The Definition of Points
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From: Lester Zick on 15 Mar 2007 18:30 On Thu, 15 Mar 2007 08:02:13 -0400, Bob Kolker <nowhere(a)nowhere.com> 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. But I think, Bob, the difference is that Euclid would willingly have adopted more appropriate definitions if they were to be had. Whereas modern mathematikers just pretend their circular definitions are true regardless and self righteously proceed accordingly. ~v~~
From: Lester Zick on 15 Mar 2007 18:42 On Thu, 15 Mar 2007 09:38:13 -0400, Bob Kolker <nowhere(a)nowhere.com> 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. So you're claiming lines are made up of points, Bob, or not? I mean if they aren't then you have no business constructing arguments based on SOAP's. But if you are then you yourself are appealing to circular regressions to support those arguments. ~v~~
From: Lester Zick on 15 Mar 2007 18:46 On Thu, 15 Mar 2007 13:59:08 GMT, Sam Wormley <swormley1(a)mchsi.com> wrote: >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. However it does not suffice for the definition of lines and arguments, proofs, and justifications based on such assumptions. Defining points is hardly essential to definition of lines based on such definitions. ~v~~
From: Lester Zick on 15 Mar 2007 18:54 On Thu, 15 Mar 2007 11:38:50 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >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. As is all of your logic, Bob. >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. As a guess not bad. As a mathematical assumption pretty awful. >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. Categorical system? What categorical system? A system with nothing more than empirical assumptions of truth to guide it? And that makes you feel warm and fuzzy, Bob? Fuzzy I can understand. Most everything you say is fuzzy. But warm? >This is a point (sic!) that Lester Zick is genetically incapable of >grasping. In your position, Bob, I might be a little more circumspect when talking genetics instead of mathematics. You're hardly qualified. ~v~~
From: Lester Zick on 15 Mar 2007 18:57
On Thu, 15 Mar 2007 12:11:36 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> 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. >> > >I think that you think that mathematikers are stupid Lazy and/or stupid. Six of one, half dozen of the other. > and it has nothing to do with lines and point. Well thanks. I certainly appreciate the liners and pointer. >I only know that they are convergent because they are limited and monotone but >this is subject for another topic :)))) Let's hope so. ~v~~ |