Jakob Bernoulli wrote in his treatise Ars Conjectandi (published posthumously in 1713) that his aim was to correct “our most frequent error” (counting things incorrectly) and to offer an art “most useful”, because it “teaches how to enumerate all possible ways in which several things can be combined, transposed, or joined with another”. This treatise stands—in my mind—for the unity inherent in mathematics, for it deals with the simplest of mathematical questions and by pursuing these questions one is led to major foundational ideas for many of our modern scientific interests.
The unity of modern mathematics, then, can already be seen in its origins, but also in what developed from those origins. The mathematician I.R. Shafarevich wrote:
Viewed superficially,mathematics is the result of centuries of effort by many thousands of largely unconnected individuals scattered across continents, centuries and millennia. However the internal logic of its development much more resembles the work of a single intellect developing its thought in a continuous and systematic way, and only using as a means a multiplicity of human individualities—much as in an orchestra playing a symphony written by some composer the theme moves from one instrument to another, so that as soon as one performer is forced to cut short his part, it is taken up by another player, who continues it with due attention to the score.
I aim to start with one elementary and beautiful theme in Bernoulli's treatise and show how it connects to broad developments in modern mathematics.