FOM: Arbitrary Objects

Vladimir Sazonov V.Sazonov at
Thu Feb 7 15:49:48 EST 2002

It was said a lot on "arbitrary objects", often very interesting. 
Of course, we could exercise in this topic a long time, playing 
with epsilon symbol or with some game theoretical explanations, 
etc. Some minor additions. 

First, why not to give the simplest and most straightforward 
explanation of what every working mathematician seems 
understand very well? Fortunately, Andrzej Derdzinski 
already gave that explanation which I would like to support. 

He wrote:

> Thus, "arbitrary" is a concept really appearing only in proofs: suppose that a
> mathematician/logician is proving a statement of the form  Ax P, where  P  is
> a fixed formula (in cases of interest, containing  x  as a free variable). In
> the course of the proof, the mathematician/logician tells himself/herself "Let
> x  be an arbitrary object" and proceeds to use that assumption to derive  P.
> What `arbitrary' means here is entirely subjective, and the intuition behind
> it may very from one person to another. 

Let me only illustrate this: Saying "let x be an arbitrary object", 
say, a triangle, or a natural number, a mathematician usually just 
draws some CONCRETE triangle (say, with the sides 3,5,7) and gives 
it a name, say, ABC. ("Press any key" ang give it a name k.) Then 
he/she presents a proof ON THIS CONCRETE TRIANGLE, but such a proof 
that actually DOES NOT DEPEND on the concrete form of this ABC. 
If we consider an arbitrary natural number, we could do the same 
thing: take some concrete one, say, 5377540941, denote it by n and 
forget which exactly was this number. Work only with its name n! 

> Note that this last property of  x  may be viewed as an "objective meaning" of
> an arbitrary object: something about which we have made no assumptions.


What I would like to add, is a comment to Charles Silver: 

>     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.  

First, I strongly believe, mathematics is a formal science always 
having, nevertheless, an informal meaning (contrary to what 
many thinkers say on formalists view to mathematics; who are 
those formalists personally? Hilbert? or some imaginary 
mathematicians?). As it is a formal science, it is SOMETIMES  
useless to ask some questions on its formalisms. They work 
well, accelerating and mechanizing our thought. Just vice versa, 
it is very useful for working mathematician to not ask what does 
it mean "let x  be an arbitrary object". It is a part of a formal 
rule (preliminary well understood, at least implicitly, by a lot 
of training). Just use it like any other mechanism. Say, when 
riding bicycle it is even dangerous to think simultaneously which 
way are you doing that. You can fall down instead of riding 
quickly (feeling intuitively how to use this wonderful 
mechanism, exactly as in mathematics)! 

Vladimir Sazonov                        V.Sazonov at 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.

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