On Sun, 10 Sep 2006 00:11:26 -0400, Tony Orlow <tony@lightlink.com>
wrote:
Well, that's Lester Zick wrote:
Mathematical SOAP Operas
~v~~
not or not not - Hi Lester :) How are you? Thanks for playing with mind
and their thoughts. You helped suck me into these various discussions. I
hope you've enjoyed my contributions.
Hey, Tony, how's it going? I wondered how long it would take you to
get back in the mix.At one time I thought we might have more in common
than we turned out to have. By the way if you want some revisionism
you might check out a recent post of mine to a collateral thread
called "Phyiscists Howl at Dark Matter". Irreverent but interesting.
The difficulty with definitions drafted in the form of "set of all
points . . ." (SOAP) is that we're still left with a bunch of points
which don't explain anything including sets and points. Finite
arithmetic is a trivial subset of mathematics and transfinite
arithmetic trivializes mathematics and mechanics by assuming
the truth of everything it describes without proof of its truth.
Well, that's a mouthfull. Point-set topology, as I pointed out some half
a year ago, has an alternative, segment-sequence topology, which
resolves the counterargument to infinite induction regarding the
diagonal line between (0,0) ad (1,1) from the infinitely-stepped but
finitely long staircase, vs. the diagonal line. Where we have a line,
there is continuity. Where we have continuity, there is infinity in any
finite space, pointwisely speaking. Where we declare logical truth,
without justification, we invent fiction, so every rule should be
justified. I hope you don;t disagree with any of those statements. :)
Hard to say, Tony. I've been away from the web for several months and
prior to that I didn't spend much time on sci.math anyway so I might
very well have missed the topic. I still prefer pointless mechanics to
describe what actually goes on in any event.
And yes, transfinitology does nothing to elucidate a tad.
An easy example of this is the vain attempt to define circles as the
"set of all points equidistant from any point" which of course defines
a sphere and not a circle at all. Then various geometric subterfuges
are employed such as the intersection of a sphere and plane without
however first defining the various geometric figures such as planes,
distance, or equidistance in terms of "sets of all points" but just
more or less relying on common sense geometric understandings.
Um, well, yeah. Like I said, the segment-sequence topology is an
alternative to the set of points.
Not sure I'd agree, Tony. Segment sequences sound like SOAP's to me. A
rose by any other name.
Indeed, while it's a little difficult
to generalize (but not impossible) the segment sequence topology to
continuous spaces, it's very clear with circles. Given whatever unit of
measure we're using, the change in horizontal position is proportional
to the vertical position, and the change in vertical position is
proportional to the additive inverse (negative, 0-x) of the horizontal
position - so the circle rotates clockwise. If they're reversed, it's
counter-clockwise. In either case, given whatever scale, it makes a
regular polygon, and in the limit, a circle.
Well the problem is the limit is a circle but is not definable in
terms of a regular polygon. If it were there would be a real number
line and we could square the circle. There's simply not a problem
defining curves if we start out with v and transverse a to begin with.
So, does it also define a sphere? Mmmm, no. That defines a circle, and
when we want to appy it to another dimension, we find that the poles are
neither east nor west, but both, and the 2D circle is not a sphere.
Instead it's a torus. Mmmmmm....
This idea of variable dimensionality is where we part company, Tony.
We don't just get to pick and choose our dimensionality. Any way you
look at it we're still stuck with fff. . . f(-).
Now it's scarcely difficult to fathom an approximate definition for
circles in terms of a SOAP. "A circle is the SOAP of common equi
distance between two points." Or "a plane is the SOAP equidistant
between two points". However even such "accurate" definitions do
absolutely nothing to make such definitions true. All we can really
say is they alleviate a chronic underlying inability of mathematikers
to figure out exactly what it is they're trying to say.
That's why the interactive definition between sine and cosine really
characterizes the circle. Like the diagonal line, it could be a polygon,
with acute vertices, and therefore have a length greater than 2*pi*r. :)
Well the problem here is that definitions for any curve including
circles as well as sine and cosine functions are transcendental to
begin with. Defining circles in terms of transcendental functions
doesn't explain the existence of transcendentals as curves in terms of
straight line segments to begin with. There's no problem defining
rationals and irrationals in terms of straight line segments defined
by points. The problem comes when we try to define transcendentals in
the same terms.
For example whereas it is true that every point on a circle is equi
distant from one and the same point, it is not true that a circle is
composed of or defined by those points. The number of points doesn't
matter. Nor would a plane be defined by those points equidistant
between two points.
It's the regular polygon in the limit, eh?
Pretty much but the limit is transcendental.
In other words dimensional figures cannot be defined by means of
points or through the absence of dimensionality. Dimensional figures
can only be defined through dimensional mechanics and dimensional
mechanisms.
Dimensions are lines, orthogonal lines, independent measures of reality.
At least it's overwhelmingly apparent that dimensionless points are
not.
~a ^ a >1, probabilistically speaking.
Not sure what this means.
~v~~
.
