Re: "Kantor dust"

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Steve Stevenson wrote in part:

>Toby Bartels wrote:

>>Floating-point reals have terrible theoretical properties;
>>they're not even a ring (not even classically).
>>This is why even after all of Kahan's good work on algorithms,
>>rounding errors are unavoidable (the "Table-Maker's Dilemma").

>Being left-handed and old, I'll propose in my dotage that we may be
>asking the wrong question. In a rewording, what constructive real
>numbers are there for the purpose of

>1. Being a model of an axiomatic characterization of the reals.
>2  Being usable in supercomputing to compute values needed for modeling
>and simulation.

I would distinguish two slightly different purposes:
1. Being usable in principle to compute values
2. Being usable in practice to compute values needed for modeling, etc.
And I'd say that constructive mathematics is inherently about (1),
although often (and like even classical mathematics, usually best when)
with an eye towards (2).  But (2) itself is something different
(applied mathematics, to give it a name; numerical analysis straddles these.)

Although I've redefined them, I think that this remains true:

>Number 1 requires that we have nice theoretical properties. Number 2
>requires something that is bounded only the dollars and life span. Those
>interested in either purpose have (presumedly) a solution for
>themselves.

Interval arithmetic, despite being more complicated than arithemetic
with either floating-point reals or Dedekind/Cauchy/whatever reals,
is an interesting subject that promises to satisfy both (1)&(2).
This is good for both: good for (1) on the general grounds
that applied mathematics usually leads to good pure mathematics
(especially, but not only, when that mathematics is constructive);
and good for (2) since you'll have theorems that you can be sure of.

>I'm willing to
>live with a demonstrably correct approximation given that we are in an
>uncertain world.

Right, and interval arithmetic promises to get us such approximations.
There's still the question of whether demonstrably correct ones
are actually calculable in practice; that depends on the application.
At some point, you have to go beyond even interval arithmetic
and start dealing with probability distributions as your values,
which is yet more complicated theoretically but matches yet more closely
what one actually has in applications.

I'm not sure how much this has to do with category theory anymore,
but as interval arithmetic is already stretching my expertise,
I don't think that I have much more to say anyway.


--Toby