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

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44
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6answers
6k views

Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?

According to Steven Krantz's Mathematical Apocrypha (pg. 186): As was custom, Weil often attended tea at [Princeton] University . Graduate student Steven Weintrab one day went about the room ...
224
votes
72answers
88k views

Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
1
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0answers
116 views

Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...
1
vote
0answers
610 views

Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...
10
votes
4answers
1k views

Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...
51
votes
27answers
7k views

Writing papers in pre-LaTeX era?

I wonder how people wrote papers in the pre-LaTeX era? I mean, when typewriters and simple computers were (60th-70th?). Did they indeed put formulas by hand in the already printed articles?
1
vote
0answers
113 views

History of categorical localization sans calculi of fractions

This question arises from a paper which I've just found and skimmed: FW Bauer, J Dugundji. Categorical homotopy and fibrations. Transactions of the American Mathematical Society, 1969 With 28 ...
42
votes
1answer
2k views

What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
6
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0answers
130 views

When did people know that all real polynomials of degree greater than 2 are reducible? [migrated]

Admittedly, this may not be a research level question, but I am deeply curious about this. Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...
13
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1answer
1k views

Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below. In ...
6
votes
0answers
173 views

$\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$ Question 1. Who first introduced the above question, and where can I find ...
13
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4answers
1k views

Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

In the Grundgesetze der Arithmetik, Frege used a number of strange characters for notation. I would be most interested to know anything about the typography (origin, usage and so on) of the strange U ...
63
votes
65answers
9k views

Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...
0
votes
2answers
242 views

Linear Algebra classic books [closed]

I'm learning linear algebra at the moment, so I'm looking for some great old classic books. Something like Fermat's or Gauss books of some great mathematians. I don't really like the nowadays books ...
11
votes
1answer
330 views

What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...
7
votes
1answer
399 views

Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?

A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already ...
20
votes
3answers
1k views

Why is the identity element of a group denoted by $e$?

The question was asked by a student, and I did not have a ready answer. I can think of the German word ``Einheit'', but since in German that is not how the identity element of a group is called, I ...
27
votes
22answers
7k views

Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any ...
19
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2answers
1k views

History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to ...
69
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22answers
7k views

Fields of mathematics that were dormant for a long time until someone revitalized them

I thought that the closed question here could be modified to a very interesting question (at least as far as big-list type questions go). Can people name examples of fields of mathematics that were ...
22
votes
0answers
433 views

Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson ...
15
votes
1answer
987 views

The list of problems for Grothendieck's thesis

Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...
12
votes
2answers
781 views

Describe the desired features of a “Mathematics Colloquium”?

I'm now a member of my department's colloquium committee. Our task is to make a great colloquium series. I thought that the first step would be to come up with an appropriate definition of ...
59
votes
15answers
6k views

Mathematical research published in the form of poems

The article Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen, Math. Z. 127 (1972), no. 1, 10-16 is written in the form of a lengthy poem, in a style similar to that of the ...
4
votes
1answer
162 views

Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...
45
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19answers
6k views

Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...
148
votes
36answers
40k 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 ...
20
votes
1answer
934 views

What is $\infty^6$?

The title of this question may make you want to close it immediately, but bear with me a moment. In several older mathematics papers (early 20th century) I have seen statements such as The ...
12
votes
4answers
2k views

What are hypergroups and hyperrings good for?

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...
1
vote
0answers
93 views

First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...
14
votes
2answers
764 views

Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus ...
11
votes
0answers
269 views

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
42
votes
4answers
1k views

How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...
54
votes
5answers
4k views

Are there any serious investigations of whether “mathematicians do their best work when they're young”?

There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious ...
19
votes
18answers
4k views

Examples of conjectures that were widely believed to be true but later proved false

It seems to me that almost all conjectures (hypotheses) that were widely believed by mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?
18
votes
5answers
899 views

Historical use of figures in geometry

I was surprised to learn from John Stillwell's comment in answer to the question, "Can the unsolvability of quintics be seen in the geometry of the icosahedron?", that There is not a single ...
10
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2answers
469 views

History of Tarski's problems on free groups

As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books. What is the original ...
66
votes
25answers
8k views

Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...
6
votes
1answer
376 views

Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?
7
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2answers
750 views

Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...
16
votes
1answer
369 views

What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
8
votes
1answer
316 views

Who first proved the fundamental theorem of projective geometry?

The following theorem is often called the fundamental theorem of projective geometry: Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of ...
14
votes
2answers
786 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
55
votes
13answers
11k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
1
vote
1answer
163 views

Non-Pythagorean proof for the square root of 2 and solution to YBC7289 [closed]

My name is J. Frederic Teubner I am an independent researcher. I wish to publish a proof for the non-Pythagorean solution to the Babylonian tablet YBC7289 and am currently inquiring as to whether or ...
2
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1answer
156 views

What kind of role has Functional Analysis played in Signal Processing? [closed]

Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?
59
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7answers
9k views

Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$ is used to represent the boundary operator in topology is that its behavior is akin to a derivative. But after ...
5
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0answers
149 views

Reference to forcing with a sigma ideal $\cong$ Cohen forcing

This is a historical question: Who was the first person to notice the following? If $V \models \kappa$ is measurable and $P$ adds $\kappa$ Cohen reals, then in $V^P$, letting $\hat{I}$ to be the ...
12
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2answers
2k views

When did the term “Lie group” first appear?

Does anyone know who was the first to coin the term "Lie group"? The following thesis from 1928 suggests that the term was already in use by that time: "Systems of Two Differential Equations from the ...
36
votes
3answers
2k views

What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...