FOM: Re: Arbitrary Objects

[charles silver]

Harvey Friedman responded:

> 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.

    Fine argues ii) vigorously.


> >     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.

    Well, you may be right.   On the other hand, I am starting to think that
'arbitrary object' should take their place alongside of 'set,' 'function,'
'number,' and all the other concepts on Steve's list that constitute the
subject matter of Foundations of Mathematics.   It's difficult for me to
believe that an answer to what arbitrary objects are is just to say that
'arbitrary' is not an adjective.   If not an adjective, what is it?   Since
mathematicians *use* arbitrary objects constantly, it seems to me there
should be some well-developed and clearly understood account of them.
Maybe such an account would eventually explain them away, as your answer
that 'arbitrary' isn't an adjective.   But, I think, considering the
frequency with which they're alluded to in mathematical proofs, that there
should be *some* detailed and satisfying account.   I'm not satisfied with
Fine's, mostly because it's so technical, though also because some of his
positions don't seem quite right to me (which may be due to my obtuseness,
since several things he says I simply don't understand).   Also, maybe one
reason so many mathematicians are Platonists is that Platonism can be
thought to provide *one* foundational answer to what arbitrary objects are.
But, that doesn't seem right to me either.
So, I'm stuck.