[FOM] Apartness topology and (constructive) nonstandard analysis

[Hendrik Boom]

On Fri, Sep 04, 2015 at 07:38:38PM +1200, W.Taylor at math.canterbury.ac.nz wrote:
> Quoting Frank Waaldijk <fwaaldijk at gmail.com>:
> 
> >In NSA one obtains continuity of a real function f by demanding: if x ~ y
> >then f(x) ~ f(y), where x ~ y means that x-y is infinitesimally
> >small. In apartness topology, one obtains continuity of a morphism
> >f by demanding:
> >if  f(x) # f(y), then  x # y.
> >
> >This latter would seem to correspond more to physics: if f(x) is observably
> >apart from f(y), then x must be observably apart from y.
> 
> Is this really a very pervasive idea in modern physics?
> 
> Given that physics has enthusiastically adopted the twin ideas that...
> 
> (a) tiny differences in conditions may be (in principle) unobservable;
> (b) tiny differences at time t can become large differences at time u > t;
> 
> ...being quantum and chaos theory respectively, it would seem to deny it.

If f is an experimental procedure, then distinguishing f(x) and f(y) is 
how you distinguish x and y.

And if we're talking about quantum states, it's not clear that we 
can even represent experimental procedures as continuous functions.

-- hendrik