FOM: Arbitrary Objects

[david ballard]

     Perhaps the intuitionists have the most robust analysis of what a
proof of (Ax)F(x) entails.   Roughly speaking, it is a function, an
operation, a method of construction which, given a relevant x, produces
a proof that said x has F-ness.   As communicators of the proof, we have
to at least present: "given x, 
[description of x’s particular
proof]
"   But to qualify as a proof of (Ax)F(x) we also have to be
convincing that the proof method applies to all relevant x.   We nudge
this impression by changing "given x
" to "given any x
" or "let x be
any
".    Still we might worry.   The audience might suspect the proof
method only applies to certain x’s, to those with strings attached.   So
we get emphatic: we say "given an arbitrary x
", "let x be an
arbitrary
"
     Thus, in my reading, our use in mathematics of "an arbitrary
object" is a shrill form of of our use of "any object" which is a
culturally necessitated feature in the selling of any - excuse me -
arbitrary proof of (Ax)F(x).
     On the other hand, significant mathematics has come mere notational
convenience or grammatical manner-of speaking.   I think of zero.   So
maybe there is some good mathematics in a theory of "arbitrary
objects".   Or maybe not.

David Ballard
Sonoma State University, California