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

Sat Apr 11 12:26:12 EDT 1998

Torkel Franzen wrote:
>
>   My comments in my last messages have only concerned the (peripheral)
> issue of a supposed conflict between the ideas of ordinary arithmetic
> and those of feasible arithmetic. In particular, I was not arguing
> that a formalization of feasible arithmetic must involve a vicious
> circle, but only responding to your suggestion that a formal system
> could in a neutral fashion represent our beliefs and intuitions. A
> formal system that represents my beliefs and intuitions must settle
> every primitive recursive equation s=t, not just those that can be
> feasibly settled.  This is not because of any use I make in
> mathematical reasoning of the decidability in the theory of such
> equations, but only because I wouldn't otherwise accept the system as
> representing my belief and intuition that every such equation has a
> determinate truth value.

It is rather difficult to me to understand what do you need to
get my point of view. I have had impression that almost
everything was done to clear it up. It is as we are speaking
sometimes in different languages or as we advocating different
ideologies or religions.  I agree that sometimes ideology is
"best set aside".  But I cannot omit at all my opinion,
especially in the case of the concept of feasible proof.

I guess that it is Peano Arithmetic or the like which must
"represent your beliefs and intuitions that every such equation
has a determinate truth value".  Of course, this is true because
PA is based on clasical logic (with the low of excluded middle,
etc.) which is particularly intended to "give" (in quotation
marks! I even would say "to give an illusion of") a
"determinate" truth value to *every* sentence, not only of the
form of s=t. It seems that you would say more: that PA allows to
deduce the truth value of any (variable free) equation s=t. As
"metamathematician" you are right, but as mathematician -- NOT!

>   >We need not "full" but only "feasible" power! For example, we
>   >even need not know that substitution of a term for all
>   >occurrences of a variable is always defined, i.e. feasible.
>
>   This may well be so, from the point of view of feasible
> arithmetic. Again, my comment was made from the point of view of
> ordinary arithmetic. There is an asymmetry here, since we both
> in fact learned about formal systems through a classical
> presentation, not through a presentation based on feasible
> formulas, feasible proofs, and so on.

Did you ever learned to write *explicitly* mathematical proofs
of infeasible length symbol-by-symbol? If you mean something
different then this looks rather as a kind of ideology.  Thus, I
would prefer to "set it aside".

Newertheless, let me continue to make the point clearer.  My
suggestion of considering formal systems with only feasibly long
formulas and proofs is indeed NEUTRAL. Moreover, this is the
ordinary everyday practice of mathematics. What you want to say
rather belongs to *a* METAmathematics. Why do you not make the
corresponding distinction which we especially need to do in the
present discussion? It is clear that children and even the
school teachers usually know (or think) not so much about
distinction between really or feasibly existing numbers and
proofs and theoretically existing ones.  But you, of course,
know it. What you call here the "classical presentation" is
nothing, but a non-critical, extremely naive TAKING THE FACT OF
MERE IGNORING OF SUCH A DISTINCTION AS A KIND OF AN APPROACH TO
ARITHMETIC. I can agree that some ARGUMENTED abstractions or
idealizations may serve as an apprach, but not an ignoring of or
non-thinking on something highly evident.  We can ignore
something until we will see that this something exists or until
it begins to play a role.

By the way, this is immediately related with my doubts that the
standard model of PA is a meaningful concept in general.

>   To sum up, I doubt that arithmetic can be learned and pursued as
> feasible arithmetic without invoking the concepts of classical
> arithmetic,

We can invoke anything, even any deviltry.  I have no problem
with this. I have no doubts of some appropriate meaningfulness
of classical arithmetic, set theory, etc. I have essentially the
same understanding of these subjects as all of us, except of
giving any ABSOLUTE value to its main concepts.  I accept any
(feasibly understood, as noted above) formal system with ANY
corresponding reasonable intuition behind it.  I am not staying
in a position that only, say, feasible mathematics (if any) has
a meaning and everything other is nonsense. I just do not
understand and do not accept some ideological points such as
"ABSOLUTELY true standard model" for PA or that, say, "CH has a
definite, ABSOLUTELY objective truth value" which we should only
try to find.  If you think that you understand this, it is your
right.  It is difficult to demand that we will not touch these
dangerous ideological points just to express the personal
opinion. But we may simply try to restrict considerably our
reaction on any such ideology and direct discussion to more
neutral and fruitful way.  We may try to extract from the
position of the opponent "the matter of fact" and to disregard,
until it is possible, his ideological points.

> and I don't think there is any merit in the idea that our
> classical understanding of arithmetic is illusory. But I don't think
> there is any great point in arguing these matters. The interest of
> feasibility lies in its positive developments, and your paper seems to
> me to be a very readable presentation of such developments. Thus
> any further comments I might make in this context will be based,
> rather, on the ideas associated with those positive developments.

Thank you very much for your positively looking evaluation of my paper.
Sorry, I cannot guarantee that I will never use the word "illusion".
But I will try. I think that some moment we will be forced to return
to this "unpleasant" subject.

possible, let us begin to discuss on the base of more concrete
things.