Biographies, philosophy of mathematics, mathematics education, recreational mathematics, communication of mathematics

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11
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2answers
384 views

History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
31
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13answers
5k 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 ...
9
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1answer
528 views

Did Cauchy think that uniform and pointwise convergence were equivalent?

I've heard that Cauchy thought he'd proved that pointwise and uniform convergence are equivalent. Is this a historical fact? If it is indeed true, I was wondering if anyone had a reference.
26
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6answers
1k views

Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...
37
votes
20answers
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 ...
87
votes
26answers
11k views

Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...
20
votes
0answers
427 views

History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...
11
votes
3answers
240 views

Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...
31
votes
5answers
3k views

What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe ...
27
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26answers
4k views

Mathematicians who made important contributions outside their own field? [closed]

It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand. I am looking for mathematicians who did just that. ...
32
votes
2answers
3k views

How did “normal” come to mean “perpendicular”?

How and when did the word "normal" acquire this meaning? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to ...
72
votes
28answers
7k views

Examples of theorems misapplied to non-mathematical contexts

For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments ...
75
votes
19answers
11k views

Do you read the masters?

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a principle that ...
6
votes
3answers
843 views

Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an english translation, preferrably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...
58
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6answers
7k views

How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...
10
votes
1answer
292 views

Elements of the method of forcing in some papers of N. N. Luzin

In the paper Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...
16
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8answers
2k views

Do there exist modern expositions of Klein's Icosahedron?

Reading Serre's letter to Gray , I wonder if now modern expositions of the themes in Klein's book exist. Do you know any?
25
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14answers
7k views

Great mathematicians born 1850-1920 (ET Bell's book ≲ x ≲ Fields Medalists)

When I was a teenager, I was given the book Men of Mathematics by E. T. Bell, and I rather enjoyed it. I know that this book has been criticized for various reasons and I might even agree with some ...
2
votes
1answer
247 views

Survey of the history of calculus?

Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has: Mathematics knows no minimum interval of ...
9
votes
2answers
399 views

Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...
6
votes
3answers
287 views

Meaning of historical fluxion notation

I've noticed that in 18th century books on calculus writers would say that 'the fluxion of $ax$ is $a\dot{x}$' and 'the fluxion of $x^n$ is $n x^{n-1} \dot{x}$'. What does this extra '$\dot{x}$' at ...
25
votes
3answers
2k views

Who invented diagrammatic algebra?

There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations ...
186
votes
72answers
76k 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 ...
11
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2answers
478 views

Riemann's quote cited by Lakatos: what is the context?

"If only I had the theorems! Then I should find the proofs easily enough." This quote is generally attributed to Bernhard Riemann. In particular, on page 9 in Proofs and refutations by Imre ...
13
votes
1answer
579 views

Who first dubbed them “expander graphs”?

Expander graphs ("sparse graphs that have strong connectivity properties") burst onto the mathematical scene around the millennium, but I have not been successful in tracing the origin of (a) the ...
20
votes
7answers
1k views

Was lattice theory central to mid-20th century mathematics?

Four years ago, I read a book on the history of mathematics up to 1970 or so. It was very interesting up until the end. The last few chapters, though, were on lattices. The author claimed that ...
48
votes
26answers
8k 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. ...
10
votes
3answers
2k views

Why are smooth numbers called “smooth”?

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.) Does anyone know what is the ...
35
votes
4answers
3k views

History of “without loss of generality”

"Without loss of generality" is a standard in the mathematical lexicon, and I am writing to ask if anyone knows where the expression was popularized. (The idea has been around since antiquity, I'm ...
18
votes
19answers
5k views

What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work?

Almost any mathematical concept has antecedents; it builds on, or is related to, previously known concepts. But are there concepts that owe little or nothing to previous work? The only example I know ...
33
votes
17answers
4k views

What are some deep theorems, and why are they considered deep?

All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
9
votes
3answers
1k 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 ...
7
votes
0answers
568 views

Original references for the homotopy groups pi_5 of SU(3) and pi_4 of SU(2)?

For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to correct my references to the original work on aspects of the homotopy groups pi_5 of SU(3) and pi_4 of SU(2). I'm not a ...
5
votes
1answer
226 views

When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things: Homotopy groups stabilize after taking suspensions ...
3
votes
1answer
362 views

Levi's book on Leibnizian calculus

Raphael Levi learned from Leibniz at a late stage in Leibniz's career. This might be a definite advantage for understanding Leibniz. Leibniz did not elaborate some of the philosophical principles ...
16
votes
4answers
679 views

Who first used the multiplication operator version of spectral theory

This is another history question. Hilbert phrased the spectral theorem in terms of resolutions of the identity. While this remained the form of Stone and von Neumann, they did also have the ...
6
votes
1answer
617 views

What are current trends/questions in algebraic logic?

What are current trends/questions in algebraic logic? I mean the research developed by Paul Halmos. Could anyone give some references for the overview of its history? Any overview of its application ...
28
votes
3answers
3k views

Did ancient mathematicians know Euler's characteristic for convex polyhedra?

The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid ...
53
votes
14answers
5k 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 ...
58
votes
10answers
8k views

What is the oldest open problem in mathematics?

What is the oldest open problem in mathematics? By old, I am referring to the date the problem was stated. Browsing Wikipedia list of open problems, it seems that the Goldbach conjecture (1742, every ...
8
votes
1answer
521 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 ...
35
votes
4answers
2k views

The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
7
votes
0answers
105 views

Origin of Lie Product Formula

I'm interested in where Lie wrote down the Lie Product formula (for finite matrices) (the precursor of the Trotter product formula; see http://en.wikipedia.org/wiki/Lie_product_formula). With a ...
4
votes
1answer
347 views

What was Seifert's contribution to the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van ...
7
votes
1answer
758 views

Is there a “big program” in mathematics at the moment? [closed]

I apologize in the event that you should find this question off topic. Please feel free to delete it if that is the case. Years ago, I studied undergrad mathematics at university. The understanding ...
53
votes
29answers
7k views

(Preferably rare) Audio/Video recordings of famous mathematicians?

Terence Tao's homepage has a link to a collection of quotes, and one among them was Hilbert's famous "We must know, we will know" quote. This quote also had an audio link to it. Now although I'm not ...
18
votes
1answer
703 views

Reference for Diagonalization Trick

There is a standard trick in analysis, where one chooses a subsequence, then a subsequence of that... and wants to get an eventual subsubsequence of all of them and you take the diagonal. I've always ...
24
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2answers
709 views

Klein's Protocols: A window into our mathematical past

Klein's Protocols in over 8,000 pages recording seminars organized from 1872 to 1913 by Felix Klein and given by Klein, his colleagues, students and other invited speakers, including luminaries such ...
52
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29answers
5k views

The half-life of a theorem, or Arnold's principle at work

Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that ...
7
votes
1answer
239 views

Analogy between Lagrange's Theorem and Rank-Nullity Theorem?

One can view view Lagrange's Theorem $$|G/H|=|G|/|H|$$ and the Rank-Nullity Theorem $$\dim(V/U)=\dim(V)-\dim(U)$$ as directly analogous. Does anyone know a high-level explanation of this analogy? I ...