The Definition of Points
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From: Lester Zick on 1 Apr 2007 18:53 On Sat, 31 Mar 2007 16:13:21 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Lester Zick wrote: >> You talk about lines as if they were made up of points. >In one model of Euclidean geometry that satisfies all of Hilbert's >Axioms the lines are made up of points. Yeah look, Bob, I really don't care about "models" and Hilbert's axioms or about undefined tables and beer bottles. So your contention that lines are made up of points is completely beside the point. > Furthermore it can be shown >Hilbert's Axioms are categorical, so all models are isometric. So a line >is made up of points in -any- model for Hilbert's Axioms. So why didn't you just jump up and say so when that guy asked for a college level citation to support your contention, Bob? I mean if Hilbert's axioms are so all fired critical to the definition of points and lines and their interrelations why didn't he define them himself instead of leaving the job to lesser mortals such as you? I mean no doubt we can all use a good laugh now and again but I don't really feel very comfortable accepting your word as gospel per say. ~v~~
From: Lester Zick on 1 Apr 2007 19:03 On Sat, 31 Mar 2007 18:15:21 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 30 Mar 2007 12:07:44 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>>>> Those aren't geometrical expressions of addition, but iterative >>>>>>> operations expressed linguistically. >> >>>>>> Which means what exactly, that they aren't arithmetic axioms forming >>>>>> the foundation of modern math? The whole problem is that they don't >>>>>> produce straight lines or colinear straight line segments as claimed. >>>>> Uh, yeah, 'cause they're not expressed gemoetrically. >>>> Well yes. However until you can show geometric expression are point >>>> discontinuous I don't see much chance geometric expression will help >>>> your case any. >>>> >>>> ~v~~ >>> What does point discontinuity in geometry have to do with anything I've >>> said? >> >> You talk about lines as if they were made up of points. >> >> ~v~~ > >I do, and the thread is picking up. And, that's not why. :) > >Or, maybe it is. No it's picking up for the reason I cited previously in my collateral reply above. By fragmenting my replies to the preceeding 800+ line message I inadvertently made it possible for mathematikers to read, reply, and lip synch slogans whilst moving their lips commensurate with their somewhat limited mentalities. However I also insulted orangutans in the process for which I sincerely apologize. > No point lies independent of any space, or it's >insignificant. No point is defined except as different in however many >directions are under consideration. Where points are so defined, they >allow for lines. Thanks for the opinion, Tony. I like Bob's better. It's so complicated it almost sounds like he knows what he's talking about for a change. ~v~~
From: Lester Zick on 1 Apr 2007 19:05 On Sat, 31 Mar 2007 18:15:58 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 30 Mar 2007 12:08:06 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>>>> So, start with the straight line: >> >>>>>> How? By assumption? As far as I know the only way to produce straight >>>>>> lines is through Newton's method of drawing tangents to curves. That >>>>>> means we start with curves and derivatives not straight lines.And that >>>>>> means we start with curved surfaces and intersections between them. >>>>>> >>>>> Take long string and tie to two sticks, tight. >>>> Which doesn't produce straight line segments. >>>> >>>> ~v~~ >>> Yeah huh >> >> Yeah indeed. >> >> ~v~~ > >I meant, "does, too". Except it doesn't even produce decent straight line segments, Tony, much less straight lines. ~v~~
From: Lester Zick on 1 Apr 2007 19:17 On Sat, 31 Mar 2007 18:36:28 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 30 Mar 2007 12:10:12 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>>>> Their size is finite for any finite number of subdivisions. >> >>>>>> And it continues to be finite for any infinite number of subdivisions >>>>>> as well.The finitude of subdivisions isn't related to their number but >>>>>> to the mechanical nature of bisective subdivision. >>>>>> >>>>> Only to a Zenoite. Once you have unmeasurable subintervals, you have >>>>> bisected a finite segment an unmeasurable number of times. >>>> Unmeasurable subintervals? Unmeasured subintervals perhaps. But not >>>> unmeasurable subintervals. >>>> >>>> ~v~~ >>> Unmeasurable in the sense that they are nonzero but less than finite. >> >> Then you'll have to explain how the trick is done unless what you're >> really trying to say is dr instead of points resulting from bisection. >> I still don't see any explanation for something "nonzero but less than >> finite". What is it you imagine lies between bisection and zero and >> how is it supposed to happen? So far you've only said 1/00 but that's >> just another way of making the same assertion in circular terms since >> you don't explain what 00 is except through reference to 00*0=1. >> >> ~v~~ > >But, I do. > >I provide proof that there exists a count, a number, which is greater >than any finite "countable" number, for between any x and y, such that >x<y, exists a z such that x<z and z<y. No finite number of intermediate >points exhausts the points within [x,z], no finite number of >subdivisions. So, in that interval lie a number of points greater than >any finite number. Call |R in (0,1]| "Big'Un" or oo., Well you'll just have to excuse me, Tony, if I don't quite seem to get it unless in accordance with me you take 00 to represent a number of infinitesimals in which case your so called "points" are not points at all but infinitesimals intead. > and move on to the >next conclusion....each occupies how m,uch of that interval? dr. ~v~~
From: Lester Zick on 1 Apr 2007 19:18
On 31 Mar 2007 16:56:16 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >On 1 Apr, 00:36, Tony Orlow <t...(a)lightlink.com> wrote: >> Lester Zick wrote: >> > On Fri, 30 Mar 2007 12:10:12 -0500, Tony Orlow <t...(a)lightlink.com> >> > wrote: >> >> >> Lester Zick wrote: >> >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> >> >>> wrote: >> >> >>>>>> Their size is finite for any finite number of subdivisions. >> >> >>>>> And it continues to be finite for any infinite number of subdivisions >> >>>>> as well.The finitude of subdivisions isn't related to their number but >> >>>>> to the mechanical nature of bisective subdivision. >> >> >>>> Only to a Zenoite. Once you have unmeasurable subintervals, you have >> >>>> bisected a finite segment an unmeasurable number of times. >> >>> Unmeasurable subintervals? Unmeasured subintervals perhaps. But not >> >>> unmeasurable subintervals. >> >> >>> ~v~~ >> >> Unmeasurable in the sense that they are nonzero but less than finite. >> >> > Then you'll have to explain how the trick is done unless what you're >> > really trying to say is dr instead of points resulting from bisection. >> > I still don't see any explanation for something "nonzero but less than >> > finite". What is it you imagine lies between bisection and zero and >> > how is it supposed to happen? So far you've only said 1/00 but that's >> > just another way of making the same assertion in circular terms since >> > you don't explain what 00 is except through reference to 00*0=1. >> >> > ~v~~ >> >> But, I do. >> >> I provide proof that there exists a count, a number, which is greater >> than any finite "countable" number, for between any x and y, such that >> x<y, exists a z such that x<z and z<y. No finite number of intermediate >> points exhausts the points within [x,z], no finite number of >> subdivisions. So, in that interval lie a number of points greater than >> any finite number. Call |R in (0,1]| "Big'Un" or oo., and move on to the >> next conclusion....each occupies how m,uch of that interval? >> >> 01oo > >So.. you (correctly) note that there are not a finite "number" of >reals in [0,1]. You think this "proves" that there exists an infinite >"number". Why? (And, what is your definition of "number")? And what is your definition of "infinite"? ~v~~ |