Re: Distinct linear orderings on Z

From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 03/28/05 

Date: Mon, 28 Mar 2005 19:13:31 GMT

On Mon, 28 Mar 2005 11:43:40 -0500, Tony Orlow (aeo6)
<aeo6@cornell.edu> in comp.ai.philosophy wrote:

>Lester Zick said:
>> On Mon, 28 Mar 2005 09:28:20 -0500, Tony Orlow (aeo6)
>> <aeo6@cornell.edu> in comp.ai.philosophy wrote:
>>
>> >Lester Zick said:
>> ><snip>
>> >> >But then, how do you define pi in more basic terms? Sorry if you've already
>> >> >explained this.
>> >>
>> >> I'm not sure what more basic terms there could be, Tony. Given spatial
>> >> dimensionality, straight lines and bisection are pretty much it.
>> >>
>> >> Regards - Lester
>> >>
>> >I mean, how do you define what numerical value you're talking about.
>>
>> The limit of length of straight line segments erected on a straight
>> line segment through bisection of angles.

>You mean dividing the circle repeatedly in half, and connecting the endpoints
>of the lines doing the bisections? How do you know where on the line segments
>to draw the "erected" lines that eventually approach curvature? Don't those
>lines extend forever, making it necessary to define a distance along them, say,
>R?

No, Tony, we don't need to divide the circle because we're trying to
define circles so we can't assume we have a circle to subdivide. What
I'm referring to is bisection of a straight line which defines radius
for the circle and erection of a normal at the point of bisection.
This gives us a triangle whose apex lies at the height of the radius
above the straight line with right angles at the base which we
subdivide repeatedly to get successively higher degree polygons
approaching a semicircular arc as their limit.
 
>> > It seems
>> >like pi is defined in terms of the circle and vice versa, and is indeed
>> >circular, which you may argue is appropriate, but where are the self-
>> >contradictory alternatives?
>>
>> Well the self contradictory alternatives lie between straight lines
>> and curves, Tony. We can define straight lines between points but we
>> can't define curves between points. That's what a curve is.

>What about bezier curves and splines? Those are curves between points......or
>arcs even.

No curve lies between points, Tony, only straight line segments.

>So the
>> regression is from one straight line segment which we can define to
>> the self contradictory alternative of a curve which we can't define.

>Now curves are self contradictory and we can't define them? I am not sure how
>anything that's not even defined can contradict anything.....

No, curves and straight lines are mutually contradictory. We can
define straight lines but not curves. Curves we define through
approximation with straight line segments. When I use the term self
contradictory above I mean the idea of a curve contradicts the
definition of straight line segments and this circumstance is self
contradictory, not either curves or straight lines. It's more on the
order of a self contradictory circumstance.

Regards - Lester

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