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Re: Patriarchy: Re: What Matriarchy?
Stephen Barnard (steve@megafauna.com)
Sat, 17 Aug 1996 11:05:54 -0800
Eric Brunner wrote:
>
> Stephen Barnard (steve@megafauna.com) wrote:
> : 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.
>
> Pity. I'm sure that if anyone noticed that a fundamental theorem in
> theoretical computer science was very closely related to a fundamental
> theorem in foundations that there would be at least one paper in the
> literature.
>
The connection is called Church's Thesis. It's the basis for a huge amount of
work in Recursive Function Theory and theoretical Computer Science.
> A single cite please, refereed, in SIAM, or AMS, or the equivalents.
>
Sorry. You'll have to do your own research. If I really wanted to embarrass you
I'd go to the library, but it's a nice day so I'm going to play tennis instead.
> : Godel's theorem is extremely interesting in that it calls into serious
> : question the foundations of mathematics...
>
> A modest appraisal, as it leaves every non-constructionist working without
> a rope over a very awkward abyss -- which we manage to live with like the
> uncomfortable cats on suspended floors of glass. We avoid vertigo by virtue
> of keeping our peepers screwed shut.
There is a common misunderstanding of the Incompleteness Theorem that you appear
to suffer from. Some people seem to think that it calls *everything* in
mathematics into question. In fact, while it's possible to construct undecideable
wffs, there are plenty of interesting theorems that can be proven, under the
assumption that the axioms are true and consistent.
A good example is the Four Color Theorem (that's its possible to color any planar
map with only four colors). This resisted proof for a long, long time, even
though there was little doubt that it was true. People began to suspect that it
was undecidable. Finally, it was proven, with the aid of huge computer
calculations. Another example is Fermat's Last Theorem, which was possibly proven
recently (the jury is still out because it's a very complex proof).
[snipping some not particularly relevant stuff]
>
> : 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.
>
> God. I should be paying for this. There are degrees of uncertainty.
Did I say that there isn't?
>
> : 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.
>
> Please, don't let me stop you Dr. Bernard. Continue until you find the
That's "Barnard".
> very last free parking spot.
>
> May I make a suggestion? Try something fundamentally easier (I've a very
> low opinion of the mathematical capabilities of all but a set of measure
> zero of all of the academic and industrial research computer scientists
> I've had the pleasure to work with (non-mathematically, numerical analysis
> doesn't count, it being only just mathematics)) over the course of my life
> after grade school, like catagory theory, or stick with the superficial
> glitter of self-similar curves.
I guess you haven't met as many computer scientists as I have. Some computational
scientists have very wide ranging interests. Because they try to find
computational metaphors and analogies from many fields, including physics,
mathematics, and even biology, they have to be polymaths, not specializing in one
narrow subject such as ergodic theory. One of the founders of computer science,
and the man who is credited with inventing the modern computer, was John von
Neumann, who was a legendary genius, and one of the most outstanding mathmaticians
of this century. For example, he contributed to the foundations of ergodic
theory. I'm sure he couldn't hold a candle to you, however.
Steve Barnard
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