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Commentary on Probabilistic Geometry

  • B. Schweizer
Chapter

Abstract

In our thirty-plus year association with Karl Menger, neither Abe Sklar nor I ever thought of asking him how, in 1942, he came to the idea of a “statistical metric”, i.e., of replacing the classical numerical-valued distance d(p, q) between two points p and q by a probability distribution function Fpq whose value Fpq(x) is to be interpreted as the probability that the “distance” between p and q is less than x. It could have been stimulated by his Rice Institute lecture “Topology without Points” [40] or, more likely, by the lectures on “The Principles of Statistical Inference” which his close friend and former student, Abraham Wald, gave at the University of Notre Dame in 1941 [78]. Or it could have just come to him; after all, he was a man who was constantly getting new ideas.

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