FOM: Arbitrary Objects

[Franklin Vera Pacheco]

>  Start out with a standard
> mathematical statement like: "Let x be arbitrary."   What is x?  

>     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?
> 
> Charlie Silver
> 

 I think that you can think in arbitrary objects as a "Rule":

   Let's translate some mathematical sentences...

" Let F:N--->R be an arbitrary biyection from N (natural numbers) to R 
(real numbers),then you can make a list with all the real numbers...." 
                    Cantor.

Translation:

 " Are given the set  F(N,R)  of all biyections from N to R , an element 
of F(N,R)^F(N,R) i.e. an enumeration of all the elements of 
F(N,R), and a "Rule" that gives in each step the elements of F(N,R) in the 
order of the enumeration." Then the rest of the proof follows with that 
element given by the "Rule".

 Then any proof using arbitrary objects can be taken as a pair of 
proof-Rule or a secuence of proof. Each time you give an element of the 
set from where you take the "arbitrary objects" there will be a proof 
without arbitrary objects with the element that you have gived.

-- 
Franklin Vera Pacheco
45 #10029 e/100 y 104
Marianao, C Habana,
Cuba.
e-mail:franklin at ghost.matcom.uh.cu
tel:2606043