Misha Gromov

Why geometers of the past were missing our $H^2$ [The hyperbolic plane]? Philosophically, because they trusted into the geometric intuition about the “real world” built into their (ego)minds. Technically, because the existence of $H^2$ is invisible if you start from Euclidean axioms - almost all of non-trivial mathematics is axiomatically (logically) unapproachable and invisible. (p. 49)

…making “an axiom” of everything which looks obvious is no good for building new mathematical structures, although the Euclidean rigor is indispensable for fencing off logical errors. (p. 50)

…there is a recorded instance of a mathematician who experiences abdominal pain upon coming across an unrigorous mathematical argument.(V. Milman , To-day I am 70, p ??? ???)

Probably, the full concept of “ergo-brain” can be expressed in a language similar to that of n-categories (see ???) which make an even more refined (than categories/functors) tool for dealing with “arbitrariness and ambiguity”. (p. 53)

Revised on May 28, 2010 10:39:13
by David Corfield
(86.158.59.137)