Re: Distinct linear orderings on Z

From: Albert Wagner (albertwagner_at_cox.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 11:44:02 -0600

Lester Zick wrote:
> On Mon, 28 Mar 2005 20:06:42 -0600, Albert Wagner
> <albertwagner@cox.net> in comp.ai.philosophy wrote:
>
>
>>Lester Zick wrote:
>><snip>
>>
>>>Well the difference is that you don't need rotation. All you need is
>>>bisection and static construction techniques to define circles by
>>>approximation. In other words my definition of a circle doesn't rely
>>>on circular rotation. That's something that has always bothered me in
>>>common geometric definitions.
>>
>>Why?
>
> Mainly because rotation presumes circular rotation which assumes the
> thing we're trying to define.

Perhaps I misunderstood. I thought you were referencing my
definition:

Circle -- a plane curve generated by one point moving at a
constant distance from a fixed point.

Which does not assume the thing I was defining.

> It also presumes an action or process as
> the basis of definition where the thing we're trying to define is not
> an action.

That makes no sense. What rule of definition disallows a fixed
thing being defined as the result of an action?

>>>My definition only relies on successively smaller subdivisions. We
>>>start off with a straight line segment and straight angle and through
>>>bisection of the segment determine radius and successively bisect the
>>>angle at a distance of the radius and take a circle to represent the
>>>curved limit of that approximation process.

Successive /means/ change in time, where one step is dependent on
the completion of a previous step. You have also relied on a
series of actions for definition, which you inconsistently use to
disallow my definition. You have only defined a way to build an
infinite set of points and could have simplified by merely
assuming that infinite set of points, as Bob does.

>>In that a point is the bisection of lines, haven't you then just
>>introduced a way of constructing points? And thereby only
>>entered the back door to a set definition of a circle?
>
> Well I consider a point the intersection of lines but not necessarily
> straight lines and not necessarily a bisection. But yes construction
> does delimit points on the perimeter of a circle. However the circle
> itself is the limit of a series of line segments between those points
> rather than the points themselves.

A distinction without a difference. The limit is only reached
after an infinite number of iterations.

> And as far as sets themselves go I don't really mind them very much.
> There are in fact a great many things we can do under the set rubric
> without going to far astray. The problem is what the mathematikers do
> with sets that make them impossible to use consistently. Even infinity
> makes sense in terms of sets of infinitessimals between cardinal
> limits. Sets in general have ordinality and certain subsets have
> cardinality and a variety of other properties.

But wasn't the problem to be solved: how to define a circle
without recourse to sets, especially infinite sets?

> Epistemologically the problem isn't with sets so much as regression of
> dimensional properties to dimensionless points.

I thought the problem was as I stated above.

> Mathematikers want to
> believe they can regress things terminologically without performing
> any kind of scientific reduction. They just adopt a new vocabulary to
> talk about cardinality, ordinality, well order, denseness, sparseness,
> etc. as if they were contributing something useful to the history of
> knowledge when all they're really doing is talking about everything
> pretty much already known in different terms. It's the same trick the
> materialists and behaviorists use in calling everything behavior as if
> they were contributing some kind of radical scientific insight without
> reducing behavior to any kind of mechanical insight worth discussing.

I think that you have used a similar bag of tricks in your
definition of a circle. 

-- 
"I know that most men, including those at ease with
problems of the greatest complexity, can seldom
accept even the simplest and most obvious truth
if it be such as would oblige them to admit the falsity
of conclusions which they have delighted in explaining
to colleagues, which they have proudly taught to others,
and which they have woven, thread by thread,
into the fabric of their lives." -
	-- Tolstoy

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... > selected as the center of the circle. ... without recourse to an infinite set of points. ... But for saying so in the hearing of others, ... In that case nothing remains of my entire castle in the air, gravitation theory included, the rest of modern physics." ...
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  • Re: Epistemology 201: The Science of Science
    ... > selected as the center of the circle. ... without recourse to an infinite set of points. ... But for saying so in the hearing of others, ... In that case nothing remains of my entire castle in the air, gravitation theory included, the rest of modern physics." ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... > selected as the center of the circle. ... without recourse to an infinite set of points. ... But for saying so in the hearing of others, ... In that case nothing remains of my entire castle in the air, gravitation theory included, the rest of modern physics." ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >> You, on the other hand, Bob, take all this into consideration and ... >> still get the definition of the circle wrong. ... > Let P be a point on the plane. ... as it does an infinite set. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >> You, on the other hand, Bob, take all this into consideration and ... >> still get the definition of the circle wrong. ... > Let P be a point on the plane. ... as it does an infinite set. ...
    (sci.math)
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