Re: Epistemology 201: The Science of Science
From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 03/19/05
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Date: Sat, 19 Mar 2005 18:00:10 GMT
On Fri, 18 Mar 2005 16:21:30 -0500, Tony Orlow (aeo6)
<aeo6@cornell.edu> in comp.ai.philosophy wrote:
>Lester Zick said:
>> On Fri, 18 Mar 2005 13:40:34 -0500, Tony Orlow (aeo6)
>> <aeo6@cornell.edu> in comp.ai.philosophy wrote:
>>
>> >Lester Zick said:
>> >> On Fri, 18 Mar 2005 10:24:43 -0500, Tony Orlow (aeo6)
>> >> <aeo6@cornell.edu> in comp.ai.philosophy wrote:
>> >>
>> >> >Lester Zick said:
[. . .]
>> >> >> Well the problem is the definition as it stands works for a sphere and
>> >> >> doesn't work for circle, Tony. Not my problem.
>> >> >>
>> >> >> Regards - Lester
>> >> >>
>> >> >Why doesn't it work for a circle? Which part of the circumference is not one
>> >> >radius from the center? You can even define a 1D circloid, on a line, as a
>> >> >center, and all points equidistant from it. Of course on a line, that's only
>> >> >two points, one radius to either side of the center point. Did you notice that
>> >> >the contained space of the 1D circloid is 2r, and the boundary measure is the
>> >> >derivative, 2? This derivative relationsip using r as a unit measure is true
>> >> >for circloids of all dimensions.
>> >>
>> >> Well, there are a couple problems here, Tony. First off it's not my
>> >> job to correct Bob's erroneous modern math definition of a circle. He
>> >> and other modern mathematikers define a sphere and call it a circle.
>>
>> >Actually, Lester, I noted years ago that the very same difnition holds for
>> >circles of any dimension, within a space of that many dimensions. If you're
>> >talking about 3 space, then you map out a sphere, and in 2 space it's a circle.
>> >The only idfference is the number of dimensions.
>>
>> Sure. That's the whole problem. Modern mathematikers just treat
>> spatial dimensionality as an arbitrary variable. That means they don't
>> have to say how we get from one dimension to another so that their
>> definitions apply to one dimensionality or another. The fact is that
>> we occupy 3 space so definitions drawn in general terms apply there
>> and not wherever we choose so as to make definitions appear general
>> when they aren't.
>But, what if you can combine the formulas for all circloids regardless of
>dimension, and form a more general formula that covers the entire family of
>shapes that share those certain characteristics? Why is it a problem to define
>a more general rule rather than deal with particular values of D as isolated
>cases? i thought you sought universal definitions?
The problem is the definitions don't describe circloids. As just
pointed out in a collateral reply, the definition "the set of all
points equidistant from any point" equally describes points, lines,
circles, and spheres depending on which dimensionality one works in.
That makes spatial dimensionality itself the independent variable and
not points or the lack of spatial dimensionality. So if you want to
regress particular definitions to definitions of greater generality
you have to do it in terms of spatial dimensionality first.
What modern math tries to do is substitute a set theory of points and
dense infinities of points for geometry and spatial dimensionality. It
doesn't work. They try to make points the independent variable when it
only represents a derivative concept. Einstein tried the same thing.
He tried to make velocity the independent variable instead of space in
universal terms and wound up with all kinds of nonsense to accommodate
the irrationalities involved.
>> >> It's just my responsibility to point out the error. And to ask why the
>> >> definition doesn't work for a circle is to ask why the definition for
>> >> a sphere doesn't work for a circle. I don't really care; for the
>> >> present it's enough to note that it doesn't. Their problem, not mine.
>> >>
>> >> A secondary issue to consider is that modern mathematikers use very
>> >> sloppy rationales to deal with dimensionality. As far as they are
>> >> concerned they can say anything they want about the dimensions and
>> >> regress any difficulties to hermit functions and Dilbert space with
>> >> impunity.They just brush off definitional implausibilities for circles
>> >> by assuming they're discussing two dimensions or whatever spatial
>> >> dimensionality will make their definition plausible.
>>
>> >If the only difference between a circle and a sphere is an extra degree of
>> >freedom, the extra dimension, then distinguishing them that way is valid, the
>> >way I see it.
>>
>> Well that's the crux of the problem with aribitrary spatial
>> dimensionality, Tony. You wind up making spatial dimensionality a
>> dependent variable of definitions in parochial terms instead of an
>> independent variable of definitions in space in universal terms.
>But space can have any number of dimensions, and if you can define a rule that
>takes that into account and combines rules that were previously particular into
>a general one, that should be a good thing. For instance, it's easy to derive
>from Pythagorean Theorem a distance formula for D dimensions in cartesian
>coordinates, as the square root of the sum of the squares of the distance along
>each of the D dimensions. Why do we need a rule that says sqrt(a^2+b^2) for 2D
>space, and another that says sqrt(a^2+b^2+c^2) for 3D space? Isn't it better to
>have a formula that covers all possibilities for D?
