The Definition of Points
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From: SucMucPaProlij on 16 Mar 2007 07:40 >>I gave a book suggestion [Sibley's geometry] and a Wikipedia link that >>mirrors what is said in Sibley, plus I already explained that there >>are undefined terms in geometry - and that 'point' is one of them. > > But a line made up of points is not one of them. > and will you share with us your secret definition for points and lines or not?
From: hagman on 16 Mar 2007 10:00 On 13 Mrz., 18:52, 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~~ Please look up the difference between "define" and "determine". In a theory that deals with "points" and "lines" (these are typically theories about geometry), it is usual to leave these terms themselves undefined and to investigate an incidence relation "P on L" (for points P and lines L) with certain properties Then the intersection of two lines /determines/ a point in the sense that IF we have two lines L1 and L2 AND there exists a point P such that both P on L1 and P on L2 THEN this point is unique. This is usually stated as an axiom. And it does not define points nor lines.
From: hagman on 16 Mar 2007 10:26 On 15 Mrz., 23:54, Lester Zick <dontbot...(a)nowhere.net> wrote: > On Thu, 15 Mar 2007 11:38:50 -0400, Bob Kolker <nowh...(a)nowhere.com> > wrote: > > >Sam Wormley wrote: > >The fact that RxR with a metric satisfies the Hilbert Axioms for plane > >geometry implies that points can be taken to be pairs of real numbers. > > As a guess not bad. As a mathematical assumption pretty awful. There's no assumption in here. "RxR satisfies Hilbert axioms for plane geometry" is provable. "Foo satisfies the axioms of a Bar object" means that all theroems of Bar theory are true when interpreted as statements about Foo.
From: SucMucPaProlij on 16 Mar 2007 10:35 "hagman" <google(a)von-eitzen.de> wrote in message news:1174053602.723585.89690(a)l77g2000hsb.googlegroups.com... > On 13 Mrz., 18:52, 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~~ > > Please look up the difference between "define" and "determine". > > In a theory that deals with "points" and "lines" (these are typically > theories about geometry), it is usual to leave these terms themselves > undefined > and to investigate an incidence relation "P on L" (for points P and > lines L) > with certain properties > > Then the intersection of two lines /determines/ a point in the sense > that > IF we have two lines L1 and L2 > AND there exists a point P such that both P on L1 and P on L2 > THEN this point is unique. > This is usually stated as an axiom. > And it does not define points nor lines. > This is interesting observation :)))) But how do you define difference between "define" and "determine"? Can "definition" determine and can "determination" define? Lester Zick has problem with "circular definitions" and you used term "point" in your "determination" to determine it. Maybe you want to say that in definition you can't use term you define to define it and in termination you can use it to determine it. I think it's time to call Determinator :)))) He is the only one who can help us! hahahahahahaha
From: SucMucPaProlij on 16 Mar 2007 10:52
> Please look up the difference between "define" and "determine". > > In a theory that deals with "points" and "lines" (these are typically > theories about geometry), it is usual to leave these terms themselves > undefined > and to investigate an incidence relation "P on L" (for points P and > lines L) > with certain properties > > Then the intersection of two lines /determines/ a point in the sense > that > IF we have two lines L1 and L2 > AND there exists a point P such that both P on L1 and P on L2 > THEN this point is unique. > This is usually stated as an axiom. > And it does not define points nor lines. > Here is one problem that is much biger that definition of point. How do you define "definition"? If you have a definition of "definition" you can't prove that it is a really stuff becouse you don't know what definition is before you defined it. I can as well say that "definition" is a big red apple and it is true by definition. You can't prove that "definition" is not a big red apple becouse you don't have definition of "definition" other then this. Since I defined "definition" first, from now on "definition" is big red apple :)))) |