X-Message-Number: 10298
Date: Fri, 21 Aug 1998 10:17:43 -0400
From: Thomas Donaldson <>
Subject: CryoNet #10289 - #10297

Hi everyone!

To Perry Metzgar:
Sorry you've never met a constructivist. Give me a little time and
I'll give you a list. The current line of thinking in constructivism
is to stop arguing about foundations and produce mathematics with this
constructive restriction. I will give you the citation of the metric
spaces book soon. That will get you started.

As you can guess, a firm constructivist would tell you that the theories
you think you have lost never existed in the first place.

Because (as I hope you know) I moved to Australia in April, and as
yet still haven't received my things --- particularly my library ---
I am unable to immediately discuss the issue of whether or not Goedel's
Theorem holds if we insist on constructivism. Constructivism means 
basically that ANY mathematical entity must come with explicit means to
construct it. (yes, PI does not exist, but arbitrary approximations to
PI do. The notion of real number can receive a definition, too, but
it will require that every number in the set of real numbers have some 
explicit construction). Proofs by contradiction are not allowed... tey
give no explicit construction. (Actually constructive proofs can be
more illuminating, sometimes, than nonconstructive ones: they tell you
much more about what's going on with the entities you are discussing).

So over to you: what are the precise hypotheses of Goedel's Theorem?
What must a mathematical system have to be "sufficiently rich"?

As for the issue of formalism, NO. That is only the way we do math in
the 20th Century. How are things changing? Well, I notice that in 
computer science there are people who "sort of" do math, too, but they
allow empirical arguments. Like: here is an algorithm which usually
finds its objective in time T, and here are the examples showing 
that it does so. To be able to state something precisely does not
require a formal system in which it is derived from axioms and 
definitions. (Not only that, but even in math, as we've progressed,
we learn the need for more and more precision in our statements,
whether we try to use axioms or not). Another famous example comes
from physics: the delta function. Yes, mathematicians did formalize
this notion, but since spaces of distributions aren't metric spaces
(though we could realize a delta function in a metric space) their
formalization is not (yet) constructive. But the real point here is
that physicists found that the delta function WORKED. and it was
because they found it helpful that mathematicians tried to formalize
it. So were the physicists using a formal system or not? Physics
actually provides a very interesting example of math done without

After all, users of math do not want to bother with unprovable 
theorems unless they must. And if they must, then they can turn to
other methods, such as experimental demonstrations that the theorem
is well-founded by experiment. What did you think would happen?
That computer scientists would simply give up when they could not
prove an algorithm will work efficiently? Are they doing mathematics?
Well, if you insist that mathematicians must use formal methods, then
mathematicians did not exist until early this century. That seems to
me a rather stringent (and useless) restriction.

			Best and long long life to all,

				Thomas Donaldson

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