The Definition of Points
in [Math]
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From: Lester Zick on 14 Mar 2007 22:24 On 14 Mar 2007 13:02:00 -0700, "PD" <TheDraperFamily(a)gmail.com> wrote: >On Mar 14, 2:13 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 14 Mar 2007 10:10:55 -0700, "VK" <schools_r...(a)yahoo.com> wrote: >> >> >On Mar 14, 1:28 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> Are points and lines not still mathematical objects >> >> > The point is ?? ?? ?? ????? ("to ti en einai") of the infinity. >> >If you want a definition based on something fresher than Aristotle >> >then: >> > The point is nothing which is still something in potention to >> >become everything. >> >IMHO the Aristotle-based definition is better, but it's personal. >> >> I don't want a definition for points fresher or not than Aristotle. >> I'm trying to ascertain whether lines are made up of points. > >Let's see if I can help. Oh that'll be refreshing for a change. >I believe Lester is asking whether a line is a composite object or an >atomic primitive. Actually I'm interested in whether vectors exist and have constituents. >One way of asking the question is whether a point sits ON a line or >whether the point is part OF the line. Like I said before you're not very good at philosophy but you're much worse at science. >Of course, since both the point and the line are idealizations, >conceptual constructions out of the human mind that don't have any >independent reality, then one could rightly ask why the hell it >matters, since there is no way to verify either statement through an >external discriminator. An external what-inator? Why don't you just call it magic and be done with it? No need to dress it up like a dog's dinner with all the philosophical badinage. You're a mystic. So what? > Lester doesn't believe in external >discriminators anyway, because that is the work of evil empirics, and >he'd rather spend his day mentally diddling away at issues like this. Whereas obviously you don't. >But to provide him with some prurient prose by which to diddle You know, sport, if you were even half as witty as I am that might indeed make you a half wit. However in this instance you're trying too hard and you wind up appearing more trying than witty. >further, let's toss him the idea that we can clearly cleave a line in >two by picking a point (either on the line or part of the line, take >your pick) and assigning one direction to one semi-infinite segment >and the other direction to the other semi-infinite segment -- >sometimes called rays. One can then take one of those rays and cleave >it again, and one of the results will be a line segment, which is >distinguished by having two end *points*. Now the interesting question >is whether those end points are ON the line segment or part OF the >line segment. Neither. The end points contain the line segment. That's how the line segment is defined. > One way to answer this is to take the geometric limit of >one end point approaching the other end point, Of course another way to answer this is to ask what defines the line segment to begin with. > and ask what the limit >of the line segment is. When it gets to zero do be sure to let us know. > That should either settle it or send Lester >into an orgasmic frenzy. Gee with another swell foop you might actually get to the calculus. Of course Newton and Leibniz and probably a thousand other wannabe's are waiting in the wings ahead of you and the other neomathematikers. >> >Now after some thinking you may decide to stay with the crossing lines >> >and hell on the cross-definition issues ;-) The speach is not a >> >reflection of entities: it is a reflection - of different levels of >> >quality - of the mind processes. This way a word doesn't have neither >> >can decribe an entity. The purpose of the word - when read - to trig a >> >"mentagram", state of mind, as close as possible to the original one - >> >which caused the word to be written. This way it is not important how >> >is the point defined: it is important that all people involved in the >> >subject would think of appoximately the same entity so not say about >> >triangles or squares. In this aspect crossing lines definition in math >> >does the trick pretty well. From the other side some "sizeless thingy" >> >as the definition would work in math as well - again as long as >> >everyone involved would think the same entity when reading it. ~v~~
From: Sam Wormley on 14 Mar 2007 22:37 Lester Zick wrote: > Look. If you have something to say responsive to my modest little > essay I would hope you could abbreviate it with some kind of non > circular philosophical extract running to oh maybe twenty lines or > less. Obviously you think lines are made up of points. Big deal. So do > most other neoplatonic mathematikers. > > ~v~~ Hey Lester-- Point http://mathworld.wolfram.com/Point.html A point 0-dimensional mathematical object, which can be specified in n-dimensional space using n coordinates. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his Elements, gave the vague definition of a point as "that which has no part."
From: Sam Wormley on 14 Mar 2007 22:40 Lester Zick wrote: > Straight lines are derivatives of curves. At least according to Newton > and his method of drawing tangents. Tell Euler et al. they can stop > rolling. Euler couldn't even get the definition of angular mechanics > right. > > Hey Lester Line http://mathworld.wolfram.com/Line.html "A line is uniquely determined by two points, and the line passing through points A and B". "A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. A line is sometimes called a straight line or, more archaically, a right line (Casey 1893), to emphasize that it has no "wiggles" anywhere along its length. While lines are intrinsically one-dimensional objects, they may be embedded in higher dimensional spaces".
From: SucMucPaProlij on 15 Mar 2007 07:11 > > Look. If you have something to say responsive to my modest little > essay I would hope you could abbreviate it with some kind of non > circular philosophical extract running to oh maybe twenty lines or > less. Obviously you think lines are made up of points. Big deal. So do > most other neoplatonic mathematikers. > I think that you think that mathematikers are stupid and it has nothing to do with lines and point. I only know that they are convergent because they are limited and monotone but this is subject for another topic :))))
From: Bob Kolker on 15 Mar 2007 08:02
Sam Wormley wrote: > > Hey Lester-- > > Point > http://mathworld.wolfram.com/Point.html > > A point 0-dimensional mathematical object, which can be specified in > n-dimensional space using n coordinates. Although the notion of a point > is intuitively rather clear, the mathematical machinery used to deal > with points and point-like objects can be surprisingly slippery. This > difficulty was encountered by none other than Euclid himself who, in > his Elements, gave the vague definition of a point as "that which has > no part." That really is not a definition in the species-genus sense. It is a -notion- expressing an intuition. At no point is that "definition" ever used in a proof. Check it out. Many of Euclid's "definitions" were not proper definitions. Some where. The only things that count are the list of undefined terms, definitions grounded on the undefined terms and the axioms/postulates that endow the undefined terms with properties that can be used in proofs. Bob Kolker |