X-Message-Number: 10276
Date: Wed, 19 Aug 1998 08:24:50 -0400
From: Thomas Donaldson <>
Subject: CryoNet #10256 - #10266

To Perry Metzgar:

Not all civilizations used formal systems as we do. Not only that, but
constructivists in math simply do not accept the existence of entities
which have no construction... and it HAS been possible to create a 
mathematics on this basis. In terms of finding a mathematics with some
relation to the rest of our existence, there's lots of merit in 

Yes, if we insist on constructivism a lot of mathematics with either
disappear (or perhaps more interesting to the mathematician!) provide
us with new and interesting problems: OK, so we must construct all 
entities we use in our proofs. Just how far can we take this? At 
present there is a theory of metric spaces developed purely on 
constructivist lines. General topology (I shall assume you know some
math) does not yet have any constructivist version. For that matter,
topological vector spaces which are not normed spaces don't have
any constructivist version --- and topological vector spaces are
basic to the idea of distributions (delta functions and all the rest,
to physicists). This does not mean that they CAN'T have a constructivist
version, it means that constructivists are working on that problem.

If you like formal systems which are NOT constructive, then no law of
God or Man prevents you from thinking about them. And yes, a lot of 
beautiful math can be developed with those assumptions. If you like it,
then all a constructivist would ask is that you accept that it is an
art form rather than science. 

As a mathematician by training, I've often asked myself what will happen
to mathematics in the future. I do not see any special reason why
formal systems will necessarily survive (other than possibly as a form
of exposition for mathematics). Math was done before anyone tried to
formalize it, even plane geometry, and it will continue to be done
if formal systems are forgotten. For that matter, the entire push for
formal systems belongs to the 20th Century, and did not exist in the
19th --- though no one would claim that mathematics did not exist then.

To Joe Strout:

Very interesting. I too would agree that we will eventually be able to
make devices capable of emulating human beings. I balk at the idea of 
calling them computers for several reasons; you may call them what you
wish, because I note that you are allowing these devices to have several
features which (at least present) computers do not have.

I will add, though, that I think the case that at least our brains can
be repaired (even with vitrification or other research lines now being
pursued by cryonicists) is much stronger than you seem to believe. Did
you ever read my ANALOG article of several years ago?

			Best wishes and long long life to all,

					Thomas Donaldson

Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=10276