FOM: Re: Arbitrary Objects

[Harvey Friedman]

Reply to Silver January 27, 2002.

>...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.

Perhaps it would be useful to mention an obvious kind of standard 
argument against their being a notion of "arbitrary object" or 
"arbitrary number", in the sense of "arbitrary" being a predicate on 
objects or numbers.

The obvious argument goes like this. If "arbitrary" is to be a 
predicate here, then it must hold of some particular thing. Imagine 
any one particular arbitrary object. Then it is either green or 
nongreen. If this one particular arbitrary object is green then by 
universal generalization, all objects are green. If this one 
particular object is nongreen then by universal generalization, all 
objects are nongreen. Neither is the case. So we have a contradiction.

Ways out of this contradiction would be

i) there is no way to imagine a specific arbitrary object.

ii) the rule of universal generalization that is used is faulty.

If "arbitrary" in "arbitrary object" is not meant to be a predicate, 
then "arbitrary object" becomes a figure of speech and one has 
essentially joined the standard anti arbitrary object position.

>
>     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.

Of course.

>What
>*are* those arbitrary objects that figure in so many mathematical proofs?
>
The standard line, of course, is that in your question, "arbitrary" 
is not an adjective.
--