The Definition of Points
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From: Virgil on 22 Mar 2007 00:40 In article <4601fa8a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460184ba(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> 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. > > > > Since TO's infinities are not like anyone else's ( longer line segments > > seem to have more points in TOmetry) "Big'Un" is no use to anyone else. > > Only by wielding the tool do we gauge its utility. Since TO's tool seems to be unwieldable, in any practical way, even by himself, we gauge it to be of no utility to anyone else. What he does with it in private is, of course, his own business.
From: Virgil on 22 Mar 2007 00:42 In article <4601fbf0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Lester Zick wrote: > > On Wed, 21 Mar 2007 14:17:16 -0500, Tony Orlow <tony(a)lightlink.com> > > wrote: > > > >> It states the specific infinite number of points in the unit interval, > >> say, on the real line. > > > > And what real line would that be, Tony? > > > > ~v~~ > > The one that fully describes the real numbers. Like, duh! The one that > exists. > > E R > 0eR > 1eR > 0<1 > xeR ^ yeR ^ x<y -> EzeR x<z ^ z<y > > 01oo TO teaching Zick is like the blind leading the blind, re mathematics.
From: Mike Kelly on 22 Mar 2007 06:32 On 22 Mar, 03:26, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 21 Mar, 19:17, Tony Orlow <t...(a)lightlink.com> wrote: > >> 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. > > > And I ask again, what does that have to do with geometry? Stephen > > already pointed out that saying "there are BigUn points in a unit > > interval" doesn't tell us anything interesting about anything. It > > doesn't add any information. It doesn't lead to any new theorems of > > any consequence. So why bother? > > > -- > > mike. > > Oh, Mike, sorry. I didn't mean to mix up statements concerning points > and lines with geometry. My apologies. > > tony. Saying "the number of points in an interval is oo * the length of the interval" doesn't add anything to geometry. Sorry. -- mike.
From: Mike Kelly on 22 Mar 2007 06:35 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.
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. |