The Definition of Points
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From: Tony Orlow on 17 Mar 2007 21:51 Bob Kolker wrote: > Tony Orlow wrote:> >> Except that linear order (trichotomy) and continuity are inherent in >> R. Those may be considered geometric properties. > > They can be defined in a purely analytic and algebraic manner starting > with the Peano axioms. While linear order is suggestive of geometric > notions, one can define such order without any geometric content > whatsoever. The order of the postive integers is more temporal than > spatial. Fist comes an integer -then- comes its successor. Etc. Etc. > > Bob Kolker > They can be defined as certain rules regarding the manipulation of certain symbols and strings thereof, such that the resulting strings translate back into geometric statements that still make sense, but the root meanings of those symbols and strings lie in geometry. The axioms concerning points and lines may be stated without explaining what points and lines are within the theory, but that doesn't mean the theory isn't about what points and lines are, as defined by their properties. As far as your idea that the order of integers is "more temporal than spatial", all I can say is that time is a dimension like the spatial ones, except that it always goes forward. It's a moving point. The integers exist all at once, and can be traversed backwards or forwards. So, they're really more spatial than temporal. You're only thinking in the manner that makes Zeno's paradox a mystery. Tony Orlow
From: Virgil on 18 Mar 2007 02:23 In article <45fc6fd6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Except that linear order (trichotomy) and continuity are inherent in R. > Those may be considered geometric properties. If one defines them algebraically, as one often does, are they still purely geometric? > > Tony Orlow
From: Virgil on 18 Mar 2007 02:28 In article <45fc7458(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Hero wrote: > > On 17 Mrz., 22:13, Bob Kolker <nowh...(a)nowhere.com> wrote: > >> Hero wrote: > >> > >>> So with Your kind of geometry You can or You can not tell, that DNA is > >>> a right screw? > >> > >> You can tell that right and left are differnt. > > > > Can You please give me a hint, where in Your geometry or in which of > > Your geometries this is axiomized or where it follows from axioms? > > Or where the plane-reflection is possible? > > > > Thanks > > Hero > > > > > > > > > > A<B -> ~B<A > A<B ^ B<C -> A<C Which is, as usual, irrelevant. Purely in the mathematics of three dimensional Euclidean or Cartesian geometry, there is no way to distinguish a right handed from a left handed system. I understand that there is some fairly esoteric experiment in physics which is alleged to distinguish between them.
From: Hero on 18 Mar 2007 05:05 Tony Orlow wrote: > Hero wrote: > > Bob Kolker wrote: > >> Hero wrote: > > >>> So with Your kind of geometry You can or You can not tell, that DNA is > >>> a right screw? > > >> You can tell that right and left are differnt. > > > Can You please give me a hint, where in Your geometry or in which of > > Your geometries this is axiomized or where it follows from axioms? > > Or where the plane-reflection is possible? > > > Thanks > > Hero > > A<B -> ~B<A > A<B ^ B<C -> A<C > This is written in a math language foreign to me. ~ means NOT -> means Material Implication ^ means AND < means ? ( 3 < 4 is three is smaller than 4) ( the only modell to Your two statements i did find: A; B, C natural numbers, ~ means minus/negative) With friendly greetings Hero
From: Hero on 18 Mar 2007 05:43
Tony Orlow wrote: > Bob Kolker wrote: > > Tony Orlow wrote:> > >> Except that linear order (trichotomy) and continuity are inherent in > >> R. Those may be considered geometric properties. > > > They can be defined in a purely analytic and algebraic manner starting > > with the Peano axioms. While linear order is suggestive of geometric > > notions, one can define such order without any geometric content > > whatsoever. The order of the postive integers is more temporal than > > spatial. Fist comes an integer -then- comes its successor. Etc. Etc. > > > Bob Kolker > > They can be defined as certain rules regarding the manipulation of > certain symbols and strings thereof, such that the resulting strings > translate back into geometric statements that still make sense, but the > root meanings of those symbols and strings lie in geometry. The axioms > concerning points and lines may be stated without explaining what points > and lines are within the theory, but that doesn't mean the theory isn't > about what points and lines are, as defined by their properties. > > As far as your idea that the order of integers is "more temporal than > spatial", all I can say is that time is a dimension like the spatial > ones, except that it always goes forward. It's a moving point. Numbers are born in a huge family, the mother was time (the counting of days into a moon cycle, displayed as the movement of the stars of Nut ) and the father was space ( with features of Geb with calculi to count the sheep and a container to measure the grain). When Bob thinks, that numbers are grown up and do not need their father any more, that they are not about spatial objects any more, so why still call ,,geometry", why not call it ,,number theory"? > The > integers exist all at once, and can be traversed backwards or forwards. Nice. > So, they're really more spatial than temporal. You're only thinking in > the manner that makes Zeno's paradox a mystery. > With friendly greetings Hero |