# 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 Kantor's book "Naming Infinity". What are some other interesting and well-sourced examples of when the work of pure mathematicians has been influenced by their cultural/social context?

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At least one review does not seem so sure of the authors' thesis: cscs.umich.edu/~crshalizi/reviews/naming-infinity –  Yemon Choi Jul 1 '12 at 9:17
Lastly for now: an obvious answer to the question in the title is "always". The more telling but harder question is: "in what ways has it been influenced?" (Even the framing of the question in terms of "pure" mathematics is deeply anachronistic.) –  Yemon Choi Jul 1 '12 at 9:21
Regarding the final question on matchings: I am pretty sure that any relation to 'marriage' is only an a posteriori expressive description of some problems so that there should be no relation whatsoever. –  quid Jul 1 '12 at 10:40
The definition of "research" level questions in mathematics seems to be very vague --- it seems pretty much anything is fair game on MO these days --- ugh! –  Suvrit Jul 2 '12 at 8:06
Meta thread tea.mathoverflow.net/discussion/1401/… for any further discussion of whether the question is appropriate, cool, etc etc –  Yemon Choi Jul 3 '12 at 23:29

MR1648209 (99h:01029)
Dauben, Joseph W.
Marx, Mao and mathematics: the politics of infinitesimals.
Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math. 1998, Extra Vol. III, 799–809.

Throughout the Cultural Revolution, Mao Ze-dong promoted Marxism and dialectics to encourage reforms in all fields of endeavor, including the sciences. In mathematics, this encouraged, as it had Marx, an appreciation (with criticism) of the infinitesimal calculus. For Chinese mathematicians, application of Abraham Robinson's newly created nonstandard analysis not only rehabilitated infinitesimals in a technical sense, but (when understood within an appropriate materialist framework) could be used to justify and promote two new fields of study in China---model theory and nonstandard analysis.

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That's great (makes me almost regret I voted to close the question). Now, that you mention it, I think in some form this came up before on MO. Maybe due to some 'curious' claims of Marx or a correspondent of his, related to Calculus. –  quid Jul 1 '12 at 17:02
For more on Mao and math, see formandformalism.blogspot.de/2011/09/… –  John Stillwell Jul 2 '12 at 5:35
Wow, ‘When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing.’ writes Lawvere on the first page of Quantifiers and Sheaves. –  Toby Bartels Jul 4 '12 at 19:59
I first learnt about Mao and infinitesimals through Adrian Mathias who also wrote this (somewhat polemical) piece on logic and Marxism: dpmms.cam.ac.uk/~ardm/stalin.pdf –  Dan Piponi Jul 5 '12 at 21:39

Perhaps a case could be made for advances in cryptography in the last half-century as driven by the social context of increased need for information security. I have in mind zero-knowledge proofs, and, say, the PCP theorem (PCP=probabilistic checkable proof).

Of course one could reject this as not the work of "pure mathematicians." Neal Koblitz might serve as a counterexample?

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I'd make this more specific: it's not just the increased need for security, but differences in relevant scenarios. For example, public key crypto is well adapted to point-to-point communication among strangers, which is crucial for things like e-commerce. By contrast, private key crypto is fine for hierarchical communication in a fixed organization, which covers many military/government/corporate applications. This is one reason why public key crypto didn't arise until the 70's. Regardless of whether you consider it pure math, it had a huge impact on areas like computational number theory. –  Henry Cohn Jul 1 '12 at 15:28
This is a good example. Going a little bit further back in the history: the work of Alan Turing and his team in breaking the German cipher was certainly heavily influenced by the society around him. The work ultimately lead to the construction of computers (in a way). Sadly, social pressure on Turing took a turn for the worse after the war. –  Jyrki Lahtonen Jul 1 '12 at 17:55

Certainly probability falls under this scope. Cardano's secretive work on the subject was done to help him in his professional gambling career. Soon after that, Pascal and Fermat worked out the notion of expected values to solve the Chevalier de Méré's gambling question. See Anders Hald's "A history of probability and statistics and their applications before 1750."

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Speaking of Cardano: I've read a somewhat melodramatic history of the solution of the cubic and quartic that places it in by a culture of public mathematical competition, which made a new solution method comparable to a prepared opening variation in a chess or checkers match. I don't know whether this is still the accepted history. (It's also a nice story that this discovery was in turn an important impetus for the Renaissance, by going way beyond the ancient Greeks, whose work had been the ultimate scientific authority; but I don't know how reliable that story is either.) –  Noam D. Elkies Jul 2 '12 at 2:29
I've heard something similar about del Ferro: that he kept his solution of the cubic a secret so that he could use it to win mathematical contests. I don't have a reference for that though. Wikipedia seems to confirm it, but they don't make it sound like gambling, more like an academic duel - loser leaves town forever. –  Zack Wolske Jul 2 '12 at 2:41
Yes, I mentioned this as an apropos to Cardano, not to gambling: he published the solution, not the discoverer who kept it to himself as a secret weapon, teaching it to Cardano but on condition that it not be revealed. The existence of such "academic duels" would be (in the original question's words) a "cultural/social context" for mathematical research quite different from what many (most? all?) of us are used to. –  Noam D. Elkies Jul 2 '12 at 5:07
From del Ferro's Wikipedia entry: "The loser in a challenge often lost funding or his university position." Makes the grant application process seem easy. –  Zack Wolske Jul 2 '12 at 5:42

Maybe an example can be the period of enlightenment. During the enlightenment period mathematics advanced a lot in part thanks to the new division of higher education in Paris. During this period many great minds like the three L`s (Lagrange, Laplace and Legendre) came up. As well as others such as Cauchy. This time period was also favored by the creation of the encyclopedia which contained many mathematical terms thanks to the work of d'Alembert.

