The Definition of Points
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From: Tony Orlow on 22 Mar 2007 09:03 Mike Kelly wrote: > On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: >>>> step...(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> step...(a)nomail.com wrote: >>>>>>> In sci.math 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.... >>>>>>> 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. >>> And yet you remain incapable of stating what these "limitations" are. >>> Why is that? Could it be because you can't actually think of any? >>> -- >>> mike. >> I already stated that the divorce between infinite set size and measure >> of infinite sets of points is a limitation, and indicated a remedy, but >> I don't expect you to grok that this time any better than in the past. >> Keep on strugglin'.... >> >> tony. > > What theorems can't be proved with current axiomatisations of geometry > but can be with the addition of axiom "there are oo points in a unit > interval"? Anything more interesting than "there are 2*oo points in an > interval of length 2"? > > -- > mike. > Think "Continuum Hypothesis". If aleph_1 is the size of the set of finite reals, is aleph_1/aleph_0 the size of the set of reals in the unit interval? Is that between aleph_0 and aleph_1? Uh huh.
From: PD on 22 Mar 2007 09:24 On Mar 21, 6:00 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > Well I can tell as soon as boobirds like Stephen come out the cause is > hopeless. At least I'm witty. Just keep telling yourself that, Lester. Oh, and perhaps you might ask yourself why you are exercising your self-acclaimed wit on a *science* group, where wit is not particularly of value. Wouldn't you be having more fun at rec.humor.arent.I.clever or alt.witty.perceived.self? PD
From: Mike Kelly on 22 Mar 2007 10:59 On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> step...(a)nomail.com wrote: > >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> step...(a)nomail.com wrote: > >>>>>>> In sci.math 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.... > >>>>>>> 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. > >>> And yet you remain incapable of stating what these "limitations" are. > >>> Why is that? Could it be because you can't actually think of any? > >>> -- > >>> mike. > >> I already stated that the divorce between infinite set size and measure > >> of infinite sets of points is a limitation, and indicated a remedy, but > >> I don't expect you to grok that this time any better than in the past. > >> Keep on strugglin'.... > > >> tony. > > > What theorems can't be proved with current axiomatisations of geometry > > but can be with the addition of axiom "there are oo points in a unit > > interval"? Anything more interesting than "there are 2*oo points in an > > interval of length 2"? > > > -- > > mike. > > Think "Continuum Hypothesis". If aleph_1 is the size of the set of > finite reals, is aleph_1/aleph_0 the size of the set of reals in the > unit interval? Is that between aleph_0 and aleph_1? Uh huh. Firstly, the continuum hypothesis is nothing to do with geometry. Secondly, division is not defined for infinite cardinal numbers. -- mike.
From: Tony Orlow on 22 Mar 2007 12:38 Mike Kelly wrote: > On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> step...(a)nomail.com wrote: >>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: >>>>>>>> step...(a)nomail.com wrote: >>>>>>>>> In sci.math 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.... >>>>>>>>> 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. >>>>> And yet you remain incapable of stating what these "limitations" are. >>>>> Why is that? Could it be because you can't actually think of any? >>>>> -- >>>>> mike. >>>> I already stated that the divorce between infinite set size and measure >>>> of infinite sets of points is a limitation, and indicated a remedy, but >>>> I don't expect you to grok that this time any better than in the past. >>>> Keep on strugglin'.... >>>> tony. >>> What theorems can't be proved with current axiomatisations of geometry >>> but can be with the addition of axiom "there are oo points in a unit >>> interval"? Anything more interesting than "there are 2*oo points in an >>> interval of length 2"? >>> -- >>> mike. >> Think "Continuum Hypothesis". If aleph_1 is the size of the set of >> finite reals, is aleph_1/aleph_0 the size of the set of reals in the >> unit interval? Is that between aleph_0 and aleph_1? Uh huh. > > Firstly, the continuum hypothesis is nothing to do with geometry. It does, if sets are combined with measure and a geometrical representation of the question considered. Is half an infinity less than itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. > > Secondly, division is not defined for infinite cardinal numbers. I'm not interested in cardinality, but a richer system of infinities, thanks. > > -- > mike. > tony.
From: Mike Kelly on 22 Mar 2007 12:57
On 22 Mar, 16:38, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > >>>>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> step...(a)nomail.com wrote: > >>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>> step...(a)nomail.com wrote: > >>>>>>>>> In sci.math 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.... > >>>>>>>>> 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. > >>>>> And yet you remain incapable of stating what these "limitations" are. > >>>>> Why is that? Could it be because you can't actually think of any? > >>>>> -- > >>>>> mike. > >>>> I already stated that the divorce between infinite set size and measure > >>>> of infinite sets of points is a limitation, and indicated a remedy, but > >>>> I don't expect you to grok that this time any better than in the past. > >>>> Keep on strugglin'.... > >>>> tony. > >>> What theorems can't be proved with current axiomatisations of geometry > >>> but can be with the addition of axiom "there are oo points in a unit > >>> interval"? Anything more interesting than "there are 2*oo points in an > >>> interval of length 2"? > >>> -- > >>> mike. > >> Think "Continuum Hypothesis". If aleph_1 is the size of the set of > >> finite reals, is aleph_1/aleph_0 the size of the set of reals in the > >> unit interval? Is that between aleph_0 and aleph_1? Uh huh. > > > Firstly, the continuum hypothesis is nothing to do with geometry. > > It does, if sets are combined with measure and a geometrical > representation of the question considered. Is half an infinity less than > itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. Everything you just said has nothing to do with the continuum hypothesis. You're terminally confused. As near as I can tell, your mish-mash of ideas all basically boil down to asserting "the 'number' of reals/points in an interval/line segment = the Lebesgue measure". What does asserting that do for us? Not much. > > Secondly, division is not defined for infinite cardinal numbers. > > I'm not interested in cardinality, but a richer system of infinities, > thanks. You brought up the continuum hypothesis. Next post, you say you don't want to talk about cardinality. Risible. -- mike. |