The Definition of Points
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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~~
From: Lester Zick on 21 Mar 2007 18:44
On Wed, 21 Mar 2007 01:25:42 -0600, Virgil <virgil(a)comcast.net> wrote: >> So, the quest is for the basic axioms that give rise to all the theorems >> we want, and more. For you, that's ZFC, but for some, that's not >> satisfying. Oh well. > >While ZFC is a nice axiomatic system, I am not at all sure that it is >what you claim for it. NBG is a nice system too, and there are a variety >of others of considerable virtue. Do any of them produce straight lines? ~v~~ |