The Definition of Points
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From: Ben newsam on 29 Apr 2007 17:41 On Sun, 29 Apr 2007 12:07:52 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Sun, 29 Apr 2007 01:06:11 +0100, Ben newsam ><ben.newsam.remove.this(a)gmail.com> wrote: >>I challenge you to show me one reference to the phrase "truth in >>mechanically reduced universal terms". Relevant literature indeed! > >Come, come, Ben. You could look up any number of my posts on the >subject. I mean this isn't rocket science. So, you can't, then. I thought not. Because there aren't any, are there? > The idea of universal truth >I'm working on is that universal alternatives to anything universally >self contradictory are perforce universally true. Self contradictions >such as "A not A" are parochially but not universally false because >they only concern "A". Self contradictions only become universally >reduced when they concern the mechanism for talking alternatives in >general as by "not" "contradiction" "differences" or "alternatives". > >Or if you want a practical applied example of mechanical reduction >just look up my "Epistemology 401: Tautological Mechanics" which shows >the origin of boolean conjunctions solely in terms of "not". You are Dagenham mad, IMO. If you want to know what that means, google on "two stops beyond barking". >>>>> You're mighty long on arbiter dicta and mighty short on >>>>>demonstrations of truth for what you say. >>>> >>>>What does "arbiter dicta" mean? Do you mean "obiter dicta"? >>> >>>"Arbiter" not "obiter". Do you imagine you're the only person in the >>>world with a dictionary? >> >>I just wondered what "arbiter dicta" meant. Not the same as "obiter >>dictum" or "obiter dicta", I wouldn't have thought. Do you think it's >>the same? > >Of course not. "Obiter dicta" refer to incidental remarks. "Arbiter >dicta" refer to "judicial dicta" in my own estimation equivalent to >"judicial fiats". However my dictionary shows no reference to the >phrase "arbiter dictum" and I'm not sure where I picked up the usage. Hahaha. Indeed.
From: Lester Zick on 29 Apr 2007 19:27 On 29 Apr 2007 14:29:12 -0700, Rock Brentwood <markwh04(a)yahoo.com> wrote: >On Mar 13, 12:52 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> The Definition of Points >> ~v~~ >> In the swansong of modern math lines are composed of points. > >In modern mathematics, lines and other manifolds are NOT defined as >point sets, but rather the opposite way from top down in terms of >their function spaces. The general formalism allows for "geometries" >that don't even have point sets (i.e. the non-commutative geometries). So lines are definitely and demonstrably not composed of points? >The structure of modern differential geometry and the modern theory of >manifolds requires only the algebra C^{infinity}(M) of the >C^{infinity} functions over a given manifold M as input. The only >place that any notion of point set ever enters into consideration is >in providing for a representation of C^{infinity}(M). >However, not even this is needed. The algebras C^{infinity}(M) for all >manifolds are equivalently defined as commtative C-* algebras. In >fact, one can go in reverse: given a commutative C-* algebra A, there >is a well-established procedure to construct from it the manifold M >for which A = C^{infinity}(M). > >However, the same formal apparatus underlying differential geometry, >manifold theory, etc.; applies regardless of where A comes from or >what kind of algebra it is -- even if A is a non-commutative algebra. I'm not sure how all this aids in the definition of points or in the definition of lines in terms of points. Last time the subject arose I was assured SOAP operas were the answer to all questions. Your comments on the subject seem to rely on "infinity" for which I have yet to see a true definition and on the idea of top down definition which strikes me as peculiar unless you begin with 3D manifolds. In other words if there is no upper limit to manifold dimensionality where is the "top" to begin the top down analysis of function spaces? Points have always appeared to provide a convenient starting or stopping place in terms of zero dimensionality regardless of whether the approach was top down or bottom up. However I don't see the bottom up approach as viable. It just seems to provide an approach parallel to the axiomatic extrapolation of natural numbers by means of Peano and suc( ) axioms by addition. Further even if top down procedures for manifold definition of lines are followed points are nonetheless defined at the intersection of lines whether or not lines are defined in terms of those points. And finally while I agree in general with the top down extrapolation, absent any absolute top you're at best left only with some dimensional stopping point and a method of consistency between and among the manifolds and function spaces which doesn't answer or even really address the question of why spaces are what they are geometrically. In fact given a point origin of zero dimensionality but no ultimate limit to dimensionality I can readily sympathsize with those who approach the problem from the ground up instead of the top down. ~v~~
From: Lester Zick on 29 Apr 2007 19:28 On Sun, 29 Apr 2007 22:35:36 +0100, Ben newsam <ben.newsam.remove.this(a)gmail.com> wrote: >On Sun, 29 Apr 2007 11:28:06 -0700, Lester Zick ><dontbother(a)nowhere.net> wrote: > >>When people sense there is something they >>should be able to spell out out can't the mind pancs and vituperation >>ensues. > >PKB! Right back atcha! ~v~~
From: Bob Kolker on 29 Apr 2007 19:51 Lester Zick wrote:> > So lines are definitely and demonstrably not composed of points? He didn't say that. He said it is possible to describe manifolds by means of function spaces. Bob Kolker
From: Lester Zick on 30 Apr 2007 13:50
On Sun, 29 Apr 2007 19:51:51 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Lester Zick wrote:> >> So lines are definitely and demonstrably not composed of points? > >He didn't say that. He said it is possible to describe manifolds by >means of function spaces. What he said in part, Bob, was: >"In modern mathematics, lines and other manifolds are NOT defined as >point sets, but rather the opposite way from top down in terms of >their function spaces . . ." And I think my question is perfectly reasonable in that context. In other words I just requested clarification as to whether lines are or can be defined as "the set of all points . . ." or not. Saying one can define manifolds by means of function spaces doesn't address that particular point that I can see. It's my understanding that you believe lines are constituted of the "set of all points . . ." and integration via the calculus applies to such points for the purpose of constituting lines. So I suppose what I'm really asking is whether what Rock is talking about represents an issue in mathematical definitions considered as pure formalisms in terms of manifolds and their function spaces or whether his comments directly concern relations between manifolds. If his comments are just directed at definitional relations between manifolds and their function spaces I don't see that it addresses relations between manifolds and that was the purpose of my question. On the other hand if his comment regarding the top-down nature of mathematical definition was just intended to refer to function spaces and manifolds alone without reference to relations between manifolds then I don't see how it addresses relations between lines and points. And simply commenting on the mathematical definition of manifolds in terms of function spaces doesn't really contribute much to resolution of that issue which was the purpose of the thread to begin with. ~v~~ |