The Virgin Birth of Points
in [Physics]
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From: Lester Zick on 14 Nov 2007 00:12 On Tue, 13 Nov 2007 21:04:31 -0500, "Robert J. Kolker" <bobkolker(a)comcast.net> wrote: >Schlock wrote: > >> Well that's okay as long as we're dealing with straight kugels. Curved >> kugels are a whole nuther kettle of fish however. > >You mean a whole other kettle of knish, don't you? Perhaps I should have said a whole nuther kettle of gefilte fish. ~v~~
From: Venkat Reddy on 14 Nov 2007 05:42 On Nov 14, 7:30 am, William Hughes <wpihug...(a)hotmail.com> wrote: > On Nov 13, 1:12 pm, Venkat Reddy <vred...(a)gmail.com> wrote: > > > > > > > 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. > > You need to define points and line segments. How do you > intend to avoid circularity? (You can't definie points in terms > of line segments and then line segments in terms of points) Let me make an attempt - A region in n-D space is a part of the space, having a non-zero n- dimensional extent, bounded fully or partly by regions of r-D (r<n). The bounding regions are called boundaries. A boundary separates a region from the other. A line segment in 1-D is an extent of a region bounded by one or two points. If a line segment is not bounded by even a single endpoint, then it coincides with the space and ceases to exist in 1-D space. A point is a part of the boundary of a line segment. In this context, it is also called an endpoint (half boundary) of the line segment. A line segment with finite extent is bounded by two endpoints. I we have a recursion here. But it is as natural as egg and chick problem. You can't have one without the other. - venkat > > But you still have problem. If you cut > [0,1] at 1/n for every natural number n, then you get > a "segment" [0,0] with length 0. > > - William Hughes- Hide quoted text - > > - Show quoted text -
From: Venkat Reddy on 14 Nov 2007 06:37 On Nov 14, 3:42 pm, Venkat Reddy <vred...(a)gmail.com> wrote: > On Nov 14, 7:30 am, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > On Nov 13, 1:12 pm, Venkat Reddy <vred...(a)gmail.com> wrote: > > > > 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. > > > You need to define points and line segments. How do you > > intend to avoid circularity? (You can't definie points in terms > > of line segments and then line segments in terms of points) > > Let me make an attempt - > > A region in n-D space is a part of the space, having a non-zero n- > dimensional extent, bounded fully or partly by regions of r-D (r<n). > The bounding regions are called boundaries. A boundary separates a > region from the other. > > A line segment in 1-D is an extent of a region bounded by one or two > points. Correction: A line segment is a region bounded by one or two points in 1-D space. > If a line segment is not bounded by even a single endpoint, > then it coincides with the space and ceases to exist in 1-D space. > > A point is a part of the boundary of a line segment. In this context, > it is also called an endpoint (half boundary) of the line segment. A > line segment with finite extent is bounded by two endpoints. > > I we have a recursion here. But it is as natural as egg and chick > problem. You can't have one without the other. > > - venkat > > > > > > > But you still have problem. If you cut > > [0,1] at 1/n for every natural number n, then you get > > a "segment" [0,0] with length 0. > > > - William Hughes- Hide quoted text - > > > - Show quoted text -- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: William Hughes on 14 Nov 2007 07:27 On Nov 14, 5:42 am, Venkat Reddy <vred...(a)gmail.com> wrote: > On Nov 14, 7:30 am, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Nov 13, 1:12 pm, Venkat Reddy <vred...(a)gmail.com> wrote: > > > > 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. > > > You need to define points and line segments. How do you > > intend to avoid circularity? (You can't definie points in terms > > of line segments and then line segments in terms of points) > > Let me make an attempt - > > A region in n-D space is a part of the space, having a non-zero n- > dimensional extent, bounded fully or partly by regions of r-D (r<n). > The bounding regions are called boundaries. A boundary separates a > region from the other. > > A line segment in 1-D is an extent of a region bounded by one or two > points. If a line segment is not bounded by even a single endpoint, > then it coincides with the space and ceases to exist in 1-D space. > > A point is a part of the boundary of a line segment. In this context, > it is also called an endpoint (half boundary) of the line segment. A > line segment with finite extent is bounded by two endpoints. > > I we have a recursion here. But it is as natural as egg and chick > problem. You can't have one without the other. > This is nothing more than a few more details added to your previous attempt. You still do not define point or line. "Recursion" is not needed. As an example of defining point independently of line, first define the real numbers R (this does not require defining a line). Then define a vector space R(k) for k a natural number (the most important cases are k=1,2,3). The vectors in R(k) are ordered k-tuples of real numbers. Still no line needed. Define a point to be a vector. A region in R(k) is a portion of a subspace, translated by a vector (i.e. a set of vectors, i.e. a set of points). A line is a one dimensional subpace, translated by a vector (i.e. a set of points). Note the definition of line depends on point, but the definition of point does not depend on line. Note that by this definition, points lines and regions have the properties you outline. However, you still have the basic problem. If you cut [0,1] at 1/n for every natural number n, then you get a "segment" [0,0] with length 0. - William Hughes
From: David C. Ullrich on 14 Nov 2007 07:33
On Tue, 13 Nov 2007 05:54:29 -0800, Venkat Reddy <vreddyp(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. Cardinality and length are two very different notions of the "size" of a set. >Whats the >"length" of a set? Why is it zero when the set contains a single >point? What Randy said. >And, to which of my statement did you negate when you said >"no"? All of them (except fot the "I agree", and the "Like I said" - yes, you did say more or less the same thing in your other post.) >- venkat > > ************************ David C. Ullrich |