The Definition of Points
in [Math]
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From: Bob Kolker on 19 Mar 2007 13:53 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... all the same. 1.0. Are you sure you comprehend what convergence is? ..999... is shorthand for the limit of the series 9/10 + 9/100 + ... etc. Do you know what a limit is? Bob Kolker
From: Bob Kolker on 19 Mar 2007 13:55 Tony Orlow wrote: > > > Geometrically incorrect. Unless there is a natural infinitely greater > than the origin, there is no infinite extent involved. He meant that the cardinality of the set of integers is infinite. Bob Kolker
From: hagman on 19 Mar 2007 14:06 On 16 Mrz., 15:35, "SucMucPaProlij" <mrjohnpauldike2...(a)hotmail.com> wrote: > "hagman" <goo...(a)von-eitzen.de> wrote in message > > news:1174053602.723585.89690(a)l77g2000hsb.googlegroups.com... > > > > > On 13 Mrz., 18:52, 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~~ > > > Please look up the difference between "define" and "determine". > > > In a theory that deals with "points" and "lines" (these are typically > > theories about geometry), it is usual to leave these terms themselves > > undefined > > and to investigate an incidence relation "P on L" (for points P and > > lines L) > > with certain properties > > > Then the intersection of two lines /determines/ a point in the sense > > that > > IF we have two lines L1 and L2 > > AND there exists a point P such that both P on L1 and P on L2 > > THEN this point is unique. > > This is usually stated as an axiom. > > And it does not define points nor lines. > > This is interesting observation :)))) > > But how do you define difference between "define" and "determine"? > Can "definition" determine and can "determination" define? > > Lester Zick has problem with "circular definitions" and you used term "point" in > your "determination" to determine it. Maybe you want to say that in definition > you can't use term you define to define it and in termination you can use it to > determine it. > > I think it's time to call Determinator :)))) > He is the only one who can help us! hahahahahahaha All I wanted to do is go along with Lester on his path of arguments as far as bearable - which is more than "as far as correct". Let me start again with a bit more sleep: In his OP, Lester talked about lines as being composed of points by definition and that the intersection of two lines determined a point. This is his private theory although he states that it were somehow usual math folklore. First imtermezzo: What is a definition? I won't define that term here rogorously, but a definition should be useful (that's just a pragmatic aspect) and make the defined term eliminable. Example: In the context of natural number we can define DEF. k is a divisior of n if there exists some m such that k*m=n. DEF. p is prime if p has eactly two divisors. These are good definitions as we can eliminate the terms "prime" and "divisor" from statements like "There are infinitely many primes" = "There are infinitely many numbers p such that there are exactly two numbers k such that there exists some m such that p=k*m" So: What is the definition of "point"? And what is the definition of "line"? Mathematical theories where both the terms "points" and "lines" are used usually go like this: A tuple G=(P,L,E) is called a "geometry" (the precise term may depend on the precise set of axioms used below and might as well be "euklidean plane" or the like) if P and L are disjoint sets and E is a relation among them, i.e. E subset PxL, such that 1. For two distinct p,q in P, there is exactly one g in L such that (p,g) in E and (q,g) in E 2. For two distinct g,h in L, there is at most one p in P such that (p,g) in E and (p,h) in E 3. For each p in P there is at least one g in L such that (p,g) not in E 4. ... Only the context of such a geometry and the complete set of axioms listed define point (and line and incidence). Thus, "Let p be a point ..." should be writen via elimination as "Assume G=(P,L,E) is a geometry and p in P ..." Note that lines are not "composed of" points. However, two (non-parallel) lines determine a point in the sense that it is a theorem (in fact an axiom) in this theory that for these lines there is a unique point incident with both lines. As lines are not composed of points, the very first sentence of the OP is nonsense. However, this nonsense can be escaped from: Given a geometry G(P,L,E), we ca define a mapping from pointson: L -> 2^P, g |-> {p in P: (p,g) in E} This mapping is injective and allows us to replace L by {pointson(): g on L} and E by element containment to obtain an isomorphic geometry. Hence one may assume wlog that lines are sets of points. However, we could alternatively have used a mapping goesthru: P -> 2^L, p |-> {g in L: (p,g) in E} and would then interprete lines as "atomic" and points as sets of lines. The first method is more useful as it is more readily generalized to other sets of points (circles etc.) In view of the above, "lines are composed of points" and "lines determine points" is no more miraculous than "sets contain elements" and "sometimes two sets have exactly one element in common"
