The Definition of Points
in [Math]
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From: Lester Zick on 15 Mar 2007 18:26 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~~
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 18:34 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~~
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~~ |