The Definition of Points
in [Math]
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From: Nam D. Nguyen on 18 Mar 2007 10:07 Bob Kolker wrote: > Hero wrote: > >> 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"? > > Like Tevyeh in -Fiddler on the Roof- says: Tradition! > > Modern math has outgrown its parents and gone far beyond them, like any > successful Son. Agree. Except that like its parents, "any" Modern math would eventually "settle down" and would have its own successful Sons, who would outgrow it. > > Bob Kolker >
From: Wolf on 18 Mar 2007 12:51 Virgil wrote: > 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. The experiment merely shows that there are in fact left- and right- handed thingummies. It does not tell you which is which. "Right" and "left" are, as VK pointed out, purely arbitrary terms. We learn early on which word to use for which hand, and as observation and experience show, it ain't easy to learn arbitrary terms. It was once explained to me that the right hand is the one with the thumb on the left, and the left hand is the one with the thumb on the right. Which nicely sums um VK's point. ;-) -- Wolf "Don't believe everything you think." (Maxine)
From: Tony Orlow on 18 Mar 2007 11:52 Virgil wrote: > 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 One may express them algebraically, but their truth is derived and justified geometrically.
From: Tony Orlow on 18 Mar 2007 11:55 Virgil wrote: > 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 a mathematical point of view, it's all relative, and arbitrary, which direction is "right" or "left". It's just a matter of transitive asymmetric order relations. This applies to "less than" in the quantitiative sense, as well as "proper subset". The statements above apply to both.
From: Bob Kolker on 18 Mar 2007 11:59
Tony Orlow wrote: > > One may express them algebraically, but their truth is derived and > justified geometrically. At an intuitive level, but not at a logical level. The essentials of geometry can be developed without any geometric interpretations or references. Similarly algebraic systems (rings) can be derived from affine spaces geometry by using similar triangles to develop products from proportions. Bob Kolker |