The Definition of Points
in [Math]
|
Prev: Guide to presenting Lemma, Theorems and Definitions
Next: Density of the set of all zeroes of a function with givenproperties
From: Lester Zick on 13 Mar 2007 18:37 On 13 Mar 2007 12:55:32 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: >On Mar 13, 1:52 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> The Definition of Points >> ~v~~ >> >> In the swansong of modern math lines are composed of points. But then >> we must ask how points are defined? However I seem to recollect >> intersections of lines determine points. But if so then we are left to >> consider the rather peculiar proposition that lines are composed of >> the intersection of lines. Now I don't claim the foregoing definitions >> are circular. Only that the ratio of definitional logic to conclusions >> is a transcendental somewhere in the neighborhood of 3.14159 . . . >> > >The modern axiomization of geometry due to Hilbert leaves >points, lines, and planes undefined. Probably just as well. I think what we have to consider however is whether lines are made up of points and the intersection of lines defined by points defines points. Or whether perhaps Hilbert and others were a little too preoccupied with problematic axioms and circular logic to ascertain the actual truth of what he didn't define. > In fact, he famously >said about this construction: "One must be able to say at >all times-instead of points, lines, and planes---tables, >chairs, and beer mugs." > >In other words, despite whatever intuition and inherent >meaning we might ascribe to these things has no effect >on the mathematical structure. > >No doubt Lester will find this approach lacking and I mainly find circular regressions pretty much meaningless and unable to support mathematikers' contention that points constitute lines. >assert he has a superior axiomization built up from "infinite >epistomological ontologies of finite tautological >regression" or something equally meaningless. Aha, Randy. As usual you lie like a flatfish. Unlike mathematikers I don't use axioms. It's just that I have an unusual penchant for truth as opposed to guesses and assumptions typifying mathematikers. ~v~~
From: Hero on 13 Mar 2007 18:37 Randy Poe wrote: > Lester Zick wrote: > > > The Definition of Points > > ~v~~ > > > In the swansong of modern math lines are composed of points. But then > > we must ask how points are defined? However I seem to recollect > > intersections of lines determine points. But if so then we are left to > > consider the rather peculiar proposition that lines are composed of > > the intersection of lines. Now I don't claim the foregoing definitions > > are circular. Only that the ratio of definitional logic to conclusions > > is a transcendental somewhere in the neighborhood of 3.14159 . . . > > The modern axiomization of geometry due to Hilbert leaves > points, lines, and planes undefined. In fact, he famously > said about this construction: "One must be able to say at > all times-instead of points, lines, and planes---tables, > chairs, and beer mugs." > > In other words, despite whatever intuition and inherent > meaning we might ascribe to these things has no effect > on the mathematical structure. > A mathematical structure, which is the same for points, lines, and planes as well as for tables, chairs, and beer mugs, seems to me not very far advanced, there is not even a difference between an object with a volume and one without. Take any object of volume, a chair. It's center of gravity is a point. Rotate the chair, the axis of rotation is a line. Let the axis spin (precession), so every part of the chair is moving with the exception of one "thing", which is at rest - a point. So points really exists, not as matter or stuff, but as an aspect of things. Just describe them. This is possible in different ways, f.e: one point is an invariant in a precessing rotation. With friendly greetings Hero PS. Lester, You claim > > ...that the ratio of definitional logic to conclusions > > is a transcendental somewhere in the neighborhood of 3.14159 . . . So definitional logic behaves like a radius extending to conclusions like half a circle. Just reverse Your way and search for the center and You have defined Your starting point. Nice. NB, why half a perimeter?
