The Definition of Points
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From: Wolf on 14 Mar 2007 11:04 Eric Gisse wrote: > On Mar 13, 9:54 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 13 Mar 2007 17:18:03 -0700, "Eric Gisse" <jowr...(a)gmail.com> wrote: >> >>> On Mar 13, 9:52 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >>>> The Definition of Points >>>> ~v~~ >>>> In the swansong of modern math lines are composed of points. But then >>>> we must ask how points are defined? However I seem to recollect >>>> intersections of lines determine points. But if so then we are left to >>>> consider the rather peculiar proposition that lines are composed of >>>> the intersection of lines. Now I don't claim the foregoing definitions >>>> are circular. Only that the ratio of definitional logic to conclusions >>>> is a transcendental somewhere in the neighborhood of 3.14159 . . . >>>> ~v~~ >>> Points, lines, etc aren't defined. Only their relations to eachother. >> So is the relation between points and lines is that lines are made up >> of points and is the relation between lines and points that the >> intersection of lines defines a point? > > No, it is more complicated than that. > > http://en.wikipedia.org/wiki/Hilbert's_axioms > >> ~v~~ > > Hey, Eric, you're actually trying to teach Zick something. IOW, you're assuming he really wants to know. But Zick doesn't want to be taught. To be taught would mean admitting that he doesn't know what he's talking about, or worse, that he cannot understand what you are explaining. For reasons we had better not examine to closely, Zick can't tolerate that admission. -- Wolf "Don't believe everything you think." (Maxine)
From: Bob Kolker on 14 Mar 2007 10:07 SucMucPaProlij wrote:> > > One can assume that there are some objects other than points but I don't think Only if one makes this assumption explicit. This means introducing objects other than points and lines into the system and it means some axiom must somehow mention and characterize this additional object or kind of object. The idea of an axiom system such as Hilbert's is to -explicitly- mention those objects which are not defined and characterize them with the axioms. Thus, given two distinct points there is one and only one line containing the points. The containment relation expressed in a number of ways is also undefined. We we say a point is on a line. A line contains a point or a line passes through a point etc.. Look at hilbert's axiom system in wiki. Bob Kolker
From: Eckard Blumschein on 14 Mar 2007 10:51 On 3/13/2007 6:52 PM, Lester Zick wrote: > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? I hate arbitrary definitions. I would rather like to pinpoint what makes the notion of a point different from the notion of a number: If a line is really continuous, then a mobile point can continuously glide on it. If the line just consists of points corresponding to rational numbers, then one can only jump from one discrete position to an other. A point has no parts, each part of continuum has parts, therefore continuum cannot be resolved into any finite amount of points. Real numbers must be understood like fictions. All this seems to be well-known. When will the battle between frogs and mices end with a return to Salviati?
From: PD on 14 Mar 2007 11:07 On Mar 14, 9:51 am, Eckard Blumschein <blumsch...(a)et.uni-magdeburg.de> wrote: > On 3/13/2007 6:52 PM, Lester Zick wrote: > > > > > In the swansong of modern math lines are composed of points. But then > > we must ask how points are defined? > > I hate arbitrary definitions. I would rather like to pinpoint what makes > the notion of a point different from the notion of a number: > > If a line is really continuous, then a mobile point can continuously > glide on it. If the line just consists of points corresponding to > rational numbers, then one can only jump from one discrete position to > an other. That's an interesting (but old) problem. How would one distinguish between continuous and discrete? As a proposal, I would suggest means that there is a finite, nonzero interval (where interval is measurable somehow) between successive positions, in which there is no intervening position. Unfortunately, the rational numbers do not satisfy this definition of discreteness, because between *any* two rational numbers, there is an intervening rational number. I'd be interested in your definition of discreteness that the rational numbers satisfy. PD > > A point has no parts, each part of continuum has parts, therefore > continuum cannot be resolved into any finite amount of points. > Real numbers must be understood like fictions. > > All this seems to be well-known. When will the battle between frogs and > mices end with a return to Salviati?
From: Bob Kolker on 14 Mar 2007 11:45
Eckard Blumschein wrote:> If a line is really continuous, then a mobile point can continuously > glide on it. If the line just consists of points corresponding to > rational numbers, then one can only jump from one discrete position to > an other. Points don't glide. In fact points don't move. You are still pushing discrete mathematics? All you will get is a means of totalling up your grocery bill. Bob Kolker |