The Virgin Birth of Points
in [Physics]
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From: Robert J. Kolker on 12 Nov 2007 17:39 John Jones wrote: > > The intersections of lines are positions, not points. There is no > precedent for creating a new metaphysical entity from the arbitrary > arrangement of lines. I would have thought it obvious. But plainly I > was mistaken. Take a pair of linear equations in two variables each of which define a line. If the equations are not linearly dependent they determine a unique solution (x,y) which is --- aha!---- the point of intersection. In a Euclidean Plane lines when they intersect at all, have a unique point of intersection. And mathematical objects are not metaphysical entities. They are brain farts the blow about in our skulls. Bob Kolker >
From: John Jones on 12 Nov 2007 17:41 On Nov 12, 9:01?pm, 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. > > The other question concerns what the intersection > of two lines is. Well, first we must define > "intersection" -- in terms of our _primitives_, > of course -- before we can answer. And the only > answer we can possibly give is in terms of > _containment_: the intersection of two lines a,b > is a point A such that a contains A and b contains > A as well, provided that such a point exists. We > can't call it a "position," since "position" is > not a _primitive_ of our theory. > > So now all we have to do is prove that if such a > point exists, it must be unique. But this follows > directly from Axiom I.1. For if there were two > points of intersection A,B, then I.1 tells us that > two points determine a line, so that AB = a and > AB = b as well, therefore a = b. So if a,b are > distinct and intersect, then they intersect in a > unique point of intersection.
From: Lester Zick on 12 Nov 2007 17:43 On Mon, 12 Nov 2007 11:35:02 -0800, John Jones <jonescardiff(a)aol.com> wrote: >On Nov 12, 6:07?pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On Mon, 12 Nov 2007 07:40:03 -0800, John Jones <jonescard...(a)aol.com> >> wrote: >> >> >On Nov 11, 9:40?pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> The Virgin Birth of Points >> >> ~v~~ >> >> >> The Jesuit heresy maintains points have zero length but are not of >> >> zero length and if you don't believe that you haven't examined the >> >> argument closely enough. >> >> >> ~v~~ >> >> >Points have zero length when construed as lying in a spatial >> >framework. However, points have no length because points are not >> >objects that arise in a spatial framework. Positions, not points, >> >arise in the spatial framework, and positions are always >> >constructions. >> >> So the intersections of lines are not points? Dearie, me. >> >> >I conclude that the question about points cannot be a logical inquiry >> >or someone here would have been able to sort it out... >> >> Logically or illogically? >> >> ~v~~ > >The intersections of lines are positions, not points. There is no >precedent for creating a new metaphysical entity from the arbitrary >arrangement of lines. I would have thought it obvious. But plainly I >was mistaken. Plainly you were, are, and will be for if a point is a metaphysical entity then so I suggest are lines which I should have thought was obvious. ~v~~
From: John Jones on 12 Nov 2007 17:45 On Nov 12, 9:01?pm, 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. > > The other question concerns what the intersection > of two lines is. Well, first we must define > "intersection" -- in terms of our _primitives_, > of course -- before we can answer. And the only > answer we can possibly give is in terms of > _containment_: the intersection of two lines a,b > is a point A such that a contains A and b contains > A as well, provided that such a point exists. We > can't call it a "position," since "position" is > not a _primitive_ of our theory. > > So now all we have to do is prove that if such a > point exists, it must be unique. But this follows > directly from Axiom I.1. For if there were two > points of intersection A,B, then I.1 tells us that > two points determine a line, so that AB = a and > AB = b as well, therefore a = b. So if a,b are > distinct and intersect, then they intersect in a > unique point of intersection. A position may well not be a primitive, but the intersections of lines construct positions, not points. Primitives are incommensurables. Points, lines, planes, etc are incommensurables which do not contain the properties of one within the other. Their 'synthesis' is not a synthesis of properties or objects, but of the frameworks that establish objects and properties (see Kant).
From: Robert J. Kolker on 12 Nov 2007 18:12
Lester Zick wrote:> > And apparently sometimes they don't? So what is it exactly you're > sometimes saying, Bobby? Parellel lines have no points of intersection. Next question? Bob Kolker |