The Definition of Points
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From: Lester Zick on 14 Mar 2007 15:12 On Wed, 14 Mar 2007 10:07:49 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >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. Well for that matter why introduce points into the system except as the intersection of lines? The obvious answer is so that mathematikers can pretend they're doing arithmetic with SOAP definitions instead of geometry. >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.. In other words you can just make the problem go away with erroneous definitions? Straight line segments don't contain points; points contain straight line segments. Hell points don't even contain curves. ~v~~
From: Lester Zick on 14 Mar 2007 15:13 On 14 Mar 2007 10:10:55 -0700, "VK" <schools_ring(a)yahoo.com> wrote: >On Mar 14, 1:28�am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> Are points and lines not still mathematical objects > > The point is ?? ?? ?? ????? ("to ti en einai") of the infinity. >If you want a definition based on something fresher than Aristotle >then: > The point is nothing which is still something in potention to >become everything. >IMHO the Aristotle-based definition is better, but it's personal. I don't want a definition for points fresher or not than Aristotle. I'm trying to ascertain whether lines are made up of points. >Now after some thinking you may decide to stay with the crossing lines >and hell on the cross-definition issues ;-) The speach is not a >reflection of entities: it is a reflection - of different levels of >quality - of the mind processes. This way a word doesn't have neither >can decribe an entity. The purpose of the word - when read - to trig a >"mentagram", state of mind, as close as possible to the original one - >which caused the word to be written. This way it is not important how >is the point defined: it is important that all people involved in the >subject would think of appoximately the same entity so not say about >triangles or squares. In this aspect crossing lines definition in math >does the trick pretty well. From the other side some "sizeless thingy" >as the definition would work in math as well - again as long as >everyone involved would think the same entity when reading it. ~v~~
From: Lester Zick on 14 Mar 2007 15:15 On 13 Mar 2007 23:21:54 -0700, "Eric Gisse" <jowr.pi(a)gmail.com> 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. Well that's certainly a relief. I thought you said "only their relations to each other". It's certainly good to know that what lines are made up of is not "only a relation" between points and lines. ~v~~
From: Lester Zick on 14 Mar 2007 15:19 On Wed, 14 Mar 2007 10:04:15 -0500, Wolf <ElLoboViejo(a)ruddy.moss> wrote: >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. Only because pedagogy is less your forte than philosophical ineptitude, Wolf. ~v~~
From: Hero on 14 Mar 2007 15:20
Lester Zick wrote: > Hero wrote: > >Randy Poe wrote: > >> Lester Zick wrote: > > >> > The Definition of Points ................. > >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. > Accepted. By Your own reasoning You've got already three points: A center, from which definitional logic starts out into two directions,and two points, where it changes into conclusions. And You can go in circular way in Your picture from conclusions to conclusions ( NB there is more than one diameter and it can be extended to a sphere). Historical, axioms are not the beginning of geometry. You start with full, complex life, understand here a bit and there, proceed from simple things to complex ones and than You look for the most simple and common structure underlying the geometry you have done so far. F.e. You shrink a sphere to it's infinitesimal minimum, which is radius ( and diameter ) zero - and like the smile of Cheshire cat - there You have, what You've looked for. Have a smile Hero |