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I was at first temporarily confused what you mean by 'ilustration period'; enlightenment period or Age of Enlightenment are I think better in English, to express Ilustración (in the meaning of the epoque). –  quid Jul 1 '12 at 14:55

I think there's two related issues here. I've heard some sociological theories suggest that the answers we get in math are socially influenced. This is clearly mistaken, because math is a purely logical discipline: once you pick axioms (and a system of logic, to be pedantic), your answers are fixed. Similarly in the physical sciences, once you pick your experiment, you have no control over the outcome.
The exception to this of course is when people disregard the strict rules of logic. Example: philosophers used to believe that Euclidean geometry was somehow 'automatic' (see e.g. Kant). There was a mathematician whose name escapes me who came extremely close to developing non-Euclidean geometry, but, after proving a number of theorems about it, concluded that there was a contradiction because the system was 'absurd' or something like that. But this is an example of somebody doing math 'wrongly.' Math done 'right' has consequences that are not socially influenced.

On the other hand, there's the question of what we choose to study. That is obviously going to be influenced by social and personal factors. For example I recall some stuff about how the ancient Greeks liked to think geometrically, whereas the ancient Chinese liked to think algebraically (and I think the ancient Arabic mathematicians as well). You can see this in the types of discoveries that they made. There are studies out there on what sorts of differences made this happen; I think there's even someone who conjectures that Westerners still tend to think more geometrically, and Easterners more algebraically, or something like that.

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The conjectures in your last para have set off my Anachronism Klaxon... –  Yemon Choi Jul 1 '12 at 20:38
Furthermore, I think your assessment in the first para is a bit too confident. It is true that mathematics can proceed logically from assumptions to conclusions, and that this process is not socially influenced in the sense claimed. On the other hand, social context (within the mathematical or professional community) can affect what goes into the assumptions and what significance is attached to the conclusions. See for instance the evolution of Elliott's classification program for (certain) $C^*$-algebras –  Yemon Choi Jul 1 '12 at 20:43
Dear Blake, The claims of the first para. don't seem to take into account that we are not automata generating lists of all possible sylogisms in proof space and attaching equal weight to each, but rather thinking beings who choose particular narrow paths to follow out of the enormous number of possible paths in proof space. (Relatedly, in physics there can be substantial debate over the interpretation of an experiment --- see e.g. such debates in the development of quantum theory and in the possible interpretations of quantum mechanics --- and so the claim about physical sciences also ... –  Emerton Jul 3 '12 at 11:17
... seems a bit overconfident to me.) Regarding the final sentence concerning mathematicians from various countries/cultures and the directions of their mathematical strengths, since some of the strongest living geometers are Hironaka, Mori, Siu, and Yau, who were preceded by Chow, Chern, and Igusa, among others, it seems like complete nonsense (as most such generalizations tend to be). Regards, Matthew –  Emerton Jul 3 '12 at 11:21
By the way, Saccheri is the name of the mathematician that escaped you: en.wikipedia.org/wiki/Giovanni_Girolamo_Saccheri –  Todd Trimble Mar 10 '13 at 2:41

Pure mathematics has never been influenced by "social context" in any meaningful way.

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Do you want us to take your word for it, or do you have something to back this up. This is not an answer, maybe a comment. –  quid Jul 1 '12 at 14:00
Nik, am I mistaken, or does that little photo of you seem to have a little bump in your cheek, right around where your tongue might be? –  Lee Mosher Jul 1 '12 at 17:00
Serious, semi-serious, tongue-in-cheek, ironic, cynical, whatever. IMO this answer adds no relevant information, and as such should not be an answer. Please limit this type of contribution to comments. I consider such an answer essentially as spam. –  quid Jul 2 '12 at 1:56
Dear Gil Kalai, in various scientific disciplines the problem to establish the independence of some thing A of some other thing B arises. (In our case, A being 'pure mathematics' and B being 'social context'.) And, there are various ways to do this, leading perhaps not to a "proof" but at least to some significant evidence. (For instance one could evaluate how mathematics developped in different not to connected systems where a different social context was present.) Please note that I did not ask for examples or detailed explanation or even 'prove' I just asked for 'something'. Furthermore –  quid Jul 4 '12 at 17:25
it would also already help in assesing the plausibility of the point of view expressed to have some information what OP considers as 'in a meaningful way'. In some sense I agree with you that the POV is reasonable, yet mainly because all this is so vague that any 'naked' point of view is reasonable, after all we have Yemon Choi's oppossing point of view that this happens always. (Which in contrast to this one, while being only a comment, indeed was accompanied by some explanation how one should understand this claim.) So that it really comes down to what 'in a meaningful way' means. –  quid Jul 4 '12 at 17:34