Re: Epistemology 201: The Science of Science

From: robert j. kolker (nowhere_at_nowhere.net)
Date: 02/09/05 

Date: Wed, 09 Feb 2005 14:17:30 -0500

Tony Orlow (aeo6) wrote:
> points contained within the point that marks its middle? These are
> equivalent to the Cantorian approach to the cardinalities of infinity.
> It's like saying all lines are the same size. IT"S COUNTERINTUITIVE
> BECAUSE IT"S WRONG.

There are no known contradictions in transfinte set theory (when it is
properly restricted).

Here is a counter intuitive result for you to ponder. Weirstrass
produced a function that is -everywhere- continuous and -nowhere-
differentiable. Our intuition is fed by drawing curves with pencils and
pens. The idea of a supremely jaggedy function is counterintuitive, but
it is correct.

There are curves that bound finite areas but which have undefined curve
length. That is counterintuitive and correct.

Almost all the constructs of fractals are counterintuitive, but correct.

In the general n-body problem using Newtonian gravity almost all initial
configurations of positions and velocities are chaotic. That is
counterintuitive but correct.

Banach and Tarski show that for dimension greater than 2 one can take a
nice compact figure, decompose it into unmeasureable sets and recompose
the sets so that the result has a greater volume than the original. That
is counterintuitive, but correct. Gooogle <Banach Tarski Paradox>

Mathematics is loaded with counterintuitives that are quite correct
(i.e. logically derived without contradiction). Like I say, if you want
to do cutting edge math get rid of your common sense intuition, for it
is more often wrong than right.

This is even more true for physics. There is nothing about quantum
theory that makes common sense. It is totally counterintuitive.

>
> By the way, if you were to argue that there may well be an infinite
> number of points in the same place on that line segment, I wouldn't
> disagree. Oh crap, now I did it! (readies snowballs and butresses fort
> for the next round).

What does "in the same place" mean. Assuming normal cartesian
co-ordinates each place or position has a unique co-ordinate. Here is
something that is counterintuitive. A unit circle with are pi consists
of an infinite set of points each having zero area. Add up all those
zeroes and get pi. Go figure. That is why we have to ditinguish measure
from cardinality.

Bob Kolker

Relevant Pages

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