[FOM] Alain Connes' approach to Analysis

[Sam Sanders]

Dear José, dear FOM readers, 

Let me chime in here: 

part of the power of Nonstandard Analysis stems from the fact that any structure studied in math can be viewed directly as 
being discrete or approximated up to infinitesimals by a discrete structure (in both cases I mean: discrete involving nonstandard numbers).  
Hence, the unification of the discrete and the continuous seems to be an accomplished fact (which I discuss in more detail below).  

So why are we having this discussion?

Some people, including Alain Connes, view the aforementioned power of Nonstandard Analysis as something unholy.  To reject Nonstandard 
Analysis based on this or similar intuitions,  a lot of smart people have come up with faulty arguments that revolve around the idea that the afore-
mentioned power must somehow make Nonstandard Analysis fundamentally non-constructive, ineffective, or not-applicable in physics 
(millennia of intuitive infinitesimal calculus in math and physics notwithstanding).  

Luckily, Terrence Tao provides some counterweight: he views Nonstandard Analysis as a tool that can
do things other tools cannot (do as well).  See his blog and associated book(s). 

I shall now discuss José example from his post:

> 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.


Let us take Nelson’s syntactical approach to Nonstandard Analysis, as embodied by IST (=an extension of ZFC with axioms governing a new predicate “is standard”).


First of all, regarding the usual notion of continuity:

In IST, there is the usual “epsilon-delta”-definition of continuity, viewing the reals (or the uncountable set X you mention) in the usual way of mathematics.  

In IST, there is also the ‘nonstandard’ definition of continuity, familiar from physics (‘≈' is infinitesimal proximity, defined in terms of ‘is standard'):  

For standard x in X and any y in X, if x ≈ y then f(x) ≈ f(y).  (*)

These two definitions are equivalent for standard functions f, by the axioms of IST, but not for nonstandard ones: there are many nonstandard functions f 
satisfying (*) that have (infinitesimal) jumps in their graph.  For instance, for a nonstandard natural number N, 1/N is infinitesimal, and define f(x)=  n/N if  n/N  < x < (n+1)/N. 
Note the discrete nature of this function; in fact, one can approximate the reals (or any uncountable set) by a discrete grid with infinitesimal mesh.  


Secondly, Mikhail Katz has previously pointed out my paper “To be or not to be constructive” in this thread. This paper provides a template for obtaining computational content from theorems of Nonstandard Analysis.  
The aforementioned definitions of continuity yield different results when applying this template, say to the theorem “a uniformly continuous function is Riemann integrable” (see Section 4.5 in that paper).  

Using usual eps-delta-continuity, one obtains the expected theorem 

“a modulus of Riemann integration is computed from a modulus of uniform continuity (via a term of Goedel’s T)”

However, using nonstandard continuity, one obtains another version, involving moduli 'up to precision' (that is, the epsilon in the eps-delta-definition cannot be smaller than the precision):

“for any natural number k, a modulus of Riemann integration *up to precision 1/k*, is computed from a modulus of continuity *up to precision a number computed in k* " 

Note that the first one is the typical continuous picture, while the second one allows discontinuous/discrete functions.  

Best,

Sam