The Definition of Points
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From: Lester Zick on 18 Mar 2007 15:25 On Sun, 18 Mar 2007 14:07:11 GMT, "Nam D. Nguyen" <namducnguyen(a)shaw.ca> wrote: >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. Yeah, Nam, I daresay modern math, philosophy, and tradition go hand in hand for those forced to rely on them instead of science. By the way a couple of months back you demanded I demonstrate my contention that conjunctions weren't needed for tautological inference and mechanics. I couldn't do it at the time but I have now in the post "Epistemology 401: Tautological Mechanics" if you're still interested, relying in part on your and/or Virgil's and Wolf's comments regarding the structural linkage between conjunctions "and" and "or" in Boolean logic. So unless you or others have further comments on the subject I consider my own comments on Bob Kolker's anti tautologies closed. ~v~~
From: Lester Zick on 18 Mar 2007 15:28 On Sun, 18 Mar 2007 00:23:39 -0600, Virgil <virgil(a)comcast.net> 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? Until the Peano and suc( ) axioms can produce straight lines out of so many straight line segments except by naive assumptions of truth they are. ~v~~
From: Lester Zick on 18 Mar 2007 15:30 On Sun, 18 Mar 2007 11:59:40 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >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. Not true, Bob, until you can show how the Peano and suc( ) axioms produce colinear straight lines out of so many straight line segments without naive assumptions of truth. > 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 ~v~~
From: Tony Orlow on 18 Mar 2007 15:59 Hero wrote: > Tony Orlow wrote: >> Hero wrote: >>> 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? > >>>> 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) > >> Hi Hero - >> >> '~' indeed means logical "not". '<' means "less than", and can be >> interpreted in a number of ways, such as "is to the left of", "is a >> smaller quantity than", or "is a proper subset of". You may be able to >> think of other examples of transitive and asymmetric relations, such as >> "is inferior to". >> >> :) > > I didn't know, that A<B -> ~B<A means A<B -> ~ ( B<A ). Sorry, I thought it was clear, with A and B numbers, that the negation '~' pertained to the statement "B<A". That's what I meant. > > But how does this fit to left and right screws and to reflection? > It's more like :) and :( Ummm, it doesn't really. I brought it up to answer the original inquiry, as I saw it, where "right and left" were suggested as interpretations of '>' and '<'. As far as I'm concerned, they're interchangeable. > > A reflection is an involution, half of an identity, so to speak: > r ( r ( A ) ) = A > And a right screw is a form, the form of DNA. > Join the four points of a tetra in a consecutive way with three lines, > You have one type of screw. And the other three edges are of the > opposite type. Still more basic is a skew tetra and it's reflection, > they are different to each other, chiral, but without further > differences one can not say which is left-type and which is right- > type. > One abstract example is a cartesian coordinate system, more abstract a > vector-space of three dimensions with a cross-product of vectors: > x cross y = z, and the other: x cross z = y. > So who of the modern math axiomatists ( Hilbert, Tarski..) of geometry > treat this subject? > > With friendly greetings > Hero > PS I was able to think of such relations as " is inferior to". If i > would be inferior to You, i would be more down to earth and Your nose > would be higher up. :) > > I suppose. I didn't mean anything personal, of course. :)
From: Tony Orlow on 18 Mar 2007 16:07
Virgil wrote: > In article <45fd6045(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > How does one prove geometrically what is only defined algebraically? example? |