FOM: Feasibility and (in)determinateness of the standard numbers

Wed Apr 8 15:02:45 EDT 1998

```Torkel Franzen wrote:

> Rather, as I explained before, I don't find any
> substance in the idea that we are in the grip of an illusion in our
> thinking about the natural numbers. This idea, to my mind, is best set
> aside in considering the interest and value of the investigation of
> feasibility.

Yes, I essentially agree. It is better if this issue will arise
when the need appears. (But for me it appears permanently.)

> The applications of the rule are indeed mostly imaginary, but what
> is indeterminate about the totality of objects obtainable (as one
> says, in principle) by the rule?

It is this unqualified "in principle" which is unclear. (In
which "principle"?) Now we are concerned with *physical* or
*real* feasibility and with corresponding version of "in
principle".

Let, e.g., we got with the help of a coin some binary string of
the length 1000 which satisfies a given property (say,
propositional formula) and then suddenly lost it. Can we assert
that this string feasibly exists "in principle"?  Can we recover
it again? Can we be sure that the lost string was indeed correct
and that there was no error? Even if we are sure that there was
no error, can we assert that any really recovered correct string
coincides with the lost one?

I think that a reasonable "feasible" version of "in principle"
should or may be rather read as follows (or the like):  "we have
a sufficiently clear idea how to feasibly create a binary
string".  Then it seems evident that corresponding "set" of
binary strings even of the fixed feasible length 1000 is indeed
very indeterminate. I am also in strong doubts that the "coin"
will help us to get an idea of completely determinate collection
of "all" strings. We rather have many different possible version
of "all".

I conclude: we realize very well whether something, when it is
given, is indeed a binary string of a given length. Nevertheless
the set of "all" these strings is indeterminate.  If we
formalize this indeterminate collection by some formal axiomatic
system (as strong as PA or as weak as a feasibility theory), an
impression (I would say -- illusion) may arise that it became
determinate because corresponding system (or "rules of a game")
is completely formal and fixed or because this formal system
seems to give us no formal occasion for doubts.  I believe that
it is more fruitful to consider this impression not very (or not
absolutely) seriously, even despite any strong temptation. If we
will try to recall or reconstruct how a mathematical notion
arose we surely will find many of "indeterminate".

>   >It seems that our beliefs and non-beliefs related with "standard
>   >model" lead us into a vicious circle. A reasonable "neutral" and
>   >fruitful solution in general would consist in representation of
>   >any beliefs and intuitions by a formal system.
>
>   Why do you think so? I don't think formal systems are one bit clearer
> or more determinate than the natural numbers. If you insist on
> introducing and using "feasible formal systems", that is hardly a
> neutral procedure.

As I have replied to another participant of FOM, each concrete
(i.e. feasible) formal proof in a concrete formal system is
rather clear (fixed, concrete, unambiguous). It is by this
reason that formal systems serve as "neutral" relative to any
philosophy or idea.  (There is nothing new here.) The set of
"all" proofs, theorems and well-formed formulas of the language
of a formal system is, of course, very indeterminate from the
feasible point of view.

>   >For example, the limit and
>   >continuity concepts were rather unclear before formalization.
>   >Also, in the framework of formal system ZFC G"odel and Cohen
>   >*actually* have demonstrated in some concrete terms why and how
>   >2^N is indeterminate.
>
>   We can introduce formal systems and their associated concepts
> without relying on limits, continuity, infinitary set theory.

Yes. But I actually said somewhat different: that we usually
introduce a formal system to *formalize* some unclear intuition
(limits, continuity, infinity, feasibility, etc.)

> We do rely on our ordinary understanding of recursively generated
> structures like the natural numbers.

But we hardly need the full power of PA to understand what is a
formal language, a formal proof rule, how to use it, etc.