The unity of mathematics, preserved

If you were asked to name two domains of mathematics which are as far apart as possible today, so far as to be essentially disconnected and living witnesses of the impossibility of believing in the unity of mathematics, it is quite possible that the answer “Applied statistics” and “Category theory” would spring to mind.

But, lo and behold: two wonderful papers by Guy Romier in the Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, published in 1969, give the missing link between these two estranged cousins (see here and there; thanks are again due to the Numdam project for making the archives of many French mathematics journals available freely). There, the foundations of statistics experiments are established on the firm footing of a whole battery of categories; in particular (Définition 1 in the first paper, page 278), the “experimental structure” of of a statistical problem is defined as a subcategory of one of those basic categories, satisfying two axioms, one of which involves projective limits of some kind.

(I must say that I rather wonder how such papers were received in the statistics community; of course, my knowledge of statistics is essentially inexistent, my excellent probability teacher having decided to devote as much time as possible in his course to Brownian motion, but somehow I don’t think this approach has stuck…).

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I am a professor of mathematics at ETH Zürich since 2008.