From: SucMucPaProlij on 13 Mar 2007 18:40 "Lester Zick" <dontbother(a)nowhere.net> wrote in message news:2t8ev292sqinpej146h9b4t4o4n9pvr8c2(a)4ax.com... > On Tue, 13 Mar 2007 20:48:34 +0100, "SucMucPaProlij" > <mrjohnpauldike2006(a)hotmail.com> wrote: > >>> In the swansong of modern math lines are composed of points. But then >>> we must ask how points are defined? However I seem to recollect >>> intersections of lines determine points. But if so then we are left to >>> consider the rather peculiar proposition that lines are composed of >>> the intersection of lines. Now I don't claim the foregoing definitions >>> are circular. Only that the ratio of definitional logic to conclusions >>> is a transcendental somewhere in the neighborhood of 3.14159 . . . >>> >> >>point is coordinate in (any) space (real or imaginary). >>For example (x,y,z) is a point where x,y and z are any numbers. > > That's nice. And I'm sure we could give any number of other examples > of points. Very enlightening indeed. However the question at hand is > whether points constitute lines and whether or not circular lines of > reasoning support that contention. > >>line is collection of points and is defined with three functions >>x = f(t) >>y = g(t) >>z = h(t) >> >>where t is any real number and f,g and h are any continous functions. >> >>Your definition is good for 10 years old boy to understand what is point and >>what is line. (When I was a child, I thought like a child, I reasoned like a >>child. When I became a man, I put away childish ways behind me.....) > > Problem is you may have put away childish things such as lines and > points but you're still thinking like a child. > > Are points and lines not still mathematical objects and are lines made > up of points just because you got to be eleven? > > ~v~~ hahahahaha the simple answer is that line is not made of anything. Line is just abstraction. Properties of line comes from it's definition. Is line made of points? If you don't define term "made of" and use it without too much thinking you can say that: line is defined with 3 functions: x = f(t) y = g(t) z = h(t) where (x,y,z) is a point. As you change 't' you get different points and you say that line is "made of" points, but it is just an expressions that you must fist understand well before you question it.
From: Clifford Nelson on 13 Mar 2007 19:46 In article <et77vp$prt$1(a)ss408.t-com.hr>, "�u�Mu�PaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: > > Bucky Fuller's kindergarten teacher gave her class semi-dried peas and > > toothpicks to build "structures". All of the kids built structures that > > had 90 degree angles like squares and cubes except Bucky. He could not > > see because he didn't have a pair of glasses yet, and felt that the > > triangle and tetrahedron were strong, but the square and cube did not > > hold their shape. He got a patent for the structure he made about 60 > > years later. He thought like a child for about 60 years and started to > > write Synergetics. 15 years later the first volume was published. > > > > it is nice story but nothing more. > It is one of the stories that fits in "how to become rich and successful" > book, > chapter "Change the way you think and all your problems will be solved" You missed the point in a discussion about points. The point is that some things are primary, first, simple. The beginning geometry text books say that the tetrahedron is advanced "solid" geometry. Bucky Fuller discovered it when he was four years old because he could not see. Geometry is taught in a way that psychiatrists would call an example of, in layman's terms, a "thought disorder". Ditto for geometry's "points". If RBF had spelled out the obvious conclusions between the lines, sections, and chapters in Synergetics, I'll bet he wouldn't have been able to get his books published at all. Cliff Nelson Dry your tears, there's more fun for your ears, "Forward Into The Past" 2 PM to 5 PM, Sundays, California time, http://www.geocities.com/forwardintothepast/ Don't be a square or a blockhead; see: http://bfi.org/node/574 http://library.wolfram.com/infocenter/search/?search_results=1;search_per son_id=607
From: Eric Gisse on 13 Mar 2007 20:18
On Mar 13, 9:52 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > The Definition of Points > ~v~~ > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? However I seem to recollect > intersections of lines determine points. But if so then we are left to > consider the rather peculiar proposition that lines are composed of > the intersection of lines. Now I don't claim the foregoing definitions > are circular. Only that the ratio of definitional logic to conclusions > is a transcendental somewhere in the neighborhood of 3.14159 . . . > > ~v~~ Points, lines, etc aren't defined. Only their relations to eachother. |