The Definition of Points
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From: Lester Zick on 17 Mar 2007 18:44 On 17 Mar 2007 10:00:26 -0700, alanmc95210(a)yahoo.com wrote: >On Mar 16, 10:48 am, "ken.quir...(a)excite.com" <ken.quir...(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!). > > Euclid established the foundation for our mathematical deduction >system. As he realized from his Axioms and Postulates, you can't >prove everything. You've got to start with some given Axioms. Lines >and points are among those basic assumptions- A. McIntire In other words there's nothing new under the sun? ~v~~
From: Tony Orlow on 17 Mar 2007 18:46 Bob Kolker wrote: > SucMucPaProlij wrote: > >>> You can develop geometry based purely on real numbers and sets. You >>> need not assume any geometrical notions to do the thing. One of the >>> triumphs of mathematics in the modern era was to make geometry the >>> child of analysis. >>> >> >> >> And it means that lines, planes and points are defined in geometry. >> Make up your mind, Bob! > > Not true. One of the mathematical systems which satisfy Hilbert's Axioms > for plane geometry is RxR , where R is the real number set. Points are > ordered pairs of real numbers. Not a scintilla of geometry there. > > Bob Kolker > >> >> Except that linear order (trichotomy) and continuity are inherent in R. Those may be considered geometric properties. Tony Orlow
From: Lester Zick on 17 Mar 2007 18:48 On Sat, 17 Mar 2007 13:50:01 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >alanmc95210(a)yahoo.com wrote:> >> Euclid established the foundation for our mathematical deduction >> system. As he realized from his Axioms and Postulates, you can't >> prove everything. You've got to start with some given Axioms. Lines >> and points are among those basic assumptions- A. McIntire > >The lines and points are undefined objects. It is the axioms concerning >lines and points that are the basic assumptions. True but pretty much irrelevant if the axioms concerning lines and points are definitive.If they are lines and points aren't reciprocally undefined. And if they're truly undefined axioms concerning lines and points aren't definitive. Six of one half dozen of the other, Bob. ~v~~
From: Lester Zick on 17 Mar 2007 18:52 On Sat, 17 Mar 2007 03:08:34 GMT, Sam Wormley <swormley1(a)mchsi.com> wrote: >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). Like I said a circular point. ~v~~
From: Lester Zick on 17 Mar 2007 19:00
On Sat, 17 Mar 2007 03:10:15 GMT, Sam Wormley <swormley1(a)mchsi.com> wrote: >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 As opposed to a universally length minimizing curve? Or as opposed to a locally length maximizing curve? Or as opposed to a universally length length maximizing curve? I have no idea what this is in aid of. Terminological regressions are a dime a dozen. In the biz they're called buzz words. Happy to use "geodesic" instead of "line" if that's all that's bothering you. There's nothing especially geo- about them. ~v~~ |