All of the math and history in this post is elaborated on in my paper here.
Are there any interesting applications of PURE math to the Social Sciences or History? Scarf's application of the Brouwer fixed point theorem to Economics is one of many examples of applying (arguably) pure Mathematics to Economics. Cartwright, Harary, and others appear to have used graph theory to model social relationships; however, on closer inspection they just used the language of graph theory. In C.P. Snow's article, The Two Cultures, he speculates that there is a cultural divide between the sciences and the humanities, which may make such collaborations difficult. This points to the lack of interaction between the sciences and the humanities being a sociological problem in itself; however, we are not going to go there.
There was an application of Ramsey theory to sociology in the 1950's. In Jacob Fox's Lecture Notes in Combinatorics he tells the the following well known story:
In the 1950's, a Hungarian sociologist Sandor Szalai studied friendship relationships between children. He observed that in any group of around 20 children he was able to find four children who were mutual friends, or four children such that no two of them were friends. Before drawing any sociological conclusions, Szalai consulted three eminent mathematicians in Hungary at that time: Paul Erdos, Paul Turan, and Vera Sos. A brief discussion revealed that indeed this is a mathematical phenomenon rather than a sociological one. (Namely R(4)=18≤20.)
This is more of an anti-application since math was used to prove there was NO interesting sociology.
Recently there was a REAL application of Ramsey Theory to History, and later of History to Ramsey Theory. We summarize the results; however, the reader should look at the link above for more details.
- Sir Woodson Kneading, a scholar of pre-christian history of England noted that, from 600BC to 400BC, whenever 6 lords were in close proximity, war broke out (with one exception). Either 3,4, or 5 of them formed an alliance against the rest, or 3,4,5, or 6 hated each other and went to war. The exception: all six formed an alliance. Kneading hired a CS grad student, H.K. Donnut, to verify the data. (Note that they really used R(3)=6.)
- Kneading noted that, between 400BC and 200BC, whenever 18 lords were in close proximity, war broke out. Either 4,...,17 of them formed an alliance against the rest, or 4,...,18 hated each other and went to war. Again, Donnut verified the data (Note that they really used R(4)=18.)
- Kneading found more results of this sort. His resuls and speculations, when translated to mathematics, are Ramsey Theoretic.
- Kneading wrote a 300 page book on this topic using the data that he colleced and Donnut verified (Donnut declined to be a co-author since, in his view, Kneading did the intellectual heavy lifting).
- Alma Rho Grand, a combinatorist, saw Kneading's book and realized that Ramsey Theory would simplify the work tremendously. Grand and Kneading wrote an article of which Kneading said My paper with Alma says cleanly in 30 pages what I said clumsily in 300 pages.
- Grand noticed that one of Kneading's examples had 48 lords in proximity but no war broke out. This was in an era where if 5 formed an alliance or if 5 hated each other then a war should happen. She verified that this configuration showed R(5)≥49. It is already known that R(5) ≤ 49. Hence she showed R(5)=49. (Note that R(5) was unknown before this time.)
This is a case where Ramsey Theory helped History and History helped Ramsey Theory. Hopefully there will be more.