FOM: Franzen on "which undecidables have determinate truth value"

Torkel Franzen torkel at
Fri Feb 27 04:18:11 EST 1998

  Hartry Field says:

  >On his critique of my attempt to use empirical assumptions to ground the
  >determinacy of 'finite': I certainly agree that anyone antecedently
  >convinced that 'finite' was indeterminate wouldn't be convinced.  But I was
  >addressing my remarks to those who (like almost everybody) is strongly
  >inclined to think that 'finite' is determinate, but who at the same time are
  >puzzled how it COULD be determinate given that (a) our meanings are
  >determined by our practice and (b) our practice with 'finite' seems
  >exhausted by what we accept and (c) what we accept has nonstandard models.

   This is a valuable clarification, which could perhaps have emerged
more clearly in the paper. Such formulations as

    ...when (as envisioned here) we discover by such reasoning that the
    concepts are determinate


    ..if we accept S, then this consequence of it allows us to extend the
    determinacy in the physical vocabulary to the notion of finiteness

suggested to me that what was intended was an argument for a statement
of the form "if [cosmological assumption] then 'finitely many' is
determinate", an argument which did not presuppose that "finitely
many" is determinate. This was the mistaken interpretation that prompted
my critical remarks to the effect that any doubts (of the nature suggested)
one may have about the determinateness of "finite" will apply equally
in the argument at issue, depriving it of force. This criticism doesn't
really have anything to do with how we are inclined, but concerns the
logic of the (supposed) argument.

  Although clarifying, your comments are at the same time
puzzling. This is in part because I now don't know what to make of
statements such as the ones quoted above. To the extent that we accept
"finite" as determinate, we will indeed also - given the assumption
that "event" is determinate - accept "finite sequence of events" as
determinate, but what sort of "extending" of the determinacy of the
physical vocabulary to the determinacy of the notion of finiteness is
then involved?

   >(The last part of the paper is an attempt to show how we could
   >learn to live with the conclusion that 'finite' is radically
   >indeterminate--something I claim we need to do if the empirical
   >assumptions fail.  This is the context for the discussion of the
   >incompleteness theorem.)

  As far as I can see, only one brief paragraph in section 4 of the
paper deals with the question how we could learn to live with the
conclusion that "finite" is radically indeterminate. I didn't comment
on that paragraph, since it seemed to me no more than a dubious
aside. Instead, as you yourself note, "the main theme of this section
is that the consideration of Godel's theorem gives no decisive reason
for thinking that some undecidable sentences have determinate truth
value". Here my comment remains that your "main theme" can hardly
be disputed, but only because of the doubtful character of the idea of
"our fullest mathematical theory" and therewith of "undecidable

Torkel Franzen
Computer science, Lulea technical university

More information about the FOM mailing list