The Definition of Points
in [Math]
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From: Lester Zick on 16 Mar 2007 21:02 On Fri, 16 Mar 2007 13:41:43 -0700, Bob Cain <arcane(a)arcanemethods.com> wrote: >Lester Zick wrote: > >> The technique of unambiguous definition and the definition of truth is >> simply to show that all possible alternative are false. Empirics and >> mathematikers generally prefer to base their definitions on >> undemonstrable axiomatic assumptions of truth whereas I prefer to base >> definitions of truth on finite mechanical tautological reduction to >> self contradictory alternatives. The former technique is a practice in >> mystical insight while the latter entails exhaustive analysis and >> reduction in purely mechanical terms. > >Can you provide a useful physical example of the latter? Hell, even a >useless one would be better than what you've provided so far. "Useful"? "Physical"? Don't know what you mean. Now if you'd just asked for a "true" example it would be a different matter. >What won't suffice is any example that defies consensus understanding. "You" being the consensus? Oh I can hardly wait. > If you provide a symbol string that only you can "understand" it >won't satisfy the challenge because it has no practical value. Are there any other rules you have in mind to determine the practicality of what you demand of me? C'mon now, stringellow, don't be shy. Last time I asked how to mechanize angular momentum as I recall you suggested a piece of string. Not very scientific but very practical.How about a piece of string for a useful, physical, example? ~v~~
From: Lester Zick on 16 Mar 2007 21:04 On 16 Mar 2007 10:48:47 -0700, "ken.quirici(a)excite.com" <ken.quirici(a)excite.com> wrote: >On Mar 13, 1:52 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> The Definition of Points >> ~v~~ >> >> In the swansong of modern math lines are composed of points. But then >> we must ask how points are defined? However I seem to recollect >> intersections of lines determine points. But if so then we are left to >> consider the rather peculiar proposition that lines are composed of >> the intersection of lines. Now I don't claim the foregoing definitions >> are circular. Only that the ratio of definitional logic to conclusions >> is a transcendental somewhere in the neighborhood of 3.14159 . . . >> >> ~v~~ > >My impression is that Euclid defined a line, not in terms of points, >and never claimed a line was made up of points, but defined a line as >a geometrical object that has only the property of extensibility >(length, >where length can be infinite). > >He uses points in his proofs specifically as intersections of lines, >if I >remember correctly, and makes no attempt at describing or >explaining their density in a line. (You gotta lot of 'splainin to do, >Euclid!). Yeah, wherever did modern math get its quaint notions? No doubt somewhere along the way. Which way of course remains an issue of historical investigation. ~v~~
From: Sam Wormley on 16 Mar 2007 23:05 Lester Zick wrote: > On Fri, 16 Mar 2007 04:05:59 GMT, Sam Wormley <swormley1(a)mchsi.com> > wrote: > >> 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? > > U.S. Naval Academy, Annapolis, MD. 1966 BSME. I'm sure they can > provide cv's to such worthy souls.Finished playing trivial pursuit now > and may we return to discussing the problem at hand or would you > prefer further essays on educational effluvia? > > ~v~~ Engineers should know better!
From: Sam Wormley on 16 Mar 2007 23:08 Lester Zick wrote: > On Fri, 16 Mar 2007 04:09:49 GMT, Sam Wormley <swormley1(a)mchsi.com> > wrote: > >> 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 > > But is it a circular point, Sam? > > ~v~~ No--it is a point (0-dimensional mathematical object) with located with UTM easting, northing, elevation and time (UTC).
From: Sam Wormley on 16 Mar 2007 23:10
Lester Zick wrote: > On Fri, 16 Mar 2007 04:13:10 GMT, Sam Wormley <swormley1(a)mchsi.com> > wrote: > >> 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". > > So instead of lines, straight lines, and curves, Sam, now we're > discussing geodesics, straight geodesics, and curved geodesics? Pure > terminological regression. Not all that much of an improvement. > > ~v~~ locally length-minimizing curve |