History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.

learn more… | top users | synonyms

11
votes
0answers
230 views

Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...
36
votes
9answers
3k views

What do named “tricks” share?

There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous tricks, a term which in this context is in no sense derogatory. Here is a list of 10 such tricks (the ...
21
votes
1answer
1k views

Arnold on Newton's anagram

Arnold, in his paper The underestimated Poincaré, in Russian Math. Surveys 61 (2006), no. 1, 1–18 wrote the following: ``...Puiseux series, the theory which Newton, hundreds of years before ...
12
votes
1answer
1k views

What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
59
votes
22answers
9k views

Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time. I am looking for a list of Major theorems in mathematics whose proofs ...
60
votes
9answers
6k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
29
votes
1answer
867 views

Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid)

Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction ...
53
votes
20answers
7k views

Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13 Archimedean solids in a lost work, later mentioned by Pappus. But it awaited Kepler (1619) for the 13 semiregular polyhedra to be ...
39
votes
14answers
5k views

Do mathematical objects disappear?

I am asking this question starting from two orders of considerations. Firstly, we can witness, considering the historical development of several sciences, that certain physical entities ...
2
votes
2answers
134 views

Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...
15
votes
0answers
323 views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
90
votes
6answers
9k views

what mistakes did the Italian algebraic geometers actually make?

It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations ...
20
votes
9answers
3k views

Was the early calculus inconsistent?

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY. George Berkeley wrote in 1734 with reference to the early calculus that ...
3
votes
1answer
354 views

Who is the original author of this simple paradoxical decomposition?

Paradoxical decompositions of sets usually require the axiom of choice; Hausdorff or Banach-Tarski are well-known examples. A paradoxical decomposition of a point set without the axiom of choice has ...
17
votes
1answer
1k views

Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...
22
votes
4answers
2k views

In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...
10
votes
2answers
515 views

Who was the first to discover that the curvature of an embedded surface is the product of the principal curvatures?

The invention of intrinsic differential geometry is usually attributed to Gauss in the context of his theorema egregium but the notion of the curvature of an embedded surface existed before. Who was ...
0
votes
1answer
396 views

The $\zeta-$word [closed]

I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder ...
72
votes
25answers
9k views

Papers that debunk common myths in the history of mathematics

What are some good papers that debunk common myths in the history of mathematics? To give you an idea of what I'm looking for, here are some examples. Tony Rothman, "Genius and biographers: The ...
84
votes
5answers
7k views

New arXiv procedures?

Recently I encountered a new phenomenon when I tried to submit a paper to arXiv. The paper was an erratum to another, already published, paper and will be published separately. I got a message from ...
10
votes
1answer
377 views

What are “Artin fractions”?

The German Wikipedia entry for Ernst Witt https://de.wikipedia.org/wiki/Ernst_Witt has a photo of his grave in Hamburg. The bottom part has a visible text "Artin Brueche" (Artin fractions) but the ...
10
votes
2answers
266 views

Did the notion of “angle” originate with Thales?

Thales (circa 600BC—roughly 50 years before Pythagoras, 200 years before Plato, and 300 years before Euclid) certainly knew and reasoned with the concept of a planar angle. Are there earlier ...
-4
votes
1answer
156 views

When do Theorems (or Algorithms or Methods) Become Celebrated? [closed]

I recently noticed that certain theorems (e.g. Tutte's 1-factor theorem or, Edmond's Blossom algorithm) are attributed celebrated. A quick search on the internet yields further examples: ...
60
votes
19answers
21k views

Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
168
votes
36answers
46k views

Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...
5
votes
1answer
116 views

Historical refererences for Castelnuovo-Mumford regularity

Does anyone know a good reference to understand the historical background of Castelnuovo-Mumford regularity? I know the backgound for the modern commutative-algebra approach (using free graded ...
21
votes
1answer
1k views

William Rowan Hamilton and Algebra as Time

This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the ...
70
votes
30answers
10k views

What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here. My concern in this question is slightly ...
1
vote
0answers
162 views

Why the Castelnuovo exact sequence is named after Castelnuovo

I have seen variations of the following exact sequence referred throughout the literature as the Castelnuovo sequence: $$0\longrightarrow \mathscr I_{X:H}(-d)\longrightarrow \mathscr ...
24
votes
5answers
2k views

History of Mathematical Notation

I would like to see a simple example which shows how mathematical notation were evolve in time and space. Say, consider the formula $$(x+2)^2=x^2+4{\cdot}x+4.$$ If I understand correctly, Franciscus ...
3
votes
2answers
256 views

Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?
18
votes
2answers
3k views

Location of Archimedes' grave in Syracuse (math/archaelogy trivia)

This is really a question for our archaelogist friends, but I could not find an "archaelogy overflow" and some mathematicians might find it interesting. In a few weeks I am giving a talk in which I ...
12
votes
1answer
744 views

Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...
19
votes
1answer
1k views

history of quaternion algebras

Who is responsible for the generalization of Hamilton's quaternions to other types of quaternion algebras, and when did this occur? In particular, Hamilton's quaternions are the 4-dimensional algebra ...
17
votes
1answer
443 views

When did “Betti cohomology” come to be used the way it is today? (and how is it used)

This is sort of a mixture of a math and history question. First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I ...
11
votes
2answers
561 views

First formulation of the Dedekind and Hasse-Weil conjectures

I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated: ...
32
votes
10answers
7k views

real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
53
votes
3answers
4k views

What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem): ...the end goal [is] to establish as ...
4
votes
1answer
194 views

Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as: Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially? ...
6
votes
1answer
150 views

How to divide a square into three similar rectangles

Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles? With a bit of algebra it can ...
9
votes
1answer
376 views

Examples of abstractions that did *not* turn out to be useful [closed]

I’ve read (but cannot find any reference now) that new abstract mathematical concepts like set theory and – not too long ago – category theory were in their time often considered too abstract to be ...
69
votes
26answers
11k views

What are some famous rejections of correct mathematics?

Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner Theorem: There are only a finite number of imaginary quadratic fields that have unique factorization. ...
8
votes
2answers
1k views

How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematics

Does anyone know how news of Gödel's incompleteness theorem spread? Did it do so little by little, or was it shouted in dramatic headlines throughout mathematical literature? If anyone can point me to ...
16
votes
2answers
410 views

Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...
11
votes
1answer
432 views

Two Vinogradovs? Is one the son of the other? [closed]

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming ...
20
votes
1answer
685 views

What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
22
votes
1answer
733 views

How and why did mathematicians develop spin-manifolds in differential geometry?

First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...
19
votes
19answers
25k views

Good books on problem solving / math olympiad

Hello, I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would ...
144
votes
136answers
30k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
11
votes
1answer
911 views

Taniyama's original conjecture

I've just read on Wikipedia that the original Taniyama conjecture about L-functions of elliptic curves over an arbitrary number field was still unproven. This made me want to know more about this ...