The Virgin Birth of Points
in [Physics]
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From: Schlock on 12 Nov 2007 18:16 On Mon, 12 Nov 2007 18:54:56 +0000 (UTC), Dave Seaman <dseaman(a)no.such.host> wrote: >On Mon, 12 Nov 2007 13:20:13 -0500, Robert J. Kolker wrote: >> Dave Seaman wrote: >>> >>> >>> I think you need to learn some measure theory. This is a question about >>> mathematics, by the way, not philosophy. > >> Zick is totally incapable of understanding either mathematics or >> physics. Robert Heinlein had some clever things to say about people who >> cannot cope with mathematics. Heinlein said they are subhuman but >> capable of wearing shoes and keeping clean. > >Zick is in my killfile. He is not the person I was responding to. Aw c'mon, Davey. Everybody knows a Seaman just loves semen.
From: lwalke3 on 12 Nov 2007 18:59 On Nov 12, 2:45 pm, John Jones <jonescard...(a)aol.com> wrote: > > 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). I'm sorry, but I must concur with Mr. Kolker here. The intersection of two lines is a point. One way to see what's going on here is to consider the standard model of Hilbert, namely R^3. (Kolker uses the notation E2 for 2D Hilbert, but let us consider the third dimension now as well.) Now we can determine what this model happens to map the primitives to. As Kolker has said, "line" is mapped to the set of ordered triples (x,y,z) satisfying a linear relation. But what about "point"? Is "point" mapped to a triple itself (an element of a line), or is it a singleton whose sole element is an ordered triple (a subset of a line)? This is, of course, closely related to what the primitive "containment" is mapped to. It could be membership (point = ordered triple) or inclusion (point = singleton of ordered triple). I believe that mapping containment to membership will be awkward. Let us recall what Hilbert wrote about containment: "Containment, three binary relations, one linking points and lines, one linking points and planes, and one linking lines and planes." So we see that lines contain points, planes contain points, and planes contain lines. And here lies the problem. If we let containment be mapped to membership, then planes would have both points and lines as distinct elements. And even if we only allowed planes to have lines as elements, which lines would be the elements of the plane anyway. For the plane z = 0, for example, are x = constant the elements of the plane, or y = constant, or all of them? So it makes much more sense to map "containment" to "inclusion." Thus points are singletons and subsets of the lines and planes that happen to "contain" them. And therefore the intersection of two lines is the set intersection -- which is exactly the "point." Of course, what about the ordered triples -- the elements of points, lines -- themselves? We may call them "positions," if we want. So the single element of a point is a "position," and the elements of a line are "positions." And so answering the OP's question, positions don't have a measure, but "points" do -- at least, in the standard model R^3 of Hilbert, where subsets in R^3 have a Lebesgue measure. Of course, this is all only in the standard model of Hilbert. In other models, "point," "line," may be mapped to something other than sets, so we can't always refer to the element of a point as a "position," because "point" may be mapped to something that doesn't have an element. To see what I mean, let us take a page from Han de Brujin's book and come up with a new model of some subset of Hilbert's axioms. (If you don't know who HdB is, it's not that important for this example.) Consider Hilbert's Axioms of Incidence only -- the ones labeled I.1 to I.7. Notice one can't prove from these axioms alone that more than finitely many points exist. Indeed, we observe I.7: "I.7: Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane." Apparently, by I.1 through I.7, we can't even prove the existence of more than _four_ points, and indeed, we can construct a model of I.1 through I.7 in which only four points exist. Now in this model, we will map our four points to natural numbers -- in particular, the natural numbers 1, 2, 4, and 8. (Those familiar with HdB should know by now where I am heading with this.) Lines contain exactly two points -- mapped to the sum of the two points that lie on them. Planes contain exactly three points -- mapped to the sum of the three points that lie on them. So we have: 1. point (1) 2. point (2) 3. line (1+2) 4. point (4) 5. line (1+4) 6. line (2+4) 7. plane (1+2+4) 8. point (8) 9. line (1+8) 10. line (2+8) 11. plane (1+2+8) 12. line (4+8) 13. plane (1+4+8) 14. plane (2+4+8) 15. space (1+2+4+8) In this model, containment is mapped to a more complicated matter -- one can try bitwise AND (or OR) to come up with the relation onto which containment is mapped. The important part is that this points don't have "positions" at all.
From: Lester Zick on 12 Nov 2007 19:52 On Mon, 12 Nov 2007 13:04:15 -0500, "Robert J. Kolker" <bobkolker(a)comcast.net> wrote: >Lester Zick wrote: > >> >> Except the main purpose of this thread is less to discuss the zero >> length of points than the heresy of maintaining self contradictory >> predicates, as in "has zero length" and "is not of zero length". > >Points do not have a length (0 or not). Some -sets- of points have >-measure-. In particular a set consisting of a single point has measure 0. > >You have manage to confuse an object with a set whose element is that >object. Hey it's not my problem, Bobby. I'm not the one who claims points have zero length but are not of zero length.Modern mathematics is a heresy. ~v~~
From: Lester Zick on 12 Nov 2007 19:53 On Mon, 12 Nov 2007 18:08:29 -0000, Igor <thoovler(a)excite.com> wrote: >On Nov 12, 12:08 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On Mon, 12 Nov 2007 05:13:10 -0800, Venkat Reddy <vred...(a)gmail.com> >> wrote: >> >> >> >> >> >> >On Nov 12, 5:02 pm, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: >> >> On Sun, 11 Nov 2007 20:57:47 -0800, William Elliot >> >> >> <ma...(a)hevanet.remove.com> wrote: >> >> >On Sun, 11 Nov 2007, Lester Zick 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. >> >> >> >Clearly points don't have zero length, they have a positive infinitesimal >> >> >length for which zero is just the closest real approximation. >> >> >> Erm, no. Points (or rather singletons) have zero length. >> >> >I agree. Also, like I said in the other post, points can only exist as >> >boundaries of higher dimensional regions. Lines, surfaces, solids etc >> >can exist as regions in their own world and as boundaries in higher >> >dimensions. When they are in the role of a boundary they are not part >> >of any regions (of higher dimension). >> >> >We can't observe life of a point as a region in its own dimensional >> >space. >> >> Except the main purpose of this thread is less to discuss the zero >> length of points than the heresy of maintaining self contradictory >> predicates, as in "has zero length" and "is not of zero length". > > >No. Your main purpose in this thread is the same as in any other of >your threads. And that is the intentional obfuscation of established >mathematical concepts. I know, Igor. I'm just making an exception in this particular case. ~v~~
From: Lester Zick on 12 Nov 2007 19:55
On Mon, 12 Nov 2007 13:23:11 -0500, "Robert J. Kolker" <bobkolker(a)comcast.net> wrote: >Igor wrote: > >> >> >> No. Your main purpose in this thread is the same as in any other of >> your threads. And that is the intentional obfuscation of established >> mathematical concepts. >> >> >He is unable to do otherwise. He cannot comprehend standard mathematical >concepts. Zick cannot cope with mathematics. Robert Heinlein had some >interesting things to say about people like Zick. Yes, I know, Bobby. He said they don't suffer heresies like modern mathematics, relativity, and quantum theory gladly. ~v~~ |