The Definition of Points
in [Math]
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From: Virgil on 21 Mar 2007 16:08 In article <46018576(a)news2.lightlink.com>, 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: > >>>> 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. It is a good deal nearer perfect than any of TO's inventions so far. > > >> 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. Not unless that "new system" has benefits the old ones lacked, and so far, TO has presented no evidence of any.
From: stephen on 21 Mar 2007 16:07 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: >> >>> 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. But you cannot identify any actual limitation with either the analytical (set base) approach, or the more traditional approach. >>> 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. All you have done is add '*oo' to the length. You have not demonstrated any ability to say anything new about geometry. >>>> 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 I thought you were addressing the shortfalls of analytical geometry? Do you have any idea what you are talking about? Stephen
From: Lester Zick on 21 Mar 2007 18:41 On 21 Mar 2007 08:56:02 -0700, "Hero" <Hero.van.Jindelt(a)gmx.de> wrote: > Lester Zick wrote: >> Hero wrote: >> > Lester Zick wrote: >> >> >.... There's topology, just the simple >> > beginning: >> > A space (mathematical) is a set with structure. >> > A point is a geometrical space without geometrical structure, but it >> > can give structure to geometry. >> > Think of a vertex or a center and so forth. >> > A line is made up of points and sets of points ( the open intervalls >> > between each two points),which obey three topological rules. >> >> Hero, exactly what makes you think the foregoing observations are >> true? >> >> >What i learned recently: >> >With adding a point to an open (open in standard topology) flexible >> >surface one can enclose a solid, with adding a point to an open line >> >one can enclose a figure, and two points are the boundary of an >> >intervall on a line. But there is no point at or beyond infinity. >> >> All very interesting but I still have no idea why any of this is >> supposed to be true. >> > >Geometry doesn't start in school. You do a lot of observations and >more over You practise geometry - just one example: a football-match. I assume by "football" you mean what we call "soccer". >This needs lots of practical knowledge about differences of directions >(angles) and different coordinate systems (while moving You shoot the >ball to another person, which is moving an a different way), not to >speak of simpler things like lines and points. >Later on, and in school, one learns analyzing and synthesizing too, >also with logical reasoning. > >And talking about sets ( not only from me) - just in the beginning of >it is the definition of a set by Georg Cantor: >"By a set we understand any collection M of definite, distinct objects >m of our perception ("Anschauung") or of our thought (which will be >called the elements of M) into a whole." > >That a line is made up of points is not sufficient, this is shown here >and else. >The topological property, that a line is made up of points and sets of >points, was never questioned. >May be one needs more, but not less for a line. >And this is not a circular definition. Well I agree, Hero,it's not a circular definition but it's really only an assumption and not a demonstration of what's true. You might proceed from other assumptions and reach different conclusions. ~v~~
From: Lester Zick on 21 Mar 2007 18:42 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~~
From: Lester Zick on 21 Mar 2007 18:43
On Tue, 20 Mar 2007 23:06:22 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <4600b8f8$1(a)news2.lightlink.com>, > 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. > >They do produce the same conclusions, and manage to produce geometric >theorems that geometry alone did not produce until shown the way by >algebra. Except mathematikers don't seem to be able to produce straight lines. >Actually, if all the axioms of one system become theorems in another, >then everything in the embedded system can be done in the other without >any further reference to the embedded system at all. ~v~~ |