The Definition of Points
in [Math]
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From: Tony Orlow on 21 Mar 2007 23:24 Lester Zick wrote: > 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~~ Purrrrr....say! Division by zero is not undefinable. One just has to define zero as a unit, eh? Uncountable bunches certainly can attain nonzero measure. :)
From: Tony Orlow on 21 Mar 2007 23:26 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.
From: Tony Orlow on 21 Mar 2007 23:28 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.
From: Tony Orlow on 21 Mar 2007 23:39 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.
From: Tony Orlow on 21 Mar 2007 23:42
Lester Zick wrote: > On Tue, 20 Mar 2007 23:47:48 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Bob Kolker wrote: >>> Tony Orlow wrote: >>> >>>> You know that's not what I mean. >>> I do? Then what do you mean. >>> >>> >>> How do you measure the accuracy of the >>>> premises you use for your arguments? You check the results. That's the >>>> way it works in science, and that's the way t works in geometry. If some >>> But not in math. The only thing that matters is that the conclusions >>> follow from the premises and the premises do not imply contradictions. >>> Matters of empirical true, as such, have no place in mathematics. >>> >>> Math is about what follows from assumptions, not true statements about >>> the world. >>> >>> Bob Kolker >> If the algebraic portions of your mathematics that describe the >> geometric entities therein do not produce the same conclusions as would >> be derived geometrically, then the algebraic representation of the >> geometry fails. Hilbert didn't just pick statements out of a hat. >> Rather, he didn't do so entirely, though they could have been >> generalized better. In any case, they represent facts that are >> justifiable, not within the language of axiomatic description, but >> within the spatial context of that which is described. > > Well Hilbert seems to have had a penchant for tables and beer bottles > in his non definitions of lines and points. > > ~v~~ Beer bottle are better on the table than the floor, that's for sure. 01oo |