The Definition of Points
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From: Lester Zick on 14 Mar 2007 01:52 On 13 Mar 2007 15:37:50 -0700, "Hero" <Hero.van.Jindelt(a)gmx.de> wrote: >Randy Poe wrote: >> Lester Zick 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 . . . >> >> The modern axiomization of geometry due to Hilbert leaves >> points, lines, and planes undefined. In fact, he famously >> said about this construction: "One must be able to say at >> all times-instead of points, lines, and planes---tables, >> chairs, and beer mugs." >> >> In other words, despite whatever intuition and inherent >> meaning we might ascribe to these things has no effect >> on the mathematical structure. >> > >A mathematical structure, which is the same for points, lines, and >planes as well as for tables, chairs, and beer mugs, seems to me not >very far advanced, there is not even a difference between an object >with a volume and one without. > >Take any object of volume, a chair. It's center of gravity is a point. >Rotate the chair, the axis of rotation is a line. Let the axis spin >(precession), so every part of the chair is moving with the exception >of one "thing", which is at rest - a point. >So points really exists, not as matter or stuff, but as an aspect of >things. >Just describe them. This is possible in different ways, f.e: one point >is an invariant in a precessing rotation. >With friendly greetings >Hero >PS. Lester, You claim >> > ...that the ratio of definitional logic to conclusions >> > is a transcendental somewhere in the neighborhood of 3.14159 . . . >So definitional logic behaves like a radius extending to conclusions >like half a circle. Just reverse Your way and search for the center >and You have defined Your starting point. Nice. >NB, why half a perimeter? Who said anything about half a perimeter, Hero? I believe the ratio pi is between the full circumference of a circle and its diameter. ~v~~
From: Lester Zick on 14 Mar 2007 01:54 On 13 Mar 2007 17:18:03 -0700, "Eric Gisse" <jowr.pi(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? ~v~~
From: Lester Zick on 14 Mar 2007 01:58 On 13 Mar 2007 18:17:55 -0700, "Tom Potter" <tdp1001(a)gmail.com> wrote: > >"Eric Gisse" <jowr.pi(a)gmail.com> wrote in message >news:1173831482.988051.220120(a)y66g2000hsf.googlegroups.com... >> 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 each other. > >Euclid's Elements > >Definition 1. >A point is that which has no part. > >Definition 2. >A line is breadthless length. > >Definition 3. >The ends of a line are points. > >Definition 4. >A straight line is a line which lies evenly with the points on >itself. > >Definition 5. >A surface is that which has length and breadth only. > >Etc. > >I suggest that the best definition of point >as far as physics is concerned, would be: >"A point is the intersection of orthogonal properties." > >In other words, >a physical point is where time, x,y, and z spaces, >charge and impedance are referenced. Fascinating. But are lines composed of points? The foregoing definitions are reasonable as far as they go however I see nothing in them that sheds light on this issue. ~v~~
From: Lester Zick on 14 Mar 2007 02:00 On Wed, 14 Mar 2007 01:23:33 GMT, Sam Wormley <swormley1(a)mchsi.com> wrote: >Tom Potter wrote: > >> Euclid's Elements >> >> Definition 1. >> A point is that which has no part. >> >> Definition 2. >> A line is breadthless length. >> >> Definition 3. >> The ends of a line are points. >> >> Definition 4. >> A straight line is a line which lies evenly with the points on >> itself. >> >> Definition 5. >> A surface is that which has length and breadth only. >> > > Hey Potter--That was a useful posting for a change! Certainly useful as far as it goes however not very useful for elucidating the basic question as to whether points compose lines. ~v~~
From: Eric Gisse on 14 Mar 2007 02:21
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~~ |