The Virgin Birth of Points
in [Physics]
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From: John Jones on 12 Nov 2007 14:35 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.
From: John Jones on 12 Nov 2007 14:37 On Nov 12, 6:21?pm, "Robert J. Kolker" <bobkol...(a)comcast.net> wrote: > Lester Zick wrote: > > > So the intersections of lines are not points? Dearie, me. > > Lines (which are sets of points) sometimes have a non-zero set > intersection which consists of a single point. > > 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. > > Bob Kolker 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.
From: Hero on 12 Nov 2007 14:42 Hero wrote: > Randy wrote: . > > Is that what you were asking? . > Yes.Thanks, Randy. > And thanks to Dave, John and Robert too. > > Now we have all we need to point to the important fact: > It is measuring sets of points, not measuring points or measuring many > points. > A point A is different from the set { A }. > > And a hint to the standard topology of the real number line, which > according to Kuratowski, gives to every two points the intervall in > between, a set of points. > I just see, that Robert wrote more or less the same, posted in the same hour too, i didn't know about. But anyhow, the truth can be expressed twice, without getting twisted. With friendly greetings Hero
From: Dave Seaman on 12 Nov 2007 15:48 On Mon, 12 Nov 2007 11:31:22 -0800, John Jones wrote: > On Nov 12, 6:05?pm, Dave Seaman <dsea...(a)no.such.host> wrote: >> On Mon, 12 Nov 2007 07:50:52 -0800, John Jones wrote: >> > On Nov 12, 3:42?pm, Dave Seaman <dsea...(a)no.such.host> wrote: >> >> On Mon, 12 Nov 2007 07:06:39 -0800, Hero wrote: >> >> > What measure will give a non-zero number/value? >> >> >> Lebesgue measure will do so, not for all possible uncountable sets, but >> >> for some. For example, the Lebesgue measure of an interval [a,b] is its >> >> length, b-a. >> >> > An interval [a,b] is composed of positions, not points. But even >> > positions are constructions, and it is not appropriate to analyse a >> > construction in spatial terms. >> >> I think you need to learn some measure theory. This is a question about >> mathematics, by the way, not philosophy. >> >> - Show quoted text - > I think you need to distinguish between a position and a point before > wildly conflating them in both a philosophical and mathematical > confusion. In my statement that you quoted, I used neither of the terms "position" or "point". I mentioned only Lebesgue measure, uncountable sets, and intervals. Exactly what is your, er, point? Why do I need to distinguish between terms that I didn't use? Neither of those is a precise mathematical term, by the way. The meaning depends on context, but to me a "point" is a member of some abstract space (possibly a vector space, or a topological space, or a metric space, or a measure space, or a Banach space, or whatever). A "position", on the other hand, suggests a point that is given in some coordinate system. That doesn't always apply. Lots of times we talk about points in situations where there are no coordinates in sight. I consider "position" to be too limited a term for that reason. -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit <http://www.abu-jamal-news.com/>
From: lwalke3 on 12 Nov 2007 16:01
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. |