FOM: ultrafinitism; objective vs. subjective

Vladimir Sazonov sazonov at
Thu Mar 19 16:30:44 EST 1998

Dear Professor Sympson,

Thanks for your questions. They are very appropriate.

>I've been rereading some of your postings on the natural number
>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?

Yes. More precisely, the main sources for me are some "vague" 
but fascinating ideas on feasibility from a paper of 
Yesenin-Volpin "On abstraction of potential feasibility" (in 
Russian) and mathematically rigorous approach of R. Parikh 
"Feasibility in arithmetic", JSL, 1971 where he presented 
"almost" consistent formalization of feasible numbers closed 
under successor and upperbounded by the "concrete" number

	exponential tower 2^{2^{...^2}} of the height 2^1000.

Of course, this is too rough bound. What I did, was changing 
appropriately the latter approach so that the upper bound could 
be taken now more realistically as 2^1000. (Note, that there 
are some unusual features of this simple formalization.)

>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.

At least now I see that this is possible in principe.

>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?

Yes and again yes. However, it is (at least partly) subjective 
matter which way we will approach this reality, which tools we 
will use. I consider mathematical theories just as such 
(potential) tools.

Then, we are using some ideas, abstract notions like the natural
numbers.  These ideas are also subjective in the sense that they
are our creations (with the roots in reality, of course) and
often vague. What is not vague, is any concrete tool we use (a
formal system with explicitly fixed rules and axioms). Some of
the ideas considered may be rather "stable" in some respects.
Then we have an illusion of some "absolute truth" and the like.
I prefer to remember always what is an illusion (also making 
accent on objective roots of it). 

>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?

Of course I would prefer to designate feasible numbers as 
standard model. However, there is tradition to include in the 
"ordinary" standard model also infeasible numbers.

I agree that physically represented numbers are the most 
concrete numbers we know. However, this "set" is very fuzzy.  
("Therefore", as I described in my previous posting to FOM, the 
natural attempt to get from feasible numbers the ordinary 
standard model gives rise to much more fuzzy "set" of "all" 
numbers.) Are its members those numbers which may be written in 
a sheet of paper or those written in a hard disk of a computer 
(of which generation?)? How to consider a number which 
corresponds to a physical quantity which is so large that it 
cannot be represented in a computer (in unary notation)? We have 
only rather vague picture.  What is known definitely is the 
"fact" that 2^1000 (and even 2^100) is not feasible. 

> > 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?

Of course, "reality-oriented, manner". We should use our 
subjective tools in exactly this manner. Thus, there are two 
sides of the medal. Instead of saying that our tools are "true" 
it is more correct to say that they are (more or less) adequate 
appropriate or, if you want, adequate truth-operating tools. 

Best wishes,
Vladimir Sazonov
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.), 
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: sazonov at 
152140, RUSSIA			|

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