Tony, this all depends on whether space and spatial dimensionality is
a dependent variable or not. If space is an independent variable it is
the universal frame of reference whose dimensionality does not depend
on dependent variables in space.
Exactly where do you think spatial dimensionality comes from? It isn't
out there in space. Space doesn't care how it's dimensioned. It's in
here and is merely something we use to describe space in comparative
terms of differences between differences etc.
>> >> I've suggested an alternative way to consider dimensionality in terms
>> >> of geometric dimensions and topological directions. There are zero,
>> >> one, two, and three spatial dimensions. No one has any demonstrable
>> >> information beyond this.
>>
>> >That's really not entirely true, as I believe I pointed out a few different
>> >ways.
>>
>> You've certainly clearly pointed out your beliefs on the subject,
>> Tony, not the justification for those beliefs. I haven't pointed out
>> the justification for my claim either, Tony, but for what it's worth I
>> also claim there is explicit justification.
>I can justify them if you like. Feel free to ask if you're interested. The
>distance formula is a no brainer. Simply consider the hypotenuse/partial
>distance, that you have calculated from the legs at any given stage in the
>calculation, to be a leg in the next stage of the calculation, ignoring the
>rest of the triangle. Then add the partial distance along the next dimension
>for the next stage as a perpendicular to that leg, and calulate the next
>hypotenuse. By the definition of dimensions as orthogonal directions, you are
>dealing with each dimension in turn, and never need to draw more than a
>triangle to see what you're doing.
Well, Tony, it's not that I'm indifferent to your rationale. It's more
that you seem to consider space as a dependent variable of some thing
or things in space that I disagree with. Space can only be dimensioned
as an independent variable according to whatever universal basis of
commensuration there may be. And that means differences between
differences as far as I am concerned and the finite tautological
regression to self contradictory alternatives allows.
>> > Much can be derived concerning spaces and shapes of more than three
>> >dimensions, including surface features of rectanguloids and trianguloids of an
>> >arbitrary number of dimensions, the distance between two points in a space of
>> >any number of dimensions as an extension of Pythagorean theorm, etc. The rules
>> >of geometry for a space of a given number of dimensions are generally actually
>> >special cases of more general rules for spaces of variable numbers of
>> >dimensions. It's kind of like saying we have no information about what happens
>> >after we take the third integral of a function.
>>
>> Spatial dimensionality is just a matter of exponential variables.
>Hmmm... it could be 2^x or x^2, a x-dimensional 2-unit object or a 2-
>dimensional x-unit object. Extra dimensions are beautifully described by
>integrals. Extra dimensions can be represented as nestings too, though I don't
>much like trying to express space in too many symbols.
The problem, Tony, is that spatial dimensionality can be none of these
things unless it is a dependent variable to begin with. Have you ever
considered that the term equidistant subsumes two critical things:
equality and distance. Spatial dimensionality is the result of how
distances are compared and equated one to another. In other words,
differences between differences applied to distances. That's where the
generalization of spatial abstractions comes from and regresses to.
>> >> If Bob wants to claim that 3D.1d circles are
>> >> the set of all points equidistant from any point on a 2D plane, that's
>> >> different from the actual claim he made. Or if Wolf want's to claim
>> >> that directionality on the surface of a 3D.2d sphere is undefined or
>> >> infinite, it's different from claiming the surface of a sphere is
>> >> unbounded, which it is not. But it's their problem and not mine and
>> >> they won't be able to solve their problem with spatial definitions
>> >> until they recognize topological directionality as distinct from
>> >> spatial dimensionality.
>>
>> >I do think that if one is specifically defining circles rather than spheres,
>> >one should specify that they are applying a rule to 2D space, not 3D or some
>> >other number of dimensions.
>>
>> The problem there, Tony, is that such definitions are not then drawn
>> in general terms of sets of all points. Then modern mathematikers have
>> to admit spatial dimensionality is not just a dependent variable of
>> their definitions and that it requires some kind of mechanization in
>> universal terms independent of sets of all points.
>I am not sure what you're saying here. Does this refer to the complications in
>the equation for a circle, if it is outside the xy plane in 3D, especially when
>tilted? It's true, things get more complicated in those ways, but if you are
>talking just about the circle, and not its relation to anything else, then that
>doesn't really matter what space it DOESN'T contain, does it? If you want to
>fully describe any circle in any space, well, in my book that means in an
>infinity-D space as well, which give you an infinitely long formula, so you
>have to only deal with the dimensions that you are really dealing with if you
>want an optimally manageable problem. I think it is justifiable to ignore
>dimensions that are irrelevant to the question at hand. Of course, it's not
>always immediately clear what's relevant.
