Re: Distinct linear orderings on Z

From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 18:12:41 GMT

On Tue, 29 Mar 2005 11:44:02 -0600, Albert Wagner
<albertwagner@cox.net> in comp.ai.philosophy wrote:

>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.

This is the same definition I learned in plane geometry, Albert. It is
what I was referencing. But it seems to me that defining a circle in
terms of circular rotation begs the question. It's true that circles
are defined in terms of equal distance from an origin on a plane. But
all this definition seems to do is say that a circle is equal distance
on a plane without saying how we get the plane rotation. A radius
could just meander in any direction. So we're really back to the
definition of a sphere unless we assume the idea of the circle to
begin with.

When I suggest definition of a circle in terms of construction, the
plane comes automatically. We first define a straight line and then
bisect it to determine a radius. We then construct a normal to that
straight line of radial length at the point of bisection as the basis
of definition. And to keep the circle on a plane we construct another
normal to both lines and proceed with angular bisection normal to both
the original straight line and the second normal.

>> 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?

I don't know as there is any specific rule. I just think a fixed thing
ought to be defined by fixed things. I will grant you that bisection
represents an action. It just doesn't define a moving thing needed to
do some other definition. The result of bisection is a static thing
used to define something else that is not the result of an action. In
the case of circular rotation the process is used to define itself.

>>>>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.

Well I hope I have explained above why my actions are allowable
and yours are not. The actions I refer to only define static straight
line segments used to define a circle. Your actions define the circle
directly and in effect make the process self defining. I am not trying
to justify my actions as opposed yours. I can't justify or define
bisection as an action either, as in effect defining a straight line
as motion in a straight line or in a direction of constant angular
velocity with respect to all points in space. I can only justify it in
terms of results as the difference between points in space.

Bob's definition wanders over the surface of a sphere unless he
assumes a plane which I don't have to do because contruction of
normals to a straight line segment defines the plane for me.

>>>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.

True. But that doesn't mean the limit is not a circle. It only means
that our ability to think through circles in terms of straight line
segments we can think through as differences between points is finite.

>> 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?

Not really. The mathematics discussion here has pretty much been
sets versus geometry. But there isn't any reason to dismiss sets as
useless just because set theorists make ridiculous assumptions
concerning set characteristics and behavior.

We originally got off on this circle definition tangent because Bob
couldn't define a circle using sets exclusively and had to go back to
Euclidean assumptions regarding planes. That doesn't mean sets are
useless. What it means is that point definitions are useless because
no dimensional definition is possible in dimensionless terms. The
problem wasn't sets in general but points in particular and collective
definition in terms of points.If we take the difference between points
we wind up with a collection of "all" points because that difference
is indefinitely divisible through bisection. But trying to go the
other way around is futile because there is no other way to define the
set of "all" points. Mathematikers pretend they can do anything they
can say or put into words. But they forget the words themselves aren't
what govern the sense or nonsense those words make in combination.

>> 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.

Well I hope I've explained a little better.

>> 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.

Well I certainly hope not, Albert. As previously stated I don't mind
being righteous but I certainly mind being self righteous. At least I
hope I've explained myself better. But regardless these are exactly
the kinds of issues that have to addressed and resolved if we're ever
going to get to the bottom of science and truth in general.

Regards - Lester

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