mainstream films about math. Now I have the following in ...
9
votes
11answers
1k views

Good examples for mathemathical problems/statements that are easely solvable/provable in one theory and hard to solve/prove in another

Let $P$ be a mathematical statement or a mathematical problem. I am looking for a couple of nice examples for $P$ that satisfy the following criteria: Given two (or more) mathematical points of view ...
32
votes
6answers
1k views

Original works of great mathematicians

In almost every mathematical text there is a line as This was first proved by Gauss or This formula first appeared in a work of Riemann, but for me it's more like My friend told me once that... For ...
5
votes
8answers
1k views

What mathematical ideas/concepts became obsolete due to technological progress?

As technology evolved, some ideas and methods became obsolete. What mathematical ideas entered this state due to technology progress? We could consider that doing some mathematical operations done by ...
6
votes
2answers
813 views

Mathematics celebrities that every mathematician should know? [closed]

As a mathematician, sometimes I meet across very embarrassing questions which were posted by who does not learn of mathematics, for example, my wife and so on. She or he always posted such questions: ...
31
votes
13answers
3k views

Research done by high-school students

I'm giving a talk soon to a group of high-school students about open problems in mathematics that high-school students could understand. To inspire them, I would like to give them examples of ...
9
votes
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 ...
5
votes
5answers
471 views

What are some unexpected things math predicts? [closed]

Once I heard about a prophet that used math to foresaw with great accuracy many events of the humanity. Today I oddly realized the time between falling drops after washing cups fit the inverse square ...
10
votes
2answers
836 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 ...
17
votes
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?
10
votes
10answers
983 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 ...
28
votes
7answers
5k views

Good books on Math History

I'm trying to find good books on the history of mathematics, dating as far back as possible. There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to ...
23
votes
9answers
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?
2
votes
4answers
564 views

What is your favorite isomorphism? [closed]

By "isomorphism" I mean any structure-preserving map with a structure-preserving inverse. (Please accept my advance apology if this question is out of bounds. I sense that it's borderline, but I'm ...