Galileo's Paradox
in [Math]
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From: Tony Orlow on 10 Dec 2006 09:50 Bob Kolker wrote: > Virgil wrote: > >> In article <457b8a2f(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >> >>> Language is a set of strings from an alphabet. >> >> >> Like Chinese and Japanese? >> >> Languages are acoustic at first, and are still learnt that way. >> Ideogramatics comes later. > > All languages (used by humans in general discourse) are acoustic because > they are primarily spoken and heard. Writing of any kind is a very > recent development (in the last 6000 years or so). You are refering to > the writing systems. It turns out that Japanese has two phonetic > alphabets (Katakana and Hirugana) but the Chinese style ideographs > (Kanjii) are preferred for traditional reasons. If the Japanese wished > to, they could use a totally phonetic graphology. They just do not want to. > > The Chinese have used ideograms for much longer, but there is a phonetic > system for representing the Mandarin pronounciation. The problem with > Chinese is there are so many dialects which are mutually incompatable > the only common written language is ideographic. The Japanese are under > no such constraint. > > The alphabet derived from the Phonecian graphology was originally > ideographic. The letters were pictures (or highly stylized > representations) of things that eventually took on conventional sound > values. > > Bob Kolker > You guys both need to take your meds, and take note of a few facts. 1) While a symbol is generally considered to be a visual pattern, it is most generally any uniquely distinguishable sensory input. It could be an auditory sound, a latin letter, arabic numberal, hieroglyph, or asian pictogram. It can be a series of beