FOM: ultrafinitism; objective vs. subjective

Stephen G Simpson simpson at math.psu.edu
Wed Mar 18 22:08:19 EST 1998

```Dear Professor Sazonov,

sequence, etc.  You seem to be advocating a form of what used to be
called ultrafinitism.  This theory was propounded by Yesenin-Volpin
and others.  There is at least one interesting article about it by
Robin Gandy.  As I understand it, the basic idea is that the natural
number sequence could or should be limited to those numbers which
physically exist in some sense, so there is a bound of something like
2^(2^10).  Am I correct in thinking that you stand in this tradition?

I find the basic idea behind ultrafinitism philosophically appealing,
because I'm interested in objective or reality-oriented mathematics.
On the other hand, I see difficulties in working out the mathematical
details in a way that would suffice for our standard mathematical
applications (building bridges, as it has been called here).  Let's
not get into those difficult issues now.

In the meantime, I want to understand your ideas better.  I'm puzzled
by one aspect which on the surface seems like a contradiction.
Namely, while you call for a theory of the natural numbers based on
physical reality, at the same time you invoke subjectivism and
indeterminateness of the natural number sequence.  My question is,
isn't physical reality objective rather than subjective?  And isn't
physical reality determinate rather than indeterminate?

For example, in your posting of 13 Mar 1998 22:23:42 you said

> By the way, can anybody here explain what is this fully
> "determinate" "standard model"?

Under your theory, wouldn't this be just the set of natural numbers
which actually exist in physical reality, bounded by 2^(2^10)?  Isn't
that model fully determinate, and doesn't it deserve to be designated
as the standard model?

> I believe that ignoring the subjective character of essential part
> of mathematical activity is also nonproductive. ...  a subjective
> activity can be purposeful successful in achieving the proper
> relation to experimental objective truth in the material world (the
> applied role and "unreasonable effectiveness" of mathematics).

I don't follow this.  If we are to have a "proper relation to
experimental objective truth in the material world", don't we need to
train our minds to operate in a ruthlessly objective,
i.e. reality-oriented, manner, rather than subjectively?

I'd appreciate any clarification that you can offer.

Best regards,
-- Steve Simpson

```