From: Lester Zick on 19 Mar 2007 14:26 On Mon, 19 Mar 2007 10:48:56 -0500, Tony Orlow <tony(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. It seems to me Lester wants to find a formula for >truth, rather than a process to detect falsehoods. 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? Tony, you know this is an interesting point in the context of physics and empirical experimentation. First let me say that my demonstration of truth in universal terms is rather limited at this point to the one proposition ~v~~ and certain mathematical implications. So I don't claim to be able to demonstrate the application of that principle or law of contradiction as I call it, to the analysis of experimentation, apart from certain quantum particle spin characterisitics, in general physical terms. At least not just yet. Having said which however just allow me to add that the particular experiment under consideration here, Michelson-Morley, and its resolution in terms of Einstein's geometric and Lorentz's material contraction hypotheses are so simple that an exhaustive mechanical analysis in fact becomes possible. The usual difficulty with abstract analysis of physical experiments is that there are so many hypothetically conceivable factors relevant to any analysis that actually analyzing such experiments is impossible. (The problem is exactly the same as I pointed out in the tautological analysis of alternatives to any specific circumstance, that "not A" is a complex combination of predicates and predicate combinations.) However in the context of Michelson-Morley there are only a couple predicates and predicate combinations involved such that we can in fact approach experimental analysis in exhaustive terms. There are the null results of the experiment, Maxwell's calculation of constant c through space independent of the experimental platform, and the analysis of the anisotropic velocity of light relative to experimental platforms, the Fitzgerald-Lorentz transforms which here I call FLT. So all we really have to look at are tautological alternatives to the various predicates and predicate combinations. Beyond this we also have to examine Einstein's geometric and Lorentz's material contraction hypotheses in exhaustive terms to determine whether or not they can in fact explain the null results of Michelson-Morley and if not exactly where the explanation has to lie. Now this may not be a simple thing to do but it is possible because in this particular instance we only have to deal with a mere handful of experimental considerations and not the welter of conceivable factors ordinary experiments would be subject to. Thus here together with particle spin characteristics we find possibly the one and only sui generis physical experiment subject to exhaustive physical analysis. >>> 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? >> >> PD >> >> >> >My two cents worth, No problem. All contributions gratefully accepted. ~v~~
From: PD on 19 Mar 2007 14:38
On Mar 19, 12:48 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 18 Mar 2007 19:27:45 -0700, "PD" <TheDraperFam...(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. And neither are you, obviously. > > >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. Well, this is hardly exhaustive, is it. For example, you haven't considered and shown false at least two other alternatives: 1) That the Earth is stationary in the ether. 2) That the ether is dragged by the Earth Secondly, you apparently do not understand the difference between "require" (as in prerequisite) and "imply" (as in postrequisite). The invariance of the speed of light does not *require* contraction, though it certainly does imply it. But let's get back to the strategy. You said you were going to demonstrate that alternatives were false. So I expect you to AT LEAST demonstrate that 4) The invariance of the speed of light is false. 5) The geometric contraction of Einstein is false. 6) The material contraction of Lorentz is false. > > 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. Sorry, if this was supposed to be a demonstration that (1)-(5) above are false, then you'll have to explain the demonstration a little more clearly. > > 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 No, it doesn't. Moreover, even in a single data run, there is no FLT applied in the measurement. Have you even read the paper? > 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. And? The wave equation arises from the case where there is *no source* (either E or M monopoles) in the volume being considered. The curl equation for the electric field has the time-dependence of the magnetic field as the *sole* source, and the curl equation for the magnetic field has the time-dependence of the electric field as the *sole* source. I'm shocked -- SHOCKED, I tell you -- that you did not know that. How did you learn electrodynamics? Moreover, your arguments FOR the plausibility of your mechanism do *nothing* to address your self-avowed strategy of determing the truth of a proposition by considering all alternatives and demonstrating their falsity. > 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. Why no, no it's not. It may be the only thing that makes sense to YOU, but then again, not much makes sense to you. Your apprehensions by no means should be taken as a measure of what is possible and what is not. > > 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. > Well, as usual Lester, if you only want to talk with yourself, you could do better. You could invent a whole new language or even a new alphabet, perfectly tailored to the expression of your mental processes. However, if your aim is communication, then there is an implicit bargain of coming to terms on certain things. There are those who think that money is generally a bad idea and do not wish to participate in the same processes that fiscal sycophants do, and that's fine as long as they are able to grow all their own food and provide their own services. However, you seem nevertheless interested in commerce, and in the commerce of ideas no less, and to use ascii text as a medium of exchange. In words that perhaps you can understand, "you can't have your fitzbold and greeble it too." PD |