Thus V. I. Arnol'd, author of Mathematical Methods of Classical Mechanics, Ergodic Problems of Classical Mechanics, Topological Methods in Hydrodynamics, etc., co-discoverer of the KAM theorem, etc., in his little book on Catastrophe Theory (pp. 109--110 of the 3rd English edition):
Unfortunately, the unsophisticated texts of Poincaré are difficult for mathematicians raised on set theory. Poincaré would have said: "The line divides the plane into two half-planes," where modern mathematicians write simply: "The set of equivalence classes of the complement R2\R1 of the line R1 in the plane R2 defined by the following equivalence relation: two points A, B \in R2\R1 are considered to be equivalent if the line segment AB connecting them does not intersect the line R1, consists of two elements" (I am quoting by memory from a schoolbook).
The reason I am not a mathematician is not that I can't understand the second version, it's that I wouldn't automatically translate it into the first.
Posted by crshalizi at August 27, 2004 06:59 | permanent link