The Definition of Points
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From: Lester Zick on 17 Mar 2007 19:34 On Sat, 17 Mar 2007 12:03:44 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Tony Orlow wrote: > >> Yes, the relationship between points and lines is rather codependent, >> isn't it? I looked at some of the responses, and indeed, one can define >> points as tuples of coordinates, but of course, that all depends on >> defining a set of dimensions as a space to begin with, each dimension >> constituting an infinite line along which that coordinate is defined. In >> language, both points and lines are taken as primitives, since their >> properties are not rooted in symbols and strings, but geometry. So, we >> may be left with the question as to what the primitives of geometry >> really are, sets of points, or sequences of lines. That's the conundrum >> right, that differences and differences between differences are lines, >> and not points? :) > >You can develop geometry based purely on real numbers and sets. You need >not assume any geometrical notions to do the thing. One of the triumphs >of mathematics in the modern era was to make geometry the child of analysis. So you can develop geometry without assuming any geometrical notions? I don't see any evidence modern math has managed to any thing of the kind. ~v~~
From: Lester Zick on 17 Mar 2007 19:36 On Sat, 17 Mar 2007 17:54:17 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >> >> You can develop geometry based purely on real numbers and sets. You need not >> assume any geometrical notions to do the thing. One of the triumphs of >> mathematics in the modern era was to make geometry the child of analysis. >> > >And it means that lines, planes and points are defined in geometry. >Make up your mind, Bob! Yeah, Bob, please define circles, planes, etc. in terms of SOAP operas without reference to geometrical notions. ~v~~
From: Lester Zick on 17 Mar 2007 19:39 On Sat, 17 Mar 2007 11:02:30 -0600, "nonsense(a)unsettled.com" <nonsense(a)unsettled.com> wrote: >SucMucPaProlij wrote: > >>>You can develop geometry based purely on real numbers and sets. You need not >>>assume any geometrical notions to do the thing. One of the triumphs of >>>mathematics in the modern era was to make geometry the child of analysis. >>> >> >> >> And it means that lines, planes and points are defined in geometry. >> Make up your mind, Bob! > >No they're not. "The locus of all points...." You mean kinda like "a circle is the locus of all points on a plane equidistant from any point", Bob? Good to know we don't need geometry anymore. I think the more pertinent question is why we need you? ~v~~
From: Lester Zick on 17 Mar 2007 19:40 On Sat, 17 Mar 2007 18:24:48 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: ><nonsense(a)unsettled.com> wrote in message >news:b71c5$45fc1f22$4fe72e0$21877(a)DIALUPUSA.NET... >> SucMucPaProlij wrote: >> >>>>You can develop geometry based purely on real numbers and sets. You need not >>>>assume any geometrical notions to do the thing. One of the triumphs of >>>>mathematics in the modern era was to make geometry the child of analysis. >>>> >>> >>> >>> And it means that lines, planes and points are defined in geometry. >>> Make up your mind, Bob! >> >> No they're not. "The locus of all points...." >> >> >> > >You can't define points and lines with numbers and sets? >Try it. It is not hard. Oh goodie. I can hardly wait. ~v~~
From: Lester Zick on 17 Mar 2007 19:41
On Sat, 17 Mar 2007 13:54:18 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >SucMucPaProlij wrote: >> >> >> You can't define points and lines with numbers and sets? >> Try it. It is not hard. > >Points (in n-dimesnsional space) are ordered n-tuples of real numbers. That's really helpful. ~v~~ |