The Definition of Points
in [Math]
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From: Sam Wormley on 16 Mar 2007 00:04 Lester Zick wrote: > 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~~ 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: Sam Wormley on 16 Mar 2007 00:05 Lester Zick wrote: > On Thu, 15 Mar 2007 13:21:19 GMT, Sam Wormley <swormley1(a)mchsi.com> > wrote: > >> 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). > > Well we can always pretend there is something better but that doesn't > necessarily make it so. I think modern mathematikers have done such a > first rate job at the pretense that it's become a doctrinal catechism. > > ~v~~ What's your formal education in mathemaitcs, Lester?
From: Sam Wormley on 16 Mar 2007 00:09 Lester Zick wrote: > On Thu, 15 Mar 2007 02:37:12 GMT, Sam Wormley <swormley1(a)mchsi.com> > wrote: > >> 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." > > Not clear what your point is here, Sam. If the so called mathematical > machinery used to deal with points is nothing but circular regressions > then I certainly agree that machinery would really be pretty slippery. > > ~v~~ Here's the point where I reside, Lester: 15T 0444901m 4653490m 00306m NAD27 Fri Mar 16 04:09:09 UTC 2007
From: Sam Wormley on 16 Mar 2007 00:13 Lester Zick wrote: > > I don't agree with the notion that lines and straight lines mean the > same thing, Sam, mainly because we're then at a loss to account for > curves. Geodesic http://mathworld.wolfram.com/Geodesic.html "A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration".
From: Eric Gisse on 16 Mar 2007 00:15
On Mar 15, 4:01 pm, Bob Kolker <nowh...(a)nowhere.com> wrote: > Eric Gisse wrote: > > On Mar 15, 2:54 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > > [...] > > > What is your background in mathematics, Lester? > > You have asked: "what is the empty set". The empty set was my only source of amusement in my proofs class. > > Bob Kolker > > |