FOM: Feasibility and (in)determinateness of the standard numbers
torkel at sm.luth.se
Fri Apr 10 06:11:51 EDT 1998
Vladimir Sazonov says:
>It seems you have agreed that in the present context we should
>not disregard feasibility. Otherwise, what are we talking about?
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.
>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.
To sum up, I doubt that arithmetic can be learned and pursued as
feasible arithmetic without invoking the concepts of classical
arithmetic, 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.
More information about the FOM