FOM: Arbitrary Objects

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Tue Jan 29 15:18:16 EST 2002

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