The Definition of Points
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From: Bob Cain on 16 Mar 2007 16:41 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. What won't suffice is any example that defies consensus understanding. If you provide a symbol string that only you can "understand" it won't satisfy the challenge because it has no practical value. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
From: PD on 16 Mar 2007 16:54 On Mar 16, 2:14 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 15 Mar 2007 16:03:14 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: > > > >> >On Mar 14, 9:51 am, Eckard Blumschein <blumsch...(a)et.uni-magdeburg.de> > >> >wrote: > >> >> On 3/13/2007 6:52 PM, Lester Zick wrote: > > >> >> > In the swansong of modern math lines are composed of points. But then > >> >> > we must ask how points are defined? > > >> >> I hate arbitrary definitions. I would rather like to pinpoint what makes > >> >> the notion of a point different from the notion of a number: > > >> >> If a line is really continuous, then a mobile point can continuously > >> >> glide on it. If the line just consists of points corresponding to > >> >> rational numbers, then one can only jump from one discrete position to > >> >> an other. > > >> >That's an interesting (but old) problem. How would one distinguish > >> >between continuous and discrete? As a proposal, I would suggest means > >> >that there is a finite, nonzero interval (where interval is measurable > >> >somehow) between successive positions, in which there is no > >> >intervening position. Unfortunately, the rational numbers do not > >> >satisfy this definition of discreteness, because between *any* two > >> >rational numbers, there is an intervening rational number. I'd be > >> >interested in your definition of discreteness that the rational > >> >numbers satisfy. > > >> That there is a straight line segment between rational numbers? > > >Well, that's of course true, provided that you're associating numbers > >with points on the line. > > I suppose it's true for line segments defined by points regardless of > whether you're associating numbers with them or not. It's modern > mathematikers who insist on associating points with numbers, not me. Ah, good. Then please sketch for me the line segment between 3/16 and 4/7, without any assocation between any points on the line segment and those rational numbers. > > > Oh, wait, you don't believe that a line > >consists of points. > > Of course not. Perhaps you can show how this has any bearing on the > discreteness of line segments defined between points regardless? > > > Perhaps you want to associate points with numbers, > >regardless of a line, and then say that there is a line segment > >between those two points. That is probably correct. > > Probably??? That's mighty swell of you. Gee, and just above you said it was true whether you associated the numbers with points defining the line segment or not. Now you say you want to associate points with the numbers. Clever strategy, Lester. Say one thing. Mock the reader for not seeing it. Say the converse. Mock the reader for not seeing that, too. > > > The question, > >though is whether there is a line segment between two points > >corresponding to rational numbers, such that no other rational number > >corresponding to a point on the line segment between those two > >endpoints. > > Funny I don't see that as the question at all. The question I see is > whether there are line segments defined between points at all. Gee, and just a few posts ago you said a line segment was defined by its two endpoints. Now you say the question is whether there are line segments defined between points at all. There's your lovely strategy again. > If > mathematikers wish to correlate numbers with those line segments. > don't put it off on me. I don't much care what you do with them. > > > Are you saying that there is such a thing? > > I think I'm saying only what I'm saying. Of course that may not be > quite what you say I'm saying. And then there's the strategy of declining to answer any questions, and then mocking those that ask the question. > > > If so, then I > >invite you to come up with two rational numbers that satisfy that > >criterion. > > First I would invite you to show that there are two rational numbers > which necessarily lie on any common line segment at all. For the life > of me I can really find nothing in the Peano and suc( ) axioms which > shows any necessity for this. You just have various rational numbers > associated with various line segments which have no connection to one > another and could presumably each run off in any various direction. > I didn't say that rational numbers necessarily existed on a line at all. The first to mention a line segment was you. Apparently you lose track of what you say hour to hour. See above or below, your choice: Me: "I'd be interested in your definition of discreteness that the rational numbers satisfy." You: "That there is a straight line segment between rational numbers?" Me: "Well, that's of course true, provided that you're associating numbers with points on the line." Blather away, Lester. Constructing a pretty sentence does not make you look appear any brighter than a charcoal briquet. If you find this entertainment, Lester, you have a tad too much time on your hands and should be letting the occupational therapist show you how to make yarn potholders. I've grown bored because of your transparent tactics and hour-to-hour self-contradictory incoherence. A schizoid that babbles in Shakespearean prose is still a schizoid. PD
From: Lester Zick on 16 Mar 2007 19:19 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~~
From: Lester Zick on 16 Mar 2007 19:31 On Fri, 16 Mar 2007 06:19:31 GMT, mmeron(a)cars3.uchicago.edu wrote: >I'll second that. Surprise. Guy can't even take the derivative of a cross product correctly and he's seconding motions already. Go figure. >Mati Meron | "When you argue with a fool, >meron(a)cars.uchicago.edu | chances are he is doing just the same" ~v~~
From: Lester Zick on 16 Mar 2007 19:32
On Fri, 16 Mar 2007 02:21:35 -0600, Virgil <virgil(a)comcast.net> wrote: >I second it! Of course you do. You're a twit. ~v~~ |