The Definition of Points
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From: Wolf on 21 Mar 2007 10:44 Tony Orlow wrote: [...]I'm not putting down axiomatization persay. [...] I'm about to pick a nit: It's not "persay", it's "per se." Latin. Means "by means of itself." The usual English rendering is "in and of itself." OK? -- Wolf "Don't believe everything you think." (Maxine)
From: Tony Orlow on 21 Mar 2007 11:47 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).
From: Tony Orlow on 21 Mar 2007 11:50 Bob Kolker 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. > > Bob Kolekr > They certainly have measure less than any finite measure, which in standard mathematics means 0, however, extensions which allow an infinite unit may define the size of the point as the infinitesimal reciprocal of that unit. Given that, the sum of this infinite number of infinitesimal atoms gives the actual measure of the object. :) Tony Orlow
From: Tony Orlow on 21 Mar 2007 11:55 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. :) There's no reason the circumference of the unit circle can't be considered to have 2*pi*oo points. > > 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. :) > > Stephen >
From: Hero on 21 Mar 2007 11:56
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. 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. With friendly greetings Hero |