Re: Patriarchy: Re: What Matriarchy?

William Edward Woody (woody@alumni.caltech.edu)
Fri, 16 Aug 1996 13:03:26 -0700

In article <3211C436.44E0@megafauna.com>, steve@megafauna.com wrote:
> 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 (WFF's) 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.
[snip]
> 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.

For all practical purposes, this is the case: 99.9% of the scientists
out there rolling up their shirt sleeves and trying to find the LD50 of
some new chemical or somesuch is not going to be affected by the notion
that any mathematical system of well-formed formulas cannot completely
describe a particular system.

But it does have an interesting conclusion that is apropos to those
early scientists you mentioned (and I snipped) which does apply to the
relm of the physical world: their hope as mathematican/philosophers
was to construct a mathematical system which completely described the
entire universe. That is, it was their hope (not to be realized anytime
next week, of course) to be able to deductively describe the entire
of reality using mathematics. And Godel sort of through a wrench into
the whole mess.

Personally, I think this was a good thing to have happen to mathematicians:
it made them realize that their ultimate goal was impossible to reach.
And made many of them set their sights onto more interesting goals.
But the approach and the underlying assumptions still live on, or at
least lived on in the professor who was my Abstract Algaebra teacher--
whose class I did poorly into because I couldn't wrap my fuzzy little
head around the notion of starting with set theory and deducing the
existance of basic mathematics, calculus, and for all I know, the
existance of lite beer by the third trimester.

(Joke: a mathematician is a system which converts beer into proofs.)

> 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.

Um. Two words: "particle physics."

Acutally, they'd *love* to test their conjectures, but the superconducting
supercollider was murdered by the Texans who stole the money, and
eventually by Congress who didn't want their pockets picked any more.
(I always wonder why that wasn't awarded to Stanford, who was going
to retrofit their collider for a fraction of the money to reach the
same voltage levels. But who didn't say politics isn't a part of
the practical side of science?)

And so the string theorists push on like the philosopher/mathematicians
whose reality was shot down by Godel.

> 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.

My personal pet theory is that nature follows mathematics because mathematics
follows nature: that is, we invented mathematics to describe what we
saw in nature, and used nature as a guidepost to tell us what part
of mathematics we should throw out, and what part of mathematics we
should keep and use.

It's easy enough to construct (based on first principles and deductive
reasoning) to construct a mathematical system with no practical purposes.
We sorta did this (or at least I *think* we did this--again, my fuzzy
little mind got lost right about here) in my Abstract Algaebra class.
But then, those systems were tossed aside for the meatier (and more
nature-following) mathematical systems. (And of course later I learned
that those mathematical systems [such as group theory] were created
by physicists by going from what they observed to something which
described what they observed. And then was later stolen by the
mathematicians who obviously had nothing better to do.)

- Bill

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