The Virgin Birth of Points
in [Physics]
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From: Venkat Reddy on 13 Nov 2007 09:36 On Nov 13, 7:17 pm, Randy Poe <poespam-t...(a)yahoo.com> wrote: > On Nov 13, 8:54 am, Venkat Reddy <vred...(a)gmail.com> wrote: > > > > > On Nov 13, 4:30 pm, David C. Ullrich <ullr...(a)math.okstate.edu> 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. > > > > Good for you. > > > > >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. > > > > Uh, no. The reason a set consisting of a single point has zero > > > length is that a - a = 0. > > > I thought a set contains zero or more elements, and the size > > (cardinality) of the set is the number of its elements. Whats the > > "length" of a set? > > That doesn't have a meaning for general sets. > > > Why is it zero when the set contains a single > > point? > > The special kind of set called a "closed interval in R" > is defined by two endpoints, a and b with a <= b. And the length > of that kind of set can be defined as b - a. > > So it's zero when b = a. > > Measure theory generalizes the idea of length to handle > arbitrary sets of points. "Measure" corresponds to length > in the case of the special kind of set called a "closed > interval in R" but again you'd have a hard time defining > length for general point sets. Thanks for educating me a little on the sets. > > > And, to which of my statement did you negate when you said > > "no"? > > Probably that "points only exist as boundaries of higher > dimensional regions". Yes, because if we consider 1D space, points can't exist without lines and lines can't exist without points. Points have to be defined in terms of line segments and lines in terms of points. Similarly, the 2D regions are defined by the boundary consisting of lines and points. These boundaries are defined by the regions and regions are defined by the boundaries. So points exist as regions in zero dimension and as boundaries in higher dimensions. I'm no expert in this, but just presenting my thoughts. - venkat
From: William Hughes on 13 Nov 2007 10:07 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. 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. - William Hughes
From: Robert J. Kolker on 13 Nov 2007 11:04 Wolf Kirchmeir wrote:> > What about the fries? Do they form a Mandelbrot set? Only if mixed with almonds. Bob Kolker
From: Dave Seaman on 13 Nov 2007 11:05 On Tue, 13 Nov 2007 05:42:56 -0800, Venkat Reddy wrote: > On Nov 13, 12:31 am, John Jones <jonescard...(a)aol.com> 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: >> > >> > Robert wrote: >> >> > >> >> In Euclidean space a set which has exactly one pont as a member has >> > >> >> measure zero. But you can take the union of an uncountable set of such >> > >> >> singleton sets and get a set with non-zero measure. >> >> > >> > 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. >> >> I think you need to distinguish between a position and a point before >> wildly conflating them in both a philosophical and mathematical >> confusion. > 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? What if no coordinate system is specified? The definition of a measure space says nothing about a coordinate system. For that matter, the important elements of a measure space are not the points (elements of the space itself), but rather the measurable sets (members of the specified signma-algebra). -- 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: Robert J. Kolker on 13 Nov 2007 11:05
Venkat Reddy wrote: > I thought a set contains zero or more elements, and the size > (cardinality) of the set is the number of its elements. Whats the > "length" of a set? Why is it zero when the set contains a single > point? And, to which of my statement did you negate when you said > "no"? Do not confuse cardinality with measure. They are quite distinct. Bob Kolker |