The Definition of Points
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From: Bob Kolker on 15 Mar 2007 08:09 Lester Zick wrote: > > Actually I'm interested in whether vectors exist and have > constituents. Yes they do, in the mathematical sense. They lead to a successful description of forces for one thing. The constituents of a vector are length and direction. > > Neither. The end points contain the line segment. That's how the line > segment is defined. That is admirably correct. And given the end points of a segment one can readily define the set of points that make up the line determined by the end points of the segment. Learn some analytic geometry to see how. > > Of course another way to answer this is to ask what defines the line > segment to begin with. A pair of points. > > >> and ask what the limit >>of the line segment is. > > > When it gets to zero do be sure to let us know. I see you are channeling Bishop Berkeley again. All of hist objections have been answered by the theory of hyperreal numbers on which non-standard analysis is based. Berkeley raised cogent objections to Newton and Leibniz which were finally and complete answered in the late 1950's by Abraham Robinson. By the way, if lines (or other curves) do not consist of the points on them, what do they consist of? Bob Kolker
From: Sam Wormley on 15 Mar 2007 09:21 Bob Kolker wrote: > 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 Give me something better, Bob, or are you arguing there isn't a better definition (if you can call it that).
From: Bob Kolker on 15 Mar 2007 09:38 Sam Wormley wrote: > > Give me something better, Bob, or are you arguing there isn't a better > definition (if you can call it that). You are asking for a definition of an undefined term. There is nothing better. If one finds a definition of point it will have to be based on something undefined (eventually) otherwise there is circularity or infinite regress. We can't have mathematics based on turtles all the way down. There has to be starting point. Here is my position. If an alleged definition is no where used in proofs it should be eliminated or clear marked as an intuitive insight. Bob Kolker
From: Sam Wormley on 15 Mar 2007 09:59 Bob Kolker wrote: > Sam Wormley wrote: > >> >> Give me something better, Bob, or are you arguing there isn't a better >> definition (if you can call it that). > > You are asking for a definition of an undefined term. There is nothing > better. If one finds a definition of point it will have to be based on > something undefined (eventually) otherwise there is circularity or > infinite regress. We can't have mathematics based on turtles all the way > down. There has to be starting point. > > Here is my position. If an alleged definition is no where used in proofs > it should be eliminated or clear marked as an intuitive insight. > > Bob Kolker > Fair enough--However, for conceptualizing "defining" a point with coordinate systems suffices.
From: Bob Kolker on 15 Mar 2007 11:38
Sam Wormley wrote: > Fair enough--However, for conceptualizing "defining" a point > with coordinate systems suffices. Yes indeed. Point is a tuple of elements from a ring. But even these have be grounded upon undefined terms. The fact that RxR with a metric satisfies the Hilbert Axioms for plane geometry implies that points can be taken to be pairs of real numbers. The fact that the Hilbert Axioms for the plane is a categorical system makes me feel warm and fuzzy about identifying a line with a set of points (number pairs) that satisfy a first degree equation in the co-ordinate variables. This is a point (sic!) that Lester Zick is genetically incapable of grasping. Bob Kolker |