The Definition of Points
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From: Lester Zick on 29 Mar 2007 17:50 On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >> Your position on science and math seems to be that either we proceed >> according to naive and mechanically unreduced and inexhaustive >> assumptions of truth or we proceed by appeals to divine revelation. >> Six of one half dozen of the other. >> > >Nah, I'll take two of the first, and five loaves...no need for the other. Hey, what can I tell you, Tony, as long as you aren't particular. ~v~~
From: Lester Zick on 29 Mar 2007 17:52 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~~
From: Lester Zick on 29 Mar 2007 17:53 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~~
From: Lester Zick on 29 Mar 2007 17:55 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~~
From: Lester Zick on 29 Mar 2007 17:56
On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >> In other words modern mathematikers just assume that because the Peano >> and suc( ) axioms produce successive straight line segments between >> numbers there is some kind of guarantee that the successive straight >> line segments will themselves line up colinearly on straight line >> segments and that we can thus just assume or infer the existence of >> straight line segments and straight lines from those axioms.Doesn't >> happen that way because even if we assume the existence of straight >> line segments between numbers that doesn't demand successive segments >> align in any particular direction colinearly along any common straight >> line segment. Same principle as above, different application. >> > >"Straight" doesn't even seem to mean anything in the context of Peano... "Straight" certainly seems to mean something to mathematikers who talk about straight lines and geometry. ~v~~ |