[FOM] Feasible and Utterable Numbers

[V.Sazonov@csc.liv.ac.uk]

Quoting Karlis Podnieks <Karlis.Podnieks at mii.lu.lv> Thu, 10 Aug 2006:

> I would like to call your attention to the following two utterable numbers
> mentioned in Doron Zeilberger's 57th Opinion
> (http://www.math.rutgers.edu/~zeilberg/Opinion57.html):
>

[snip]

> 1. What is the difference between using of very large numbers in physics and
> in combinatorics? Formulas produced by combinatorists - what could be their
> "physical" meaning?

If a result from combinatorics is intended to be applied in the real 
world then we should probably be interested in the "physical" meaning. 
Otherwise it is just mathematics. I do not know what else could be 
answered here.

>
> 2. Do you think that the notion of utterable numbers is not significant
> because it can be defined simply as "values of feasible size expressions" ,
> or "results of (provably halting) computations via feasible programs"?

Why not significant? Of course, some traditional complexity theoretic 
questions related with utterability can be considered.  But if you 
really want to relate this with the idea of feasible number you should 
pay especial attention to formalisation of this notion and then to 
incorporate this in some way into the concept of "utterable".


>
> Exponential expressions represent, perhaps, the simplest structure, in which
> utterable numbers "divorce" from feasible ones. Another remarkable fact:
> comparing of numerical values of these expressions is (probably) an
> absolutely intractable task. Thus, equivalence classes

equivalence classes with respect to "feasibly provable equality"?

of exponential
> expressions (by their numerical values) represent, indeed, a "fuzzy" notion.

If the term "fuzzy" (because of participating "feasible"?) will remain 
non-formalised then what will you do with all of this? What EXACTLY do 
you mean by "fuzzy equivalence class"?

For me this is the key question to all considerations related to  
feasibility. Do you intend to make a rigorous mathematics from these 
considerations or not? If yes, the HOW???


Vladimir Sazonov

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