The Virgin Birth of Points
in [Physics]
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From: Venkat Reddy on 13 Nov 2007 08:54 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? Why is it zero when the set contains a single point? And, to which of my statement did you negate when you said "no"? - venkat
From: Wolf Kirchmeir on 13 Nov 2007 09:00 Robert J. Kolker wrote: > Amicus Briefs wrote: > >> On Mon, 12 Nov 2007 14:45:25 -0800, John Jones <jonescardiff(a)aol.com> >> wrote: >> >> >>> A position may well not be a primitive, but the intersections of lines >>> construct positions, not points >> >> >> So if we change the name of "points" to "positions" we'll solve the >> problem? > > Better still call them potatoes. Two potatoes determine a kugel. Two > kugels intesect on a potatoe. > > Bob Kolker > What about the fries? Do they form a Mandelbrot set?
From: Venkat Reddy on 13 Nov 2007 09:08 On Nov 13, 4:41 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On Mon, 12 Nov 2007 20:56:04 -0500, "Robert J. Kolker" > > <bobkol...(a)comcast.net> wrote: > >Lester Zick wrote: > > >> Well I know enough of mathematics to have convinced you there is no > >> real number line. > > >So what. The theory of real numbers can and is developed without any > >geometric content of all. Any geometrical associations with real numbers > >are merely aids to intuition, not logical necessity. > > Sure, sure, Bobby. That's why the expression "real number line" pops > up all over the place. That's why you talk incessantly about lines, > points, and lattices etc. > > >In the nineteenth century a purely analytic foundations for the theory > >of real and complex variables was developed. Geometry was purged as a > >logical necessity. Of course, geometry can be very helpful for the > >right-brain operations associated with discovering new theorems to prove > >or new mathematical systems. > > Of course geometry can be very helpful because there is a geometry of > arithmetic but no arithmetic of geometry. The problem is there is no > set of real numbers. If there were you could dance on geometry. But > you can't define any set of real numbers because there is no single > modality for real numbers for the definition of any single set.Pi lies > on circular arcs as Archimedes showed and doesn't lie anywhere else. Values, quantities do not need any geometrical entities to lie in or lie at. I can count 5 apples without using any geometrical space. So where does the value "5" lie? We should not try to model the continuum with the patch work of different kinds of numbers and keep on worrying about the holes they leave. Basically we faltered somewhere while moving from counting numbers to real numbers and then applying them to continuum. - venkat
From: Randy Poe on 13 Nov 2007 09:17 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. > And, to which of my statement did you negate when you said > "no"? Probably that "points only exist as boundaries of higher dimensional regions". - Randy
From: Venkat Reddy on 13 Nov 2007 09:27
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. > > 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. |