[FOM] Alain Connes' approach to Analysis

José Manuel Rodriguez Caballero josephcmac at gmail.com
Wed Aug 29 13:43:45 EDT 2018

> Tim wrote:
> I know very little about noncommutative geometry, but my superficial
> impression is that this is another case of "f.o.X." where the main point
> has very little to do with f.o.m., or at least with f.o.m. as f.o.m. is
> commonly perceived.

Connes is doing new f.o.m., inspired from noncommutative geometry, at least
in the following two points:

(A) Unification between continuous variable and discrete variable as
self-adjoint operators on the (unique) separable Hilbert space.

(B) Definition of (noncommutative) infinitesimals as compact self-adjoint
operators on the separable Hilbert space (a particular case of a variable).

Reference: A VIEW OF MATHEMATICS (Alain Connes)
Paper: http://www.alainconnes.org/docs/maths.pdf
Video: https://www.youtube.com/watch?v=t_4hRuNvDmU

Connes' argument for accepting (A) in mathematics rather than the
traditional approach is as follows:

A real variable is a map f: X --> R (from a set X to the real line). This
> definition does not allow compatibility between discrete variables and
> continuous variables. Indeed, if the variable is discrete, then X will be
> countable and if the variable is continuous, the set X will be uncountable.
> So, they cannot coexist.

Kind Regards,
Jose M.
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