The Definition of Points
in [Math]
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From: Lester Zick on 18 Mar 2007 14:21 On Sun, 18 Mar 2007 00:43:44 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >"Lester Zick" <dontbother(a)nowhere.net> wrote in message >news:ilrov2l07utt9j1ma26lm7stptkh28a9rg(a)4ax.com... >> On 17 Mar 2007 09:40:22 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> wrote: >> >>>Every abstract concept exists as a concept. What does the >>>size of the universe have to do with that? They don't >>>take up any space. >> >> What does the space concepts take up have to do with their truth? >> > >Every idea/concept/definition/axiom must be expressed in some physical form - it >must be formulated. > > > >You can't express concept that has more elements/parts/whatever than number of >atoms in universe. > > > >If you have concept that takes all but one atom then you can't tell if it is >true or not - you don't have enough resources to do so but you still have one >extra atom that you probably can't use for anything :))))) This would seem to be pretty much a recapitulation of Muekenheim's perspective on definitions etc. (whether original with him or not I can't say). I disagree with it because definitions do not require physical implementation to define whatever they refer to. For example the ratio between the diameter and circumference of a circle is well defined without any conceivable physical implementation. This seems absurd to me on the face of it. ~v~~
From: Hero on 18 Mar 2007 14:23 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 ). But how does this fit to left and right screws and to reflection? It's more like :) and :( 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. :)
From: Virgil on 18 Mar 2007 14:57 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?
From: Virgil on 18 Mar 2007 15:01 In article <45fd60e3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > 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. TO misses the point once more! There is no such thing as a left-handed or right-handed system in 2 dimensions, the difference between them requires 3 dimensions.
From: Virgil on 18 Mar 2007 15:05
In article <45fd63b5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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. > > > > Bob Kolker > > > > That is like saying your mind has outgrown your body, so you no longer > need to eat or breathe. The language of math is the more abstract > aspect, but the geometry of it is still the basis of its truth. The natural numbers, with their arithmetic, do not owe anything to geometry. Even Euclid separated them. |