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 constructivism. 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