The Definition of Points
in [Math]
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From: Tony Orlow on 30 Mar 2007 13:07 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? 01oo
From: Tony Orlow on 30 Mar 2007 13:08 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 01oo
From: Tony Orlow on 30 Mar 2007 13:10 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. 01oo
From: Tony Orlow on 30 Mar 2007 13:11 Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Equal subdivisions. That's what gets us cardinal numbers. >>> >> Sure, n iterations of subdivision yield 2^n equal and generally mutually >> exclusive subintervals. > > I don't know what you mean by mutually exclusive subintervals. They're > equal in size. Only their position differs in relation to one another. > > ~v~~ Mutually exclusive intervals : intervals which do not share any points. 01oo
From: Tony Orlow on 30 Mar 2007 13:13
Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>> It's the same as Peano. >>> Not it isn't, Tony. Cumulative addition doesn't produce straight lines >>> or even colinear straight line segments. Some forty odd years ago at >>> the Academy one of my engineering professors pointed out that just >>> because there is a stasis across a boundary doesn't necessarily mean >>> that there is no flow across the boundary only that the net flow back >>> and forth is zero.I've always been impressed by the line of reasoning. >> The question is whether adding an infinite number of finite segments >> yields an infinite distance. > > I have no idea what you mean by "infinite" Tony. An unlimited number > of line segments added together could just as easily produce a limited > distance. > > ~v~~ Not unless the vast majority are infinitesimal. If there is a nonzero lower bound on the interval lengths, an unlimited number concatenated produces unlimited distance. 01oo |