The Virgin Birth of Points
in [Physics]
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From: Robert J. Kolker on 13 Nov 2007 21:04 Schlock wrote: > Well that's okay as long as we're dealing with straight kugels. Curved > kugels are a whole nuther kettle of fish however. You mean a whole other kettle of knish, don't you? Bob Kolker
From: Robert J. Kolker on 13 Nov 2007 21:06 John Jones wrote: > > I think many problems could be solved by not conflating point and > position. A position is not a point, nor a point a position. A > position is an object in a framework or construction; while a point, > like a line, is a framework for the construction of objects such as > positions. Frameworks are incommensurables while objects (positions) > are not. So the rules for frameworks are quite different to that of > the rules for objects. Perhaps Hibert rudimentally envisaged this when > he described points and lines as 'primitives'. You don't say. Bob Kolker >
From: William Hughes on 13 Nov 2007 21:30 On Nov 13, 1:12 pm, Venkat Reddy <vred...(a)gmail.com> wrote: > On Nov 13, 8:07 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Nov 13, 9:27 am, Venkat Reddy <vred...(a)gmail.com> wrote: > > > > On Nov 13, 2:01 am, lwal...(a)lausd.net wrote: > > > > > On Nov 12, 11:37 am, John Jones <jonescard...(a)aol.com> wrote: > > > > > > On Nov 12, 6:21?pm, "Robert J. Kolker" <bobkol...(a)comcast.net> wrote: > > > > > > Once again you do not distinguish between objects and the sets of which > > > > > > the objects are elements. Another evidence that you cannot cope with > > > > > > mathematics. > > > > > A line is not a set of points because sets are indifferent to order. > > > > > However, if you care to order points we still do not have a minimal > > > > > definition of a line. > > > > > I've been thinking about the links to Euclid's and Hilbert's > > > > axioms presented in some of the other geometry threads: > > > > >http://en.wikipedia.org/wiki/Hilbert%27s_axioms > > > > > These last few posts are posing the question, is a > > > > point an _element_ of a line, or is a point a > > > > _subset_ of a line? > > > > > The correct answer is neither. For let us review > > > > Hilbert's axioms again: > > > > > "The undefined primitives are: point, line, plane. > > > > There are three primitive relations: > > > > > "Betweenness, a ternary relation linking points; > > > > Containment, three binary relations, one linking > > > > points and lines, one linking points and planes, > > > > and one linking lines and planes; > > > > Congruence, two binary relations, one linking line > > > > segments and one linking angles." > > > > > So we see that line is an undefined _primitive_, > > > > and that there is a _primitive_ to be known as > > > > "containment," so that a line may be said to > > > > "contain" points. > > > > > Notice that the primitive "contain" has _nothing_ > > > > to do with the membership primitive of a set > > > > theory such as ZFC. Why? Because this is a > > > > geometric theory that is not even written in > > > > the _language_ of ZFC. > > > > > So both "a point is an element of a line" and "a > > > > point is a subset of a line" are incorrect. > > > > Excellent! This settles my question in the main thread. > > > Hardly. There is more than one way of defining lines > > and points. Certainly you can take "line" > > be a primative. In this case > > a line is not composed of anything, it just is. > > > However, your argument is not only "it is possible > > to define a line as not being composed of points", > > but also that "it is impossible to define a line > > as being composed of points". The latter statement > > is false. > > > It is certainly possible to define points without > > reference to lines and then to define a line > > as a particular set of points. > > My humble thoughts - Given a pair of end points, a line segment is > uniquely defined in 1D. Conversely, given a line segment its two end > points are uniquely defined. You need to define points and line segments. How do you intend to avoid circularity? (You can't definie points in terms of line segments and then line segments in terms of points) But you still have problem. If you cut [0,1] at 1/n for every natural number n, then you get a "segment" [0,0] with length 0. - William Hughes
From: Traveler on 13 Nov 2007 21:58 On Tue, 13 Nov 2007 18:30:29 -0800, William Hughes <wpihughes(a)hotmail.com> wrote: >You need to define points and line segments. Nobody can define them in any way that does not lead to an infinite regress. The truth is that there are no such things as points, lines, distance, size, surfaces, etc... They are all illusions of perception. There exist only particles and these have no size. Having no size is not synonymous with having zero size. Size simply does not exist. It is not a property of nature. There is no law that requires anything to have size. Yet particles have properties such as position, orientation, energy, etc... Size is abstract, being the abstract vector difference between two positions. Distance is thus an illusion. It is conceivable that, in the future, we will have technologies that will aloww us to move from any position to any other, instantly. We already have evidence of this in the ophenomenon known as quantum jumps. For more on the non-existence of space, see the link below. Nasty Little Truth About Space: http://www.rebelscience.org/Crackpots/nasty.htm#Space Louis Savain
From: William Hughes on 13 Nov 2007 22:09
On Nov 13, 9:58 pm, Traveler <trave...(a)noasskissers.net> wrote: > On Tue, 13 Nov 2007 18:30:29 -0800, William Hughes > > <wpihug...(a)hotmail.com> wrote: > >You need to define points and line segments. > > Nobody can define them in any way that does not lead to an infinite > regress. Piffle. -William Hughes |