FOM: Arbitrary Objects

[Kanovei]

From: "charles silver" <silver_1 at mindspring.com>
Date: Fri, 25 Jan 2002 15:39:47 -0600
Subject: FOM: Arbitrary Objects

In the initial letter on this topic Silver considers the 
following scheme of arguments: 

IF 1) Ex F(x) and 2) Ax (F(x)--> C)
THEN C

(assuming that C does not contain x). 

Part 2) begins, normally, with the phrase like 
"suppose that x is an arbitrary object with F(x), then ... 
(and towards C)". 

The question what is *arbitrary object* in this argument 
is, to me, quite similar to the question what is dx 
in \int f(x)dx, 
and the most meaningful answer to the both is: a figure 
of speech. 
Despite of this, the question can be quasi-answered by 
reduction to other figures of speech, sometimes 
(and individually) more appropriate. 

The best such a reduction is the following. 

Let X, Y be two mathematicians. 

X: I know that  Ax (F(x)--> C)

Y: I know that  Ex F(x) 

X: then I claim that C

Y: why ? 

X: give me any x with F(x) and I'll show you 

At this moment, either Y submits x as required with which 
X maintains the argument F(x)--> C 
or Y refuses and quits, then X wins by default. 

Thus, the trick is to charge the opponent with all burden 
of thinking about what is an arbitrary object. 

V.Kanovei