[FOM] Apartness topology and (constructive) nonstandard analysis

[W.Taylor at math.canterbury.ac.nz]

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.

Bill Taylor


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