FOM: Re: Re: Arbitrary Objects

[charles silver]

> On Januari 28 Charlie Silver wrote:
>
> >    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.
>
> And he wondered:
>
> >  What
> >*are* those arbitrary objects that figure in so many mathematical proofs?
> >
Ron Rood:


> One might wonder whether in saying "let x be an integer", x is meant to be
an
> arbitrary object or an arbitray integer. It seems that in this respect
James
> R. Brown touched upon an interesting point in his FOM message from january
> 29. Contrary to Fine, Brown seems inclined to think that we should choose
> for the second option.

    Okay, just ask the question in terms of arbitrary integer.


> Upon closer inspection, however, it seems that Fine wants to think so
because
> he attempts, among other things, to understand some specific inferential
moves
> mathematicians are supposed to make in practice. For example, inferential
> moves
> like elimination of the universal quantifier, that is, an inference from
> "(Ay)(y is an F)"
> to "x is an F" . The latter sentence would then be the first-order
counterpart
> of "let
> x be an F". I take it that Fine thinks that if we understand "x" as
referring
> to an
> arbitrary object, then we can in some way illuminate or increase our
> understanding
> of some inferential practices adopted by mathematicians.

    No, Fine doesn't care so much about this particular inference at all.
It's UG and EI that grab his attention.

>Notice that the original sentence "let x be
> an F" is
> interpreted by Fine as "x is an F", where, again, "x" is taken to refer to
an
> arbitrary
> object.

    In this case, x would be an arbitrary F.



Charlie