FOM: arbitrary objects
Eylem Ozaltun
ozaltune at boun.edu.tr
Thu Feb 14 03:59:49 EST 2002
---------- Forwarded message ----------
Date: Thu, 14 Feb 2002 08:19:10 +0000
From: eylem özaltun <ozaltune at hotmail.com>
To: ozaltune at boun.edu.tr
Subject: arbitrary objects
fom at math.psu.edu
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Name: Eylem Ozaltun
Position: T.A. in mathematics, working on my thesis for a M.A. degree in
philosophy.
Interests: philosophy of mathematics and logic, the philosophies of Kant
and Frege
I think that a Kantian perspective might throw light on Charles Silver’s
question on the arbitrary objects. Ron Rood, in his mails of January 28
and 31, emphasized the imperative nature of the statement “Let X be an F”
thus pointing at the cognitive role of this statement. I agree with him
that we are forced to cognize something in this step. I believe that a
Kantian interpretation of the notion of ecthesis (the part of the step
“Let X be an F” in a Euclidean demonstration) promises some answer to the
question that Charles Silver posed. This interpretation may enable us to
see the epistemological and cognitive function of arbitrary objects.
Hopefully it will also provide hints to the ontological status of such
objects.
I want to compare existential instantiation and ecthesis. Suppose I am to
prove a universal statement about triangles. I should start with “Let X
be an arbitrary triangle”. Here we are not referring to any concrete
object by X, but we are giving ourselves an object that is supposed to
have all and only the properties that are necessarily connected to the
concept triangle. The object that we gave to ourselves does not refer to
a triangle but stand for the concept triangle, that is, it is a machinery
that possesses the singularity of a particular object and generality of a
concept at the same time. And it represents the concept triangle without
making any claim about the existence of triangles. Here X is an objective
representation of the concept triangle. It is this representation that we
call an arbitrary object. It is in this sense that we appeal to
“arbitrary objects”: as objective representations of the concepts about
which we are to cognize something nontrivial. Surely, this
interpretation of arbitrary objects cannot be captured in the first order
logic. Still, I believe that this interpretation provides an explanation
for the cognitive function of arbitrary objects in the practice of the
working mathematician.
On the other hand, suppose I am given the existential statement that
“there exists a prime number between 10 and 15” Then in order to use this
statement I will start with “Let x be a prime number between 10 and 15”.
Here I am not referring to the concept prime number but to a particular
object that exists. Thus by X I am discursively representing an object,
namely, either 13 or 11. Here X is not an arbitrary object. X does not
stand for the concept prime number so I cannot generalize from anything
pertaining to this X to the concept prime number. I think this comparison
makes clear why we use arbitrary objects in proofs of universally
generalized statements but not for instantiations of existentially
generalized statements.
In the very first posting Charles Silver mentioned that a kind of
arbitrariness is involved even in EI. On February 4, Jay Halcomb wrote:..
“arbitrary” is elliptic for “arbitrarily chosen”. In the above
comparison “Let X be an arbitrary triangle” is not elliptic for “Let X
be arbitrarily chosen triangle” since this X does not stand for an object
in the class of triangles. But “Let X be a prime number between 10 and
15” is surely elliptic for “Choose X arbitrarily among the prime numbers
between 10 and 15” since in both expressions X refers to a particular in
the set of “prime numbers between 10 and 15”, not to a concept. Hence
“arbitrarily chosen” is elliptic for arbitrariness involved in the
applications of EI but not of the use of the term arbitrary object in
ecthesis.
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