Gian-Carlo Rota

"The spectacular results in the fluctuation theory of sums of independent random variables, obtained in the last 15 years by , , , , , , and others, have gradually led to the realization that the nature of the problem, as well as that of the methods of solution, is algebraic and combinatorial. After Baxter showed that the crux of the problem lay in simplifying a certain operator identity, several algebraic proofs (, , Wendel) followed. It is the present purpose to carry this algebraization to the limit: the result we present amounts to a solution of the for s. The solution is not presented as an algorithm, but by showing that every identity in a Baxter algebra is effectively equivalent to an identity of symmetric functions independent of the number of variables. Remarkably, the identities used so far in the combinatorics of fluctuation theory "translate" by the present method into classical identities of easy verification. The present method is nevertheless also useful for guessing and proving new combinatorial identities: by way of example, it will be shown in the second part of this note how it leads to a generalization of the Bohnenblust-Spitzer formula for the action of arbitrary ."

"It has been observed that whereas s and s are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are at variance with those of artistic beauty. For example, courses in “art appreciation” are fairly common; it is however unthinkable to find any “mathematical beauty appreciation” courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term “beauty” as it is currently used by mathematicians."

"It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in s at some time in their lives: Cantor, , Gödel, Peano, and are some. was one of the saner among them, though in some ways his behavior must be classified as strange, even by mathematicians' standards. He looked like a cross between a and a large owl. He spoke softly in complete paragraphs which seemed to have been read out of a book, evenly and slowly enunciated, as by a . When interrupted, he would pause for an uncomfortably long period to recover the thread of the argument. He never made casual remarks: they did not belong in the baggage of ."

"The more experimental scientists and s are, the more common sense they have, and so on until you get to the mathematicians, who are totally devoid of common sense."