26
votes
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 ...
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
votes
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. ...
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 ...
50
votes
8answers
3k views

Have you solved problems in your sleep? [closed]

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 ...
4
votes
1answer
131 views

What is the early history of the concepts of probabilistic independence and conditional probability/expectation?

In the 1738 second edition of The Doctrine of Chances, de Moivre writes, Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards ...
1
vote
0answers
419 views

Is it possible to give a fair assessment of the influence of Bourbaki's “Eléments de mathématique”? [closed]

Well, I apologize if this "soft-question" (related to the "Arnold-Serre" debate) is considered as irrelevant for MO, and for possible misunderstandings in the two earlier versions of this post (which ...
65
votes
9answers
7k views

Analogues of P vs. NP in the history of mathematics

Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...
24
votes
1answer
5k views

Who made the famous error in calculation that 'wasted' the final years of his life?

Sorry, I am merely a Middle School maths teacher at an Australian secondary school. I remember reading years ago about a famous mathematician (18th or 19th Century?) who calculated table upon table of ...
11
votes
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 ...
5
votes
1answer
385 views

Origin of the term “weight” in representation theory

In representation theory, there are the related concepts of weights and roots. Since both are kinds of generalised eigenvalues, and eigenvalues are roots of e.g. the characteristic polynomial, the ...
18
votes
2answers
2k views

Where are Georg Cantor's Original Manuscripts?

Georg Cantor is famous for introducing transfinite numbers and set theory. A main part of his mathematical point of view about this new type of "numbers" and this new "realm of mathematics" cannot be ...
37
votes
4answers
2k views

The Arnold – Serre debate

I have read (but I cannot now find where) that Arnold & Serre had a public debate on the value of Bourbaki. Does anyone have more details, or remember or know what was said?
8
votes
1answer
2k views

What is the source of this E̶r̶d̶ő̶s̶ quote?

Namely, the following one "All problems appeared once in the [American Mathematical] Monthly." I remember reading it several years ago... When I first posed the question, I believed that I had ...
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 ...
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 ...
5
votes
1answer
250 views

Origins of Axiomatic Reasoning

Is there any evidence that axiomatic reasoning has been used prior to Thales of Milet (624-547BC), who is generally credited for the "invention" of axioms. In this context I understand axioms in the ...
6
votes
0answers
203 views

History of the characterization of commutative Artin rings

When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as ...
6
votes
7answers
902 views

famous papers/results by non professional mathematicians [duplicate]

Possible Duplicate: What recent discoveries have amateur mathematicians made? Dear overflowers Out of curiosity: do you know any famous papers and/or results by non professional ...
58
votes
24answers
6k 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 ...
9
votes
1answer
319 views

Discrete Morse theory and chess

There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. ...
28
votes
13answers
2k views

Great mathematics books by pre-modern authors

Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...
13
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 ...
15
votes
1answer
554 views

Mendelson's *Mathematical Logic* and the missing Appendix on the consistency of PA

A very soft question, but I hope not out of order here. In the first edition of Elliott Mendelson's classic Introduction to Mathematical Logic (1964) there is an appendix, giving a version of ...
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 ...
31
votes
31answers
6k views

Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...
11
votes
7answers
1k views

What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far: No-wandering-domain Theorem (Sullivan) Theorems of Thurston ...
7
votes
9answers
988 views

Examples where adding complexity made a problem simpler

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples: $S^n$ is never contractible, but $S^{\infty}$ is. The ...
3
votes
0answers
237 views

A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...
6
votes
0answers
171 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
2answers
828 views

Mathematical computer desk [closed]

D. Gibb, from the Mathematical Laboratory, University of Edinburgh, describes a Computer Desk in his book A course in interpolation and numerical integration for the mathematical laboratory, G. Bell ...
16
votes
19answers
2k views

History Question: AUTObiography of Mathematicians

According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator. An autobiography offers the author the ability to recreate ...
25
votes
65answers
7k views

Fiction books about mathematicians? [closed]

What are some fiction books about mathematicians? It seems to me rather difficult for writers to create good books on this subject. Some years ago I thought there were no such books at all. There ...
7
votes
6answers
2k views

When has pure mathematics been influenced by the social context of mathematicians?

I recently learned that the Moscow school of descriptive set theory (Egorov, Lusin, etc.) was deeply influenced by the religious movement of Name Worshiping in Russia, as recounted in Graham and ...
16
votes
1answer
845 views

Raoul Bott's quote on Morse Theory cited by Bestvina and Kahle: where is it from?

I wanted to properly cite the following awesome quote: Every mathematician has a secret weapon. Mine is Morse theory. - Raoul Bott Now this has been attributed to Bott in precisely two places ...
18
votes
3answers
2k views

Contacting an eminent mathematician

I have recently started a PhD. and am researching an area that two now eminent mathematicians devoted considerable time to in the 1980s. However, there appears to have been fairly moderate focus on ...
13
votes
0answers
825 views

Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question: Who was the first to prove the Nerve Theorem?
3
votes
1answer
503 views

What does the 'V' in 'V-manifold' stand for?

The story of how the name 'orbifold' came about is pretty well-documented, but I can't find any explanation as to why Satake originally named orbifolds 'V-manifolds'. The 'manifold' part is clear ...
15
votes
17answers
3k 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?
8
votes
1answer
752 views

Hausdorff and Naive Set Theory

Erhard Scholz, in his article "Felix Hausdorff and the Hausdorff edition" writes the following: "Hausdorff considered the contemporary attempts to secure axiomatic foundations for set theory as ...
11
votes
1answer
921 views

Quote about errors in math writing

I'm searching for the original reference of a quote that went something like: "Errors in a mathematics text add an element of surprise to an otherwise predictable plot." I believe it may have been ...
18
votes
1answer
1k views

What has happened to Lang's Files and other political texts?

For some background on Lang and his files, one can read the first part of Lang's obituary in the AMS Notices at http://www.ams.org/notices/200605/fea-lang.pdf. The book "Challenges" was published in ...
6
votes
2answers
499 views

Where is the Euler/Goldbach correspondence?

I know that there is a 1965 volume containing the Euler/Goldbach correspondence, but I'm interested in looking at the original manuscripts. I'm not finding anything at University of Basel or ...
4
votes
0answers
659 views

What did Hilbert do on Hilbert spaces to deserve his name?

This question is just curiosity. When I had my first course in Functional Analysis, most of basic theorems about Banach spaces were presented to me as attributed to Banach (Hahn-Banach, ...
19
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 ...
8
votes
1answer
462 views

What is the etymology of zero-sharp?

I have wondered for a while what gave rise to the notation $0^\sharp$. According to wikipedia this is due to Solovay in 1967, but (perhaps unsurprisingly) there's no discussion of why that notation ...
16
votes
21answers
6k views

Famous mathematicians with background in arts/humanities/law etc [closed]

I've been motivated by this question about starting to study mathematics at an unusually advanced age. It would be nice to know examples of people who successfully switched from a very different field ...
11
votes
2answers
2k views

What's tropical about tropical algebra?

Please allow me to ask a potentially dumb question (or maybe more precisely, a question floating on clouds of ignorance): Why is a max-plus algebra called a tropical algebra?
3
votes
2answers
1k views

Historical basis and mathematical significance of Riemann surfaces [closed]

It is written in Riemann Surfaces (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that: "[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination ...
5
votes
2answers
604 views

About Kodaira's book on deformations

I happened to read the following sentence in the blog by the physicist Jacques Distler: "What makes Kodaira’s Complex Manifolds and Deformation of Complex Structures such a delight to read is that ...