[FOM] The empty set

[Stephen Pollard]

In an earlier post, I offered (without endorsement) an argument for  
the conceptual truth of "something has no members." Profs. Heck and  
McKay pointed out, quite reasonably, that the argument takes for  
granted that the class abstract "{x: not x=x}" refers to something.  
Is this assumption warranted? Bear with me while I briefly digress.

Let alpha be a complex cube root of 1. Use alpha to form a ring of  
cyclotomic integers. Following Kummer, we introduce gcd(3,alpha-4)  
(the greatest common divisor of 3 and alpha-4) by means of the  
following stipulation: gcd(3,alpha-4) divides a cyclotomic integer f 
(alpha) if and only if 3 divides f(4). We now freely use "gcd 
(3,alpha-4)" and other such definite descriptions to reach  
conclusions about our cyclotomic integers.

Without actually condemning this procedure, Dedekind observed that it  
could "easily evoke mistrust." We just assume that "gcd(3,alpha-4)"  
refers to something, but we give no indication of what this thing is  
or even could be. Dedekind's solution: let gcd(3,alpha-4) be the set  
of cyclotomic integers f(alpha) such that 3 divides f(4).

Some people (Hermann Weyl and Harold Edwards, for example) think that  
Dedekind solved a non-existent problem. They think our use of "gcd 
(3,alpha-4)" was already fully justified mathematically even though  
we at no point fixed a referent for it or even argued that such a  
referent is fixable. If this is correct, we would seem, without any  
help from Dedekind, to have been justified in asserting the  
conceptual truth of certain propositions about gcd(3,alpha-4): for  
example, gcd(3,alpha-4) divides 3.

What is required to justify mathematically the introduction of a  
class of definite descriptions? I consider this a deep and unsettled  
question in the philosophy of mathematics (even though the triumph of  
Dedekind's set theoretic outlook has made it a non-issue in most  
mathematical circles). My conclusion: it is not clear that the  
argument about the empty set is fallacious.

Stephen Pollard
Professor of Philosophy
Division of Social Science
Truman State University
spollard at truman.edu