Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!usc!orion.oac.uci.edu!jhess
From: jhess@orion.oac.uci.edu (James Hess)
Newsgroups: comp.ai.philosophy
Subject: Re: emergent properties
Message-ID: <27115C45.8738@orion.oac.uci.edu>
Date: 9 Oct 90 05:12:37 GMT
References: <1990Sep29.213139.2876@watdragon.waterloo.edu> <3499@media-lab.MEDIA.MIT.EDU>
Reply-To: jhess@orion.oac.uci.edu (James Hess)
Organization: University of California, Irvine
Lines: 17

Speaking of Penrose, who wants to re-introduce Platonic ideals through the 
limitations of axiomatic mathematics and the failure of reductionism:

Let the set of axioms be the sticks of various links in a set of tinkertoys,
and the allowable operations be defined by the holes in the wheel-shaped hub 
pieces.  As we operate on the sticks we begin to build structures.  Let these
be our theorems.  Now some structures will be in the set of all allowable 
structures, and some will not.  By experimenting with the axioms, operations, 
and theorems we will discover which are part of the set.  Can we specify in 
advance whether we will be able to join two sticks of given length at a given 
angle at some point in space with reference to our origin?  If we cannot, does
this mean there is some Platonic set of tinkertoy structures that we are 
discovering, or that this set is determined by our specification of the axioms
and operations?  If we change an axiom in an interesting way, we change the 
set of possible structures.  Some sets will be highly interesting, some won't.

Just how arbitrary is mathematics, anyway?