The problem is that you're not "just working" in one or the other of
the spatial dimensions. You're "working" in all three all the time.
You can't just get rid of the dimensions that muck up your specialized
assumptions and then pretend they aren't there at all. What I'm saying
above is that spatial dimensionality is the independent variable and
not points in spatial dimensionality. That's why space itself is
universal in nature and not merely some localized warpable variable.
>> > The unbounded nature of the surface of a sphere
>> >simply refers to the fact that there is no identifiable end to the surface, no
>> >perimeter to the 2D space.
>>
>> And no explanation as to how we get into 2D space when the object
>> itself is 3D and not 2D. Little shadow people is not an explanation.
>It's a 2D boundary to the 3D space. A circle has one boundary. If you look at
>things like polyhedra, they are 3D, and are bounded by 2D faces, which are
>bounded by 1D edges, which are bounded by 0-D points. The pattern that an
>object of D dimensions is defined by a boundary of D-1 dimensions is pretty
>clear.
Except that it's incorrect if extended willy-nilly beyond three
dimensions. Just saying it can be and illustrating the extension by
analogy with shadow people in three dimensions doesn't show how it's
actually supposed to happen just because shapes defined in 3D have 2D
faces defined by 1D lines defined by 0D points.
>> > It's not supposed to signify that it's infinite. I
>> >had equated "unbounded" with "unending" and "infinite", and then the suggestion
>> >was made that I was suggesting the surface was infinite, since I claimed
>> >"bounded" was related to "finite".
>>
>> The problem is that all these things relate to one concept: definition
>> or lack thereof.
>So, how do you define a circle and a sphere? What kind of definition are you
>looking for?
I don't and don't have to. I could say the intersection of a sphere
and plane but my concern is with definition or the lack of definition
and the conventional definition for a circle is the definition for a
sphere and not the definition of a circle. It's someone elses problem.
>> > Of course I don't think the surface of a
>> >finite sphere has infinite surface area; it's 4 pi R^2, the derivative of the
>> >finite volume with respect to r. (does this sound a little like your particle
>> >theory yet? okay, well, you're integrating for some reason, but, I'm probably
>> >just moving relatively bass-ackwards to you). I do think that the unboundedness
>> >of the sphere is related to its definition as the limit of a regular polyhedron
>> >of n faces as n goes to infinity. I can see three such infinite regular
>> >polyhedra which correspond to the triangle, square and hexagon tiling patterns.
>> >
>> >You seem to believe very strongly in orthogonal straight dimensions as the only
>> >real kind, and don't seem to like talking about spaces that aren't the normal
>> >spacetime we live in.
>>
>> Tony, I make it a practice not to believe in anything when I'm dealing
>> with science wherethe only criterion I maintain is universal knowlege.
>> Either space is a dependent variable or it is an independent variable.
>> Either way it has to be explained in mechanical terms dimensionally
>> and not merely extrapolated one way or another analogically. It's not
>> a question of whether I believe in finite orthogonality but what we
>> can prove universally true of spatial orthogonality finite or not.
>Well, take another look at the universal orthogonal space distance formula
>based on Pythagoras. Isn't that a significant universal result concerning
>dimensionality as a dependent variable? Do you really think specific spatial
>dimensions are a priori? I certainly have my doubts about that.
I think 3D spatial dimensionality is definitely a priori for unstated
mechanical reasons related to the general nature of commensuration.
We've already seen that definitions for circles are not general so I
have no reason to believe that definitions for spatial dimensionality
in general will prove any more insightful in mechanical terms.
>> > I can see that the surface of a sphere is truly curved
>> >around 3D, but I can also see how it locally seems flat, and that to discern
>> >the 3D-ness of the space from within the 2D surface (as a flat being, like a
>> >swirl in a soap bubble) one would have to travel some distance across the
>> >surface and detect non-linearities in the form of violations of euclidean
>> >geometry. The same holds in our space. We can only really get any definitive
>> >direct evidence of the overall geometry of space by traversing it and measuring
>> >it. In the end, I am not sure we need direct evidence, as I think the geometry
>> >of the universe will be found to be tied to, and exhibited in, behaviors at the
>> >smallest level.
>>
>> If you were to recast the foregoing in purely analogical allegorical
>> terms of what you find plausible, I think you can see that it depends
>> heavily on what you find plausible and not what is or is not true.
>>
>> Regards - Lester
>>
>Not really. It's derived from facts about geometry, not metaphors or
>allegories. I mean, allegores don't allow you to calculate anything.
Well the analogies are derived from facts about geometry but the
reasoning inside the analogies is still analogical. It becomes
allegorical if we start telling little stories about shadow people to
illustrate what we can't explain in mechanical terms.
Regards - Lester
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