The Definition of Points
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From: Lester Zick on 21 Mar 2007 15:12 On Wed, 21 Mar 2007 00:57:05 +0000 (UTC), stephen(a)nomail.com wrote: >In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> PD wrote: >>> >>> No one says a set of points IS in fact the constitution of physical >>> object. >>> Whether it is rightly the constitution of a mentally formed object >>> (such as a geometric object), that seems to be an issue of arbitration >>> and convention, not of truth. Is the concept of "blue" a correct one? >>> >>> PD >>> > >> The truth of the "convention" of considering higher geometric objects to >> be "sets" of points is ascertained by the conclusions one can draw from >> that consideration, which are rather limited. > >How is it limited Tony? Consider points in a plane, where each >point is identified by a pair of real numbers. The set of >points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. >In what way is this description "limited"? Can you provide a >better description, and explain how it overcomes those limitations? So in arithmetic, Stephen, what is a plane? Or is your arithmetic definition of a plane only assumed per say instead of true per se? ~v~~
From: Lester Zick on 21 Mar 2007 15:17 On 21 Mar 2007 02:15:20 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >On 21 Mar, 05:47, Tony Orlow <t...(a)lightlink.com> wrote: >> step...(a)nomail.com wrote: >> > In sci.math Tony Orlow <t...(a)lightlink.com> wrote: >> >> PD wrote: >> >>> No one says a set of points IS in fact the constitution of physical >> >>> object. >> >>> Whether it is rightly the constitution of a mentally formed object >> >>> (such as a geometric object), that seems to be an issue of arbitration >> >>> and convention, not of truth. Is the concept of "blue" a correct one? >> >> >>> PD >> >> >> The truth of the "convention" of considering higher geometric objects to >> >> be "sets" of points is ascertained by the conclusions one can draw from >> >> that consideration, which are rather limited. >> >> > How is it limited Tony? Consider points in a plane, where each >> > point is identified by a pair of real numbers. The set of >> > points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. >> >> That's a very nice circle, Stephen, very nice.... >> >> > In what way is this description "limited"? Can you provide a >> > better description, and explain how it overcomes those limitations? >> >> There is no correlation between length and number of points > >Oh. And I suppose there *is* such a correlation in "real" geometry? There is if you approach the definition of points on a line through real geometric subdivision instead of trying to glom line segments together into a straight line ala Frankenstein's monster per say. ~v~~
From: Tony Orlow on 21 Mar 2007 15:17 Mike Kelly wrote: > On 21 Mar, 15:47, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 21 Mar, 05:47, Tony Orlow <t...(a)lightlink.com> wrote: >>>> step...(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> PD wrote: >>>>>>> No one says a set of points IS in fact the constitution of physical >>>>>>> object. >>>>>>> Whether it is rightly the constitution of a mentally formed object >>>>>>> (such as a geometric object), that seems to be an issue of arbitration >>>>>>> and convention, not of truth. Is the concept of "blue" a correct one? >>>>>>> PD >>>>>> The truth of the "convention" of considering higher geometric objects to >>>>>> be "sets" of points is ascertained by the conclusions one can draw from >>>>>> that consideration, which are rather limited. >>>>> How is it limited Tony? Consider points in a plane, where each >>>>> point is identified by a pair of real numbers. The set of >>>>> points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. >>>> That's a very nice circle, Stephen, very nice.... >>>>> In what way is this description "limited"? Can you provide a >>>>> better description, and explain how it overcomes those limitations? >>>> There is no correlation between length and number of points >>> Oh. And I suppose there *is* such a correlation in "real" geometry? >>> -- >>> mike. >> In my geometry |{x in R: 0<x<=1}| = Big'Un (or oo). > > What does that have to do with geometry? > > -- > mike. > It states the specific infinite number of points in the unit interval, say, on the real line.
From: Tony Orlow on 21 Mar 2007 15:20 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> PD wrote: >>>>>>> No one says a set of points IS in fact the constitution of physical >>>>>>> object. >>>>>>> Whether it is rightly the constitution of a mentally formed object >>>>>>> (such as a geometric object), that seems to be an issue of arbitration >>>>>>> and convention, not of truth. Is the concept of "blue" a correct one? >>>>>>> >>>>>>> PD >>>>>>> >>>>>> The truth of the "convention" of considering higher geometric objects to >>>>>> be "sets" of points is ascertained by the conclusions one can draw from >>>>>> that consideration, which are rather limited. >>>>> How is it limited Tony? Consider points in a plane, where each >>>>> point is identified by a pair of real numbers. The set of >>>>> points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. >>>> That's a very nice circle, Stephen, very nice.... >>> Yes, it is a circle. You knew exactly what this supposedly >>> "limited" description was supposed to be. >>> >>>>> In what way is this description "limited"? Can you provide a >>>>> better description, and explain how it overcomes those limitations? >>>> There is no correlation between length and number of points, because >>>> there is no workable infinite or infinitesimal units. Allow oo points >>>> per unit length, oo^2 per square unit area, etc, in line with the >>>> calculus. Nuthin' big. Jes' give points a size. :) >>> How is that a limitation? You knew exactly what shape the >>> set of points described. There is no feature of the circle >>> that cannot be determined by the above description. There is >>> no need to correlate length and number of points. Neither >>> Euclid or Hilbert ever did that. > >> Gee, I guess it's a novel idea, then. That might make it good, and not >> necessarily bad. Hilbert also didn't bother to generalize his axioms to >> include anything but Euclidean space, which is lazy. He's got too many >> axioms, and they do too little. :) > > Then you should not be complaining about the "set of points" approach > to geometry and instead should be complaining about all prior approaches > to geometry. Apparently they are all "limited" to you. Of course > you cannot identify any actual limitation, but that is par for the course. > I wouldn't say geometry is perfected yet. >> There's no reason the circumference of the unit circle can't be >> considered to have 2*pi*oo points. > > But what is the reason to consider that is does? All you are doing > is multiplying the length by oo. You are not adding any new information. > You are not learning anything. > You are when you equate infinite numbers of points with finite measures, and develop an system of infinite set sizes which goes beyond cardinality. >>> So where is your better description, and where is the explanation >>> as to why it is better? What more can you say about a circle >>> centered at (3,-4) with a radius of sqrt(10)? > >> It's got 2*pi*sqrt(1)*oo points, as a set, which is greater than the >> number of points in the unit interval. :) > > And what good does that do? You are just giving a new name to "length". > You have not added anything. > > Stephen > > To sets, I have. Tony
From: Lester Zick on 21 Mar 2007 15:20
On Wed, 21 Mar 2007 07:40:46 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Tony Orlow wrote: >> >> There is no correlation between length and number of points, because >> there is no workable infinite or infinitesimal units. Allow oo points >> per unit length, oo^2 per square unit area, etc, in line with the >> calculus. Nuthin' big. Jes' give points a size. :) > >Points (taken individually or in countable bunches) have measure zero. They probably also have zero measure in uncountable bunches, Bob. At least I never heard that division by zero was defined mathematically even in modern math per say. ~v~~ |