FOM: Re: Arbitrary Objects
silver_1 at mindspring.com
Sun Jan 27 09:40:08 EST 2002
As I have received a number of replies sent to me personally about
questions I raised pertaining to arbitrary objects, I think I should try to
clarify a few things. First, forget about proving A is a subset of B.
Just imagine a proof that everything has F. Start out with a standard
mathematical statement like: "Let x be arbitrary." What is x? Kit Fine
thinks that x is what he calls a "distinctive object." For Fine, along
with ordinary numbers, there are arbitrary numbers. Along with ordinary
men, there are arbitrary men. He knows full well that such an idea has
fallen into disrepute, and defends "arbitrary objects" against such
luminaries as Frege, Church, Tarski, Lesniewski, Quine, Lewis, and others.
In fact, he thinks that the reason many others haven't joined the chorus of
opposition to them is that it is now completely agreed that arbitrary
objects do not exist. So, he proposes to argue *against* this accepted
view and defend their existence. In fact, he thinks that a theory of
arbitrary objects best explains what we do when we universally generalize
and also what we do when we reason with an instance of an existential
statement. His entire book constitutes a defense of arbitrary objects.
Another point of possible unclarity is the citation of an arbitrary
object having G after one has proved "something has G." But, it is fairly
standard to say that "y has G" and then to reason about y. In this case, y
is arbitrary too, though it ranges only over the G's.
It seems that mathematicians do not to want to scrutinize exactly what
they're doing when they say "let z be...", where, even if it's not
explicitly mentioned, z is intended to be "arbitrary". They just reason
with arbitrary objects as a matter of course. By the way, a solution Fine
rejects is the so-called "ambiguous names approach" (that seems to have
originated with Suppes, 1957). According to this (rejected) approach the
*name* 'z' in a proof is "ambiguous." Note here the switch to discussing
the *letter* 'z' rather than whatever (if anything) 'z' denotes. What
*are* those arbitrary objects that figure in so many mathematical proofs?
I said earlier:
> I have started puzzling over what an "arbitrary object" is. Suppose
> you want to prove A is a subset of B....
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