The Definition of Points
in [Math]
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From: Lester Zick on 14 Mar 2007 01:36 On Wed, 14 Mar 2007 02:22:35 -0000, "OG" <owen(a)gwynnefamily.org.uk> wrote: > >"Lester Zick" <dontbother(a)nowhere.net> wrote in message >news:758ev21t8r8ch5sjuoasdim467bfjvk06q(a)4ax.com... >> On Tue, 13 Mar 2007 16:16:52 -0400, "Jesse F. Hughes" >> <jesse(a)phiwumbda.org> wrote: >> >>>"PD" <TheDraperFamily(a)gmail.com> writes: >>> >>>> Interestingly, the dictionary of the English language is also >>>> circular, where the definitions of each and every single word in the >>>> dictionary is composed of other words also defined in the dictionary. >>>> Thus, it is possible to find a circular route from any word defined in >>>> the dictionary, through words in the definition, back to the original >>>> word to be defined. >>> >>>The part following "Thus" is doubtful. It is certainly true for some >>>words ("is" and "a", for instance). It is almost certainly false >>>for some other words. I doubt that if we begin with "gregarious" and >>>check each word in its definition, followed by each word in those >>>definitions and so on, we will find a definition involving the word >>>"gregarious". >>> >>>Here's the start: >>> >>>gregarious >>> adj 1: tending to form a group with others of the same kind; >>> "gregarious bird species"; "man is a gregarious >>> animal" [ant: ungregarious] >>> 2: seeking and enjoying the company of others; "a gregarious >>> person who avoids solitude" >>> >>>(note that the examples and antonym are not part of the definition!) >> >> An interesting point. One might indeed have to go a long way to >> discern the circularity. However my actual contention is that this >> variety of circularity is quite often used by mathematikers to conceal >> an otherwise orphan contention that lines are constituted of points. >> > >What you call 'orphan' is in fact 'abstract', as points necessarily are. You mean points are abstract from the intersection of lines? Or that the composition of lines is abstract from points? Curious I must say. ~v~~
From: Lester Zick on 14 Mar 2007 01:44 On Wed, 14 Mar 2007 01:37:21 +0100, "�u�Mu�PaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >> 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. >> > >And I am still missing the point. You can't learn all at once. If someone tells >you that line is made of points and point is intersection of two lines you can >accept it if you don't know anything better. > >We know better that this and we don't have to accept this definition of point >and line. You don't have to accept anything. It might be nice however if you had some tenable alternative to suggest. Are you suggesting lines are not made up of points and the intersection of lines does not define a point? Or are you suggesting we just ignore the problem because modern mathematikers are too lazy or stupid to resolve it? ~v~~
From: Lester Zick on 14 Mar 2007 01:46 On Wed, 14 Mar 2007 02:20:02 +0100, "�u�Mu�PaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >> Bucky Fuller quoted an author who said: science is an attempt to put the >> facts of experience in order. > >And I agree with this. > >>Does the tetrahedron create 4 vertexes, 6 >> edges, and 4 faces, or is it created by them? The axiomatic method of >> classical Greek geometry begins with the point. Bucky rejected the >> axiomatic method. He said you can't begin with less than the tetrahedron. >> > >I really don't know if you can't begin with less than the tetrahedron but I know >that you must begin somewhere. Beginning is just one point of your journey and >after you choose from where to begin you can go in any direction. > >You can start from the point and create tetrahedron or you can analyze >tetrahedron and get to point. At the end you will have both tetrahedron and >point. Yes but those represent the intersection of lines. What I'm asking is whether lines are composed of points. ~v~~
From: Lester Zick on 14 Mar 2007 01:50 On Tue, 13 Mar 2007 23:40:39 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: > >"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. Which is all just swell. So now the question I posed becomes are abstract lines made up of abstract points? >Is line made of points? >If you don't define term "made of" and use it without too much thinking you can >say that: Why don't you ask Bob Kolker. He seems to think lines are "made up" of points, abstract or otherwise. I'm not quite clear about how he thinks lines are "made up" of points but he nonetheless seems to think they are. >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. Frankly I prefer to question things before I waste time learning them. ~v~~
From: Lester Zick on 14 Mar 2007 01:52
On 13 Mar 2007 15:37:50 -0700, "Hero" <Hero.van.Jindelt(a)gmx.de> wrote: >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? Who said anything about half a perimeter, Hero? I believe the ratio pi is between the full circumference of a circle and its diameter. ~v~~ |