[FOM] Apartness topology and (constructive) nonstandard analysis

[Frank Waaldijk]

Bill Taylor and Sam Sanders both replied to my earlier posting, so let me
reply briefly. Bill 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.
>

Hendrik Boom replied to this, and his reply mirrors my thoughts. But Bill
may still have a point here. I certainly have no ready reply, just the same
observation that Hendrik made that  (a) and (b) together also imperil the
implication `if x ~ y' then f(x) ~ f(y)'.

Sam pointed out that MP implies the equivalence of the two implications.
Still, at first glance to me it is not immediate that MP should hold in a
finitist setting (I at least have always seen MP as an unbounded search
principle, what happens if we change our perspective on `unbounded' is
unclear to me). Anyway, it seems to me that Sam is far more knowledgeable
on NSA than I am, perhaps he also could comment on this.

In reply to Sam: clearly there are a lot of constructive developments. In
the meantime I've looked at some of your references and my posting was
perhaps a bit off the cuff in this respect. On the other hand, the
references will take me some time to digest, since I do not easily find a
tailor-made answer to some of my questions. I realize this is most likely
due to my lack of understanding, so if you allow I will ask you for some
clarification in private.

Finally, the title of your article `The unreasonable effectiveness of
Nonstandard Analysis' of course reminds me of the well-known paper `The
unreasonable effectiveness of mathematics in the natural sciences'. Let me
finish by saying that -even though my interest lies largely in constructive
mathematics- my conviction is that all mathematics is in some way about
`structures' (often beautiful structures...like ultrafilter products). If
we manage to match any such structure to phenomena in the outside world,
then this is special (in my eyes, regardless of whether the structure was
defined constructively or not).

But sometimes I also feel that the structures of constructive mathematics
get less attention than they deserve (probably a lot of mathematicians feel
that way about their field...) and so I thought it might add something to
compare IST with apartness topology.

Best wishes,
Frank
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