The Virgin Birth of Points
in [Physics]
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From: Robert J. Kolker on 13 Nov 2007 11:06 Venkat Reddy wrote:> > The position of a point is relative to the reference coordinate > system. So, position is an attribute on a point to locate it with > reference to the given coordinate system. > > Does it make some sense? In a way. Position is a name we give to points to identify them uniquely. Bob Kolker
From: Lester Zick on 13 Nov 2007 11:40 On Tue, 13 Nov 2007 05:48:58 -0800, Venkat Reddy <vreddyp(a)gmail.com> wrote: >On Nov 13, 6:44 pm, Randy Poe <poespam-t...(a)yahoo.com> wrote: >> On Nov 13, 6:31 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> > On Mon, 12 Nov 2007 20:57:25 -0500, "Robert J. Kolker" >> >> > <bobkol...(a)comcast.net> wrote: >> > >Lester Zick wrote: >> >> > >> 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. >> >> > >Neither does any one else. You have created a straw man here. >> >> > Horseshit, Bobby. I didn't create the straw man. I can cite chapter >> > and verse. >> >> Lester Zick citing a reference other than himself? Only for the purpose of ridicule. I cite you all the time. >> I'd like to see that. Exact quote, please, that says >> points have zero length but not zero length. >> > >He must be referring to the post by William in the thread "Lines >composed of points?". Sure. And when I inquired into the contradiction of predicates involved I received the reply from him that it was obvious and he couldn't discuss the subject with anyone who didn't find it obvious. ~v~~
From: Randy Poe on 13 Nov 2007 12:07 On Nov 13, 11:40 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > On Tue, 13 Nov 2007 05:48:58 -0800, Venkat Reddy <vred...(a)gmail.com> > wrote: > > > > >On Nov 13, 6:44 pm, Randy Poe <poespam-t...(a)yahoo.com> wrote: > >> On Nov 13, 6:31 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > > >> > On Mon, 12 Nov 2007 20:57:25 -0500, "Robert J. Kolker" > > >> > <bobkol...(a)comcast.net> wrote: > >> > >Lester Zick wrote: > > >> > >> 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. > > >> > >Neither does any one else. You have created a straw man here. > > >> > Horseshit, Bobby. I didn't create the straw man. I can cite chapter > >> > and verse. > > >> Lester Zick citing a reference other than himself? > > Only for the purpose of ridicule. I cite you all the time. OK, then. Your offer to "cite chapter and verse" was not a serious one. Thought not. - Randy
From: Venkat Reddy on 13 Nov 2007 13:12 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. There is no other way to find a point except as an end point of a line segment. Generalizing, if possible, a region is defined by its boundary and a boundary is defined by its region. Within a fixed boundary, the region could be curved into higher dimensional spaces, and the extent of the region could vary based on its curvature. But we do not consider it while observing from within the same dimensional space. For example, a sphere bounded by a given surface could have varying volume based the curvature of its 3D region in higher dimensions. If I can imagine wild, a point could have finite or infinite extent as a region in zero dimensional space and have zero extent in higher dimensional spaces. We could apply Same rules for n-D regions. A line has zero extent in 2D and a surface has zero extent in 3D. > If we do this > then we need a definition of "extent" for a set > of points. Note there can be more than > one "size" for a set. The cardinality is one > size but the cardinality does not have the > properties we want. However, the Lebesque measure > does. If we define the extent of a set > to be the Lebesque measure, then the extent of > a point (formally the extent of the singleton > containing the point) is 0, but the extent of a set of > points may not be 0. The fact that you do > not like this will not make it go away. Well, my like and dislike will have infinitesimal impact on the tons of math literature existing out there. I'm just trying to learn by questioning. Thanks for helping me. - venkat
From: John Jones on 13 Nov 2007 16:27
On Nov 12, 11:59?pm, lwal...(a)lausd.net wrote: > 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. If points are not positioned, as you say, then the line that has the minimal requirement of two points by virtue of 1.7, is a concept, or if you like, a set. The set has fractured and unfractured lines as its members. The set has one of its line members fractured when two points fall on the same position. Other fractured lines arise when 'before' and 'after' points fall contrary to their expected positions. Interesting. |