Subject: categories: Re: One-stop version of my postings

From: Vaughan Pratt <[email-masked]>
Date: Thu, 23 Mar 2006 20:31:47 -0800

Robert J. MacG. Dawson wrote:

> ...fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his
> axioms assert that any pair of circles whatsoever do so.)

In axiomatic mathematics, everything that is not forbidden is permitted.
Circles can climb trees and drape themselves over branches in Dali's
axiomatization of geometry, not because he says they can but because he
is not the strict disciplinarian that Euclid is.  Euclid insists that
his circles shape up or else.  This creates many problems for the
circles but few problems for mathematicians as their managers.  Dali
runs a looser ship, which lets the circles lead a less structured life
while creating more problems for mathematicians.  This is a win-win
situation: the circles end up with fewer neuroses while the
mathematicians thrive, problems being their lifeblood.  Thank god for
Lobachevsky and the others we can't remember because Tom Lehrer didn't.

Vaughan Pratt

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From: "Robert J. MacG. Dawson" <[email-masked]>
Date: Mon, 27 Mar 2006 09:56:58 -0400

> In axiomatic mathematics, everything that is not forbidden is permitted.

	Yes, in a sense... but theorems along the lines of "there exists a
model of X such that Y" were a long, long way in the future <grin>

	-Robert