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

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