The Definition of Points
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From: Randy Poe on 19 Mar 2007 12:41 On Mar 19, 12:12 pm, Tony Orlow <t...(a)lightlink.com> wrote: > Randy Poe wrote: > > On Mar 19, 11:32 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> Virgil wrote: > >>> In article <45fd9bf...(a)news2.lightlink.com>, > >>> Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Virgil wrote: > >>>>> In article <45fd6...(a)news2.lightlink.com>, > >>>>> Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> Virgil wrote: > >>>>>>> In article <45fc6...(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>> Except that linear order (trichotomy) and continuity are inherent in R. > >>>>>>>> Those may be considered geometric properties. > >>>>>>> If one defines them algebraically, as one often does, are they still > >>>>>>> purely geometric? > >>>>>>>> Tony Orlow > >>>>>> One may express them algebraically, but their truth is derived and > >>>>>> justified geometrically. > >>>>> How does one prove geometrically what is only defined algebraically? > >>>> example? > >>> That the set of naturals is infinite. > >> Geometrically incorrect. Unless there is a natural infinitely greater > >> than the origin, there is no infinite extent involved. > > > The naturals don't have physical positions, since they are not > > defined geometrically. But at any rate, this is your own private > > misconception, and it stems from your insistence that there must > > be a member of N which is "at the end" of N. > > You're within your own orb. > > Position the point of your compass at 0, and traverse the line - that's > 1. Now position it at 1, and traverse the line in the same direction ... > mark those points sequentially.... No such construction is part of the *definition* of the naturals. > >>> That the cardinality of the symmetric group on n elements is n!. > >> Isn't that demonstrable geometrically? > > > No. There is nothing geometrical in that statement. > > Arrange them in a convex polygon as vertices, and draw lines between > them.... Arrange what? The symmetric group on n elements? At any rate, no such construction is part of the definition of anything in that statement. > >>> That a polynomial over the reals of degree n has no more than n real > >>> zeroes. > >> Also geometrically demonstrable. > > > Please describe such a demonstration. > > Let me think about that, among other things... > > In the meantime, let Virgule explain how I can't... I don't believe you can. There is only one way to demonstrate that you can, which is to do it. > >>> That a square matrix always 'satisfies' its characteristic polynomial. > >>> Etc. > >> Not sure what that means, but excuse me if I take "square" to be a > >> geometrical concept.... > > > It's not, in this context. > > Oh. Define the geometrical definition of square, that's not "analytical".... > What does that have to do with the question? The statement was about square matrices. The definition of a square matrix has nothing to do with geometry. But at any rate, the *geometrical* definition of a square, which has nothing to do with square matrices, is not "analytical" whatever private meaning you ascribe to that word. - Randy
From: Brian Chandler on 19 Mar 2007 12:51 Tony Orlow wrote: > Eckard Blumschein wrote: > > On 3/15/2007 7:07 PM, Bob Kolker wrote: > > > >> 9/10 + 9/100 + etc converges to 1.0 > >> > >> Bob Kolker > > > > > > Not to 1.0, > > but why not to 1.00... as 1 + 0/10 + 0/100 + etc converges to 0.99... > > > > Hi Eckard - Long time.... > > In the adics, what is successor to ...999? ...000. In an uncountable > ring*, Three questions, Tony... (1) What is an uncountable ring? (2) What does the asterisk mean? (3) Can you remind me what the third question was? (4) Or was it four? Brian Chandler http://imaginatorium.org
From: G.E. Ivey on 19 Mar 2007 09:07 > On Tue, 13 Mar 2007 20:24:01 +0100, "SucMucPaProlij" > <mrjohnpauldike2006(a)hotmail.com> wrote: > > > > >"PD" <TheDraperFamily(a)gmail.com> wrote in message > >news:1173810896.000941.35900(a)q40g2000cwq.googlegroups > .com... > >> On Mar 13, 12:52 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~~ > >> > >> Interestingly, the dictionary of the English > language is also > >> circular, where the definitions of each and every > single word in the > >> dictionary is composed of other words also defined > in the dictionary. > >> Thus, it is possible to find a circular route from > any word defined in > >> the dictionary, through words in the definition, > back to the original > >> word to be defined. > >> > >> That being said, perhaps it is in your best > interest to find a way to > >> write a dictionary that eradicates this > circularity. That way, when > >> you use the words "peculiar" and "definitional", > we will have a priori > >> definitions of those terms that are noncircular, > and from which the > >> unambiguous meaning of what you write can be > obtained. > >> > >> PD > >> > > > >hahahahahahaha good point (or "intersections of > lines") > > And it might be an even better point if it weren't > used to justify > mathematikers' claims that lines are made up of > points. > > ~v~~ Could you give a reference in which a mathematician (not a high-school geometry book- I would accept a college geometry book) states that lines are made up of points? In every text I have seen "points" and "lines" are undefined terms. I believe it was Hilbert who said that "If you replace points and lines by beer steins and tables, every statement should still be true."
From: PD on 19 Mar 2007 13:10 On Mar 19, 10:48 am, Tony Orlow <t...(a)lightlink.com> wrote: > PD wrote: > > On Mar 18, 6:08 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > >> On 18 Mar 2007 10:36:04 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: > > >>>> All of a sudden you want to talk about original posts? I mean like the > >>>> original post where in response to your specific questions I spell out > >>>> the combined vector analysis pertinent to Michelson-Morley and you > >>>> just ignore it but subsequently pretend there is no combined vector > >>>> analysis relevant to Michelson-Morley? > >>> Actually, no, I didn't ignore it. Others could see my posts, but you > >>> (and to all evidence) you alone said you could not. Then you claimed > >>> that I was "channeling" through someone else, who plainly could see my > >>> posts and was responding to them. You, of course, assumed that the > >>> problem was not yours, and that whatever was happening was by my > >>> choice or design. > >> Well if not by design a rather peculiar lacuna in any event since you > >> subsequently asked me to repeat my analysis of Michelson-Morley. > > > No, I asked you to do what you *claim* to do about your analysis of M- > > M. > > What you did in your "analysis" of M-M was propose (guess) a > > polarization dependency of the speed of light, which you supposed > > accounted for the null result. > > But what you *claim* to do to establish truth of a proposal is to > > catalog all alternatives and to demonstrate that they are false. This > > you simply have not done in any explicit manner. If you have all those > > in your notes somewhere in your bottom drawer, do please draw them out > > and explicate them. > > Of course, PD, you know it's impossible to enumerate all possible > alternative explanations for a phenomenon, and that's why science works > the way it does. Agreed. Lester thinks not. Read that previous sentence any way you want. > It seems to me Lester wants to find a formula for > truth, rather than a process to detect falsehoods. Agreed. Lester is apparently uncomfortable with the lack of certitude afforded by science, and though he claims to have a path to the certitude he craves, he finds it much easier to spend his time issuing polemics and diatribes than in getting anything done. > I don't see his > vision in that respect, but I do agree with his disagreement regarding > sets of points as full descriptions of geometric and physical objects, > as far as he understands it himself. Right, Lester? > No one says a set of points IS in fact the constitution of physical object. Whether it is rightly the constitution of a mentally formed object (such as a geometric object), that seems to be an issue of arbitration and convention, not of truth. Is the concept of "blue" a correct one? PD
From: Lester Zick on 19 Mar 2007 13:48
On 18 Mar 2007 19:27:45 -0700, "PD" <TheDraperFamily(a)gmail.com> wrote: >On Mar 18, 6:08 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 18 Mar 2007 10:36:04 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: >> >> >> All of a sudden you want to talk about original posts? I mean like the >> >> original post where in response to your specific questions I spell out >> >> the combined vector analysis pertinent to Michelson-Morley and you >> >> just ignore it but subsequently pretend there is no combined vector >> >> analysis relevant to Michelson-Morley? >> >> >Actually, no, I didn't ignore it. Others could see my posts, but you >> >(and to all evidence) you alone said you could not. Then you claimed >> >that I was "channeling" through someone else, who plainly could see my >> >posts and was responding to them. You, of course, assumed that the >> >problem was not yours, and that whatever was happening was by my >> >choice or design. >> >> Well if not by design a rather peculiar lacuna in any event since you >> subsequently asked me to repeat my analysis of Michelson-Morley. > >No, I asked you to do what you *claim* to do about your analysis of M- >M. >What you did in your "analysis" of M-M was propose (guess) a Certainly empirics and mathematikers are no strangers to guessing. >polarization dependency of the speed of light, which you supposed >accounted for the null result. Surpassing strange but what I thought I'd proposed was that the E polarization vector needed to be analyzed in combination with the bidirectional average velocity of light relative to Michelson-Morley experimental platform in accordance with FLT to determine a net dilation in relative c, which would vary with polarization direction. >But what you *claim* to do to establish truth of a proposal is to >catalog all alternatives and to demonstrate that they are false. This >you simply have not done in any explicit manner. If you have all those >in your notes somewhere in your bottom drawer, do please draw them out >and explicate them. Well that's easy enough to do. Einstein's geometric contraction postulate and Lorentz's material contraction hypothesis both rely on contraction of one type or another. And without the possibility of a single uniform contraction factor neither can explain null results. In this regard both require contraction to explain the null results of Michelson-Morley in any single frame of reference. However both forms of contraction operate over space, Einstein's explicitly in geometric terms and Lorentz's to the extent experimental platforms occupy space. However comparable experiments at different velocities are possible interstitially in space overlapping any reference experiment to which different contraction factors would have to apply for exactly null results to be achieved for all experiments. Thus different contraction factors would have to apply to one and the same region of common space and it must be concluded that different Michelson-Morley type relative motion experiments must succeed to some extent in different but spatially overlapping frames of reference. Now my recollection of your arguments against this is that you insist FLT applies to two frames of reference not one as against which I simply noted Michelson-Morley occupies a single frame of reference not two and the null results of that experiment have to be explained in those terms not what you might imagine FLT applies to. Your second complaint is that there is also an M vector in addition to an E polarization vector associated with beams of light and why not apply FLT to both instead of just the E vector? To which I replied the E vector actually has motion along it associated with the generation of light since we know of E monopoles but we know of no M monopoles. To which I should have added there is no motion along the M vector since it is just a nominal polar vector reflecting E vector dynamics. And a third complaint I've certainly heard at least from others is that Einstein's posutlate doesn't require geometric contraction at all which is probably about the most bizarre objection I can imagine since geometric contraction as the result a second order velocity dependent temporal anisometry is the only possible explanation for Einstein's postulate in the context of single frames of reference. To which I can imagine we should add a further generalized anxiety objection that I just don't know what I'm talking about in the context of SR because I decline to talk about it in the same terms sycophants insist on, which seems to be the most common complaint to my line of reasoning and to which I'm happy to plead guilty as charged. >> Curiously I've never had the specific kind of problem you presented in >> this regard and I think you're being just a little too clever by half. >> >> >> Or the original post wherein I >> >> point out that points making up lines and the interesection of lines >> >> defining point is circular logic? Do tell which original posts exactly >> >> did you have in mind? >> >> >Yes, I believe I answered that post as well. In fact, mine was the >> >first response. Your memory is apparently dismal. >> >> I seem to recollect some kind of remarks but nothing I considered >> substantive. There's a huge difference between posting a reply and >> addressing the subject itself in terms responsive to those employed. >> >> However I have a further notion. I'd assumed when you said you were >> bored that I'd seen the last of you. If you wish to offer constructive >> criticism by all means do so. Just please get to the point. Brevity is >> the soul of wit and at this juncture you're neither. >> > >Brief enough for you? Much. ~v~~ |