Galileo's Paradox
in [Math]
|
From: Virgil on 9 Dec 2006 16:16 In article <457afaf0$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tonico wrote: > >> Tony Orlow ha escrito: > >> > >>> You can call me a troll if that makes you feel better. You seem to need > >>> to bolster your ego by piling it on top of others. Hopefully you'll work > >>> that out eventually. I won't concern myself with your spiritual > >>> development too deeply, but I do have some questions. > >> ***************************************************** > >> Lemme see: troll....yup, it makes me feel better....ah. > > > > If a troll has to know they are trolling, then I don't think Tony is a > > troll. He thinks he is merely posting what is correct. Of course, it > > would be better if he was a troll. He's more a crank. > > > > Thank you, David. I'm definitely more of a crank than a troll. :) > > Tony Does that make TO a "crall" rather than a" croll"? Or is it a "trank" rather than a tronk"?
From: Virgil on 9 Dec 2006 16:18 In article <457afd08(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Lester Zick wrote: > >>> On Tue, 05 Dec 2006 11:53:13 -0500, Tony Orlow <tony(a)lightlink.com> > >>> wrote: > >>> > >>> [. . .] > >>> > >>> You know, Tony, I got to thinking last night... > > > > Oh dear, that was Lester... <snip> > > > >>> Now there is actually a precedent in conventional mathematics for this > >>> situation. With complex numbers you actually have two component > >>> numbers: one conventional algebraic and one imaginary. Thus we can't > >>> say that r+ni is actually larger than r unless n is even. > >> That's true,... > > > > Is it? One can't say that r+i is larger than r, but we can say that > > r+2i _is_ larger than r. I'm totally baffled by this, and wonder if you > > can explain to us mathematikers what Lester is on about now? > > > > > > Actually I wasn't going to nitpick that "even" statement, since the > point is that the 2D plane isn't inherently ordered linearly, and that > imaginary numbers don't lie on the same metric line as reals. The "even" > condition doesn't make sense to me either. Or once we agree, TO. So it appears unanimous that Zick is senseless.
From: Tony Orlow on 9 Dec 2006 16:23 Lester Zick wrote: > On Fri, 08 Dec 2006 14:20:32 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Tue, 05 Dec 2006 11:53:13 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>> [. . .] >>> >>> You know, Tony, I got to thinking last night there may be a way to >>> avoid this whole paradoxical situation. Let's say on the one hand we >>> have what I would call cardinal algebra by which I just mean the >>> conventional algebra dealing with finites such as r+y=z and so on. >>> Then we wish to ascertain the nature and properties of such >>> expressions as r+dr. And I have maintained that the addition of >>> infinitesimals such as dr doesn't alter the size of finites such as r. >> Sorry it's taken me a few days to get back - phone issues. I love loving >> near the county line - sometimes. Sure, there's a way out of any >> paradox. It all depends if an infinitesimal is something other than zero. :) > > If it isn't we've got problems, Houston. > In standard math, there are no such things. Arbitrarily close means equal. They have been banished by "rigorous axiomatization" as "ghosts of departed quantities". So, yeah, we got problems. :) >>> Now there is actually a precedent in conventional mathematics for this >>> situation. With complex numbers you actually have two component >>> numbers: one conventional algebraic and one imaginary. Thus we can't >>> say that r+ni is actually larger than r unless n is even. >> That's true, and exactly like the problem of "ordering" the points in >> the Cartesian plane. However, one can fist offer the dimensions, and >> then order each dimension linearly, and achieve a linear 2D space, if >> you can grok that. > > Not if you're talking set "theory" Tony. They can't order anything > without first assuming what they're supposed to be ordering. > I was just point out to David that I don't see how the order inherent in a linear sequence can be derived from pure set membership. Is that what you mean? >>> Now what I propose is something I'll call phase array algebra where in >>> addition to conventional imaginary components we have other phases as >>> well. We also have infinitesimal components such as dr. In other words >>> to express a number such as c+ni+kdr we have not only the classical >>> finite cardinal algebraic phase c and the neoclassical imaginary phase >>> ni, we also have an infinitesimal phase kdr. >> You might want to look into Internal Set Theory, a partial >> axiomatization of Nonstandard Analysis. Both infinitesimal and infinite >> values are "nonstandard", and no reference to "standard" values is >> allowed in the definition of any set. > > No idea what this means. > Look it up. Robinson's Nonstandard Analysis is THE first rigorous treatment of infinitesimals, apparently, and Internal Set Theory is a set-theoretic axiomatization of it, apparently. :) >>> So in effect we have no way to say c+ni+kdr is larger than c alone >>> unless ni or kdr contribute something further to the magnitude of c. >>> This is despite the presence of the "+" sign because "+" may not mean >>> exactly what it means in the context of classical finite cardinal >>> algebra alone. >> That almost makes sense, except that infinitesimals don't lie on an >> orthogonal dimension. They lie between indistinguishable numbers. > > Not if the derivative is 0dr, Tony. If you want to use dr's you have > to realize they exist in a different phase from cardinal numbers just > like complex components of cardinal numbers until and unless they're > definitely finitely integrated and thereby transformed into cardinals. > The derivative of a constant is 0. So, I don't know what any of this means. Sorry. >>> The interesting thing about phase array algebra is that you can make >>> up phase array components all day long but unless we can phase one >>> component into the c phase there is no impact on the c phase. These >>> I'll just call phase transitions or rules for converting one phase >>> into another. >> I'll have to google "phase array algebra" (cuts and pastes) > > Don't bother, Tony. It's a name I just made up to describe what I'm > talking about here. Either you won't find it or it won't refer to what > I'm talking about. > Does what you're talking about have a name that is a little more well known or self-explanatory, or does it mean anything at all anyway? >>> Now we're all aware of phase transition rules for imaginary numbers. >>> c+ni is no greater or less than c alone unless n is even in which case >>> there is a transition from the i phase to the c phase and c+ni becomes >>> larger than c alone. >> I am not aware of that. Please give me a good link that explains it. It >> might prove quite important. Thanks. > > Just complex algebra, Tony, where powers of i contribute to cardinal > magnitudes according to whether they're even or not. > Uh, why don't I see any powers of i in c+ni? Do you mean c+i^n? Yes, there's a funny interaction between the real and imaginary components of complex numbers which maps to angular and linear operations on the radius of a circle. What does this have to do with infinitesimals? They are not produced by calculus, but used to form the basis of it. A number infinitesimally close to x is considered equal to x in standard math. >>> The same is true of infinitesimal numbers such as kdr only the phase >>> transition rules are different. As long as k is finite there is no way >>> to say that c+kdr is larger than c alone in finite terms. However once >>> definite integral calculus is invoked there is a transition between >>> the infinitesimal phase and the finite cardinal algebraic phase which >>> allows us to state that c+kdr is greater than c alone. But only then. >> Eh. We can say a finite plus or minus an infinitesimal is or is not >> equal to itself, like an infinite plus or minus a finite. I say it is >> not, if the infinitesimal is nonzero in the finite realm, or the finite >> is nonzero in the infinite realm. That depends on N. > > Not if the infintesimal has not been definitely integrated, Tony. > Until then it exists in a different phase like the complex part of a > complex number. > Definitely integrated? >>> In theory I suppose every number and numerical concept has a number of >>> concomitant but otherwise unrelated phases associated with it linked >>> to the c phase only through phase change transitions rules which >>> relate any one phase to the c phase in which conventional algebra is >>> done. At least that's the way I read the situation. Thus when I say >>> that c+kdr is no larger than c alone because k is finite I'm just >>> saying kdr is not in the same phase as c. >> That's not much different from saying it's "incommensurable". Was that >> your word? > > Yes in a way. It doesn't change the magnitude of finite r until and > unless it's definitely and finitely integrated. > In the case of your circle, of course, the change is always at right angles to the position, and the change in change at right angles to the change, but that's just with circles, eh? On the line, there are an infinite number of infinitesimal intervals within any finite interval. >>> Now what you can recognize in all this is the misinterpretation and >>> even perhaps the misrepresentation of the "+" sign when someone says >>> that 1+dr=1 or 00+1=00 for that matter. It doesn't mean the same as >>> the "+" sign in classical finite algebra any more than it would mean >>> the same with complex numbers until some phase transition occurs >>> between phases which renders the result of one phase consistent with >>> classical c phase algebra. >> It means, if you go to the right "dr" times, you have reached a >> different point. Have you? > > Not finitely no. > Infinitesimally? >>> In fact recalling a suggestion I offered you a couple of weeks back >>> there is an interesting parallel. Then I suggested that rather than >>> trying to cram finites and infinites onto one real number line you >>> might consider putting finites on one line and infinites on another >>> dimensional line which is exactly the way imaginaries are conceived. >>> So in effect we have a c algebraic phase concentrated on one line and >>> other kdr infinitesimal phases concentrated on other lines normal to >>> it with the only real difference being phase transition rules between >>> the two. >>> >> I'm sorry Lester, but I see that as rather kludgy and without >> justification. Measure is measure. Is distance distance? Where's the >> origin point? > > Is i a finite measure, Tony? Not until it's evenly compounded in terms > of itself. Until then it's just in a different phase from finite > measures. > Oh, i is most definitely finite, not infinite or infinitesimal. It's just not real. You can have real infinitesimals or imaginary infinitesimals, or, I'm sure, infinitesimal quaternions and such. >>> At least I hope this clears up the situation for you in terms of >>> classical algebraic arithmetic relations between infinites and >>> finites. I think this analysis is pretty much definitive but who >>> knows? We may yet have to try again if this doesn't work. >>> >>> ~v~~ >> Definitive in what sense? Actually addressing the linearity of the >> reals, including the infinites and infinitesimals? I don't really see >> definition. But, this is a discussion, not a proof. > > Definitive in the sense of comprehensive, Tony, not in the sense of > providing an exact definition. The idea just occurred to me and I > thinks it's pretty reasonable and comprehensive in the sense of > explaining the mechanics involved in associating infinitesimals with > cardinal numerics in various contexts analogous to the mechanics of > complex numbers. > > ~v~~ I think the analogy is unfounded, personally. Ah well. 01oo
From: Virgil on 9 Dec 2006 16:33 In article <457aff26(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > >> What,in mathematics, has a solution which is neither a real measure, or > >> the measure of truth of a statement, 0, 1, or somewhere in between? > > > > "Find all pairs of distinct naturals (x,y) such that x^y = y^x". The > > solution to which is the set {(2,4), (4,2)}, which doesn't appear to be > > a "real measure", nor a "measure of the truth of a statement" (as far > > as I can understand your meaning of the terms). > > Those are specifications for two points in a 2D array of naturals, the > values within each pair denoting the distance of each point in each of > the two directions, from the origin. Are you saying x and y and 2 and 4 > are not taken to be quantities? That's a rather strange position to take. Where does TO get such peculiar misinterpretations? CBrown says 'the set {(2,4), (4,2)}, which doesn't appear to be a "real measure", nor a "measure of the truth of a statement"'. I also, do not regard the set {(2,4), (4,2)} as a"real measure", nor a "measure of the truth of a statement", but that does not mean that I regard 2 or 4 as anything other than natural number quantities. > > > > > I assume that you offer more than the trivial observation that all > > mathematical statements P, including the statement "2^4=4^2", are > > examples of the (boolean) truth valued statement "it can be proved that > > P". If so, I would claim that the mathematical question is "Find a > > proof of P, or proof of not P". And the solution is not "0" or "1"; the > > solution is either a proof of P, or a proof of not P. > > First of all, that trivial observation is plenty. All logic is subsumed > under math as a calculation of truth values between 0 and 1. But one "embed" logic within mathematics without having logic a priori with which to build that model of logic. As a practical matter, logic is a priori to every axiom system. One can model logic within mathematics, but that is not at all the same thing. > > That's fine. I am still of the opinion that math boils down to measure, > the language of measure, and the operations allowed on that language. Fortunately, TO's opinion is of no weight.
From: Tony Orlow on 9 Dec 2006 16:24
David Marcus wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <MPG.1fe4233fd1946032989a0c(a)news.rcn.com>, >>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>> >>>> Tony Orlow wrote: >>>>> Okay, a "potential" infinite set is one where each element, like the >>>>> naturals, has a specific string associated with it, which has a >>>>> left-hand end. >>>> What do you mean "a specific string associated with it", and what is a >>>> "left-hand end"? >>> If TO means a string having a first character but not a last one, the >>> set of all such is uncountable, so cannot represent a merely >>> "potentially infinite" set. >> I mean a finite string. By a specific string, I mean one which differs >> from all other strings in the language in at least one position. >> >> Countably infinite means potentially infinite. > > If you wanted to say "potentially infinite" means countably infinite, > why didn't you just say so? I don't think anyone else who has been using > the phrase "potentially infinite" recently shares your meaning. > No? Why don't you ask them. It seems pretty clear to me that that's a pretty common conception. Tony |