# Re: Natural numbers

sowa <sowa@turing.pacss.binghamton.edu>
Date: Mon, 22 Nov 93 17:42:38 EST
From: sowa <sowa@turing.pacss.binghamton.edu>
Message-id: <9311222242.AA10112@turing.pacss.binghamton.edu>
To: cmenzel@kbssun1.tamu.edu, sowa@turing.pacss.binghamton.edu
Subject: Re: Natural numbers
Cc: fritz@rodin.wustl.edu, interlingua@ISI.EDU,
phayes%cs.uiuc.educg@cs.umn.edu

Chris,
By the term "Cantor & Co." I meant the whole crew who believed that
it was somehow "desirable" or "enlightening" to base all of mathematics
on logic and set theory. That includes Frege and Russell, among other.
Leopold Kronecker didn't live long enough to see the full implications
of Russell's attempt to reduce all of mathematics to logic, but I am
sure that he would have been one of the people to disapprove violently
of the entire project. Although I strongly believe in the desirability
of using logic as a formal definition language, I also believe that
you have to put something in (e.g. ontology) if you want to get anything
out. Russell put set theory into the pot. I believe that the integers,
as a system, are much simpler and more basic than set theory. Therefore,
it would be a better idea to base the rest of mathematics on the integers
than on sets. That is why Kronecker's approach appeals to me.
Besides the integers, you need some grouping theory for talking about
collections. That is why mereology appeals to me. It is also a very
much simpler theory (having only one operator part-of, instead of the
two operators member and subset of set theory). The integers plus
mereology give you a simpler basis for constructing all of the usable
(i.e. applicable) mathematics than the various constructions based on
set theory.
In any case, I am not going to stop anyone from using set theory,
if they like. I am not even going to try to stop them from using
the transfinite extensions of it. I just do not want to put the set
theory into the core part of the language. I would prefer to make it
one of the optional ontologies that you can take or leave, redefine
or replace, as you please.
John