Re: Patriarchy: Re: What Matriarchy?

Stephen Barnard (steve@megafauna.com)
Wed, 14 Aug 1996 04:19:02 -0800

Eric Brunner wrote:
>
> Personally, I'm waiting on the advocates of High Scientism to explain the
> work of a mathematician... Kurt Godel...
>
> I can wait, of course, I'm not a strict constructionist (mathematically
> speaking that is).
>

I'm not an advocate of High Scientism, whatever that is, but I'll take a
crack at it.

Godel's famous Incompleteness Theorem proves that any formal deductive
system of sufficient power (at least the power of Peano arithmetic,
which isn't that powerful) will be incomplete. That is, it will be
possible to state theorems in that system that cannot be proven true or
false. This is very closely related to Turing's Halting Problem, but I
won't go into that.

Godel's theorem is extremely interesting in that it calls into serious
question the foundations of mathematics. Not so long ago, before Godel,
it was thought and hoped that all of mathematics could eventually be
founded on a set of consistent axioms, from which *any* formal
mathematical statement could be proven to be true or false. Godel
showed that it ain't so, which was a real shock to a lot of people.

Godel's theorem has very little to do with physical science, however,
because physical science is based on inductive reasoning. People look
at evidence, make hypotheses, test the consequences of these hypotheses
against new evidence, discard or refine the hypotheses, debate one
another in often heated terms about their hypotheses (often calling one
another nasty names), attempt to confirm others' hypotheses (when they
often really want to refute them), and so on.

This bears no resemblance to formal logical deduction, which is not a
method for getting at the truth. Formal deduction is merely a way of
transforming old truths (axioms) into different truths (theorems). Or,
working backward, it's a way of transforming suspected truths
(conjectures) into actual truths (theorems), thereby proving them.
Sometimes science makes use of formal logical deduction to explore the
consequences of it's conjectures, but these consequences always remain
conjectures because they are not based on anything like real axioms.

What some people seem to be missing here is that there is no absolute
certainty in science. Uncertainty is meat and drink to scientists.
They have to deal with it all the time.

A much more interesting question, IMHO, is why nature seems to follow
mathematics. That is, why are physical theories expressed in
mathematical form so effective? There doesn't seem to be any a priori
reason why this should be so, and a lot of scientists and philosophers
are very puzzled by it. This should warm the cockles of the hearts of
those who have an anti-science agenda, but, as far as I know, they
haven't picked up on it yet.

Steve Barnard