FOM: Objectivity of logical/mathematical truth?
Torkel Franzen
torkel at sm.luth.se
Fri Dec 19 04:18:04 EST 1997
Solomon Feferman says:
> Perhaps this pushes the question back to a wider and more
>puzzling question, but if one takes the possibility of such more or less
>objective communication as a given, then the question rather becomes: what
>is it about the conceptual and inferential structure of mathematics that
>makes it such a distinctive and supremely objective part of human
>objective communication?
The perspective sketched above doesn't in any obvious way cover the
question of the objectivity of mathematical *ignorance*. To use the
specific example I introduced earlier, when it is assumed that a
theory is consistent, it is usually assumed to be consistent "as a
matter of mathematical fact", not necessarily as a matter of what can
be communicated, proved, or agreed upon. Such an interpretation of the
assumption is presupposed in the very question
Assuming ZFC to be consistent, is there necessarily any way in which
it can be proved to be consistent?
Intuitionistically, such a question can't be posed, since assuming
ZFC to be consistent *means* assuming that there is a canonical proof
(as intuitionistically conceived) of the consistency of ZFC. But this
intuitionistic interpretation does not correspond to how people in
fact talk and reason about consistency.
It would appear that if we are to resolutely stick to a view that
mathematics is "socially constructed", there cannot be any other truth
to a mathematical assumption than that which is agreed upon in a
particular society. In particular, if we were to agree that ZFC is
(perhaps "clearly") consistent, it would make no sense to speak of
the possibility of our being mistaken.
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