X-Message-Number: 10313 Date: Mon, 24 Aug 1998 10:45:43 -0400 From: Thomas Donaldson <> Subject: CryoNet #10307 - #10312 HI everyone! Perhaps Metzgar and I should continue our discussion elsewhere, but I must answer his message on Cryonet, still. I will also suggest that if he wishes further discussion, I can be reached at: In any case, there are 2 separate issues. The first of them is constructivism. You see, Perry, what constructivists are saying is that if you cannot produce a constructive proof then NO PROOF EXISTS. The only thing you are doing when you find a proof by contradiction is playing with words. You are saying nothing about the world. As for being a formal system, yes, constructivists have a formal system. The second issue, perhaps the more interesting one for cryonicists, who hope to be around for centuries, is that of whether or not we will continue to use formal methods in our mathematics. Physics and physicists provide an excellent case study on this issue. If you define mathematics as requiring a formal system, then OK, physicists aren't doing mathematics. It seems to me that a more illuminating definition would not insist on formal systems for a study to be mathematical. Among other issues, if we make such an insistence, then the day may come when no one is doing mathematics. What all those physicists doing calculations, and engineers using various algorithms to design buildings etc are doing would then have to be something else. I will point out that it is quite possible to scrutinize your own reasoning, or the reasoning of someone else, without reference to any formal system. We do it all the time. For that matter, most mathematics is NOT taught as a formal system. Theorems are proved, yes, and definitions are made, but the original axioms are generally not referred to. I will venture to give a reason for that, as someone who once taught math at university level: we are trying to describe and discuss some mathematical phenomena. Just what formal system lies behind those phenomena is not really of interest; if we find that one formal system does not include what we want to discuss, then we can happily devise another. It is as if we are trying to discuss phenomena in the world. Open a textbook on vector spaces and see just how much attention is paid to formal systems. The world of thought does not collapse into incoherence if we cease to be interested in formal systems. And finally, you make an error about prime numbers and constructivism. We can prove some things about them. The Euclidean Algorithm, very old, tells us that if we have a set of prime numbers we can construct another prime number not in the set (I refer here to a constructible, finite set). We can then make a definition: a set of numbers is infinite if for any finite subset we can construct by a given explicit algorithm another number not in the set. Yes, not every infinite set your language games tell you exists will exist or be infinite by this definition. Sorry! If you wish to take up spiritualism, then no law prevents you from doing so, but we might do better to discuss REAL entities rather than engage in language games. As for Goedel's Theorem, my library will arrive here (I am told) in about 2 weeks. I will then examine Goedel's Theorem with an eye to whether or not constructivism escapes the problems it raises. It may or then again it may not --- what you say is unconvincing. It has been years since I read the theorem and I'll have to say that I've forgotten its details. If it does escape, that will be interesting; if not that will be interesting also. Best and long long life to all, and thanks for patience from those who don't see the relevance of this discussion.... Thomas Donaldson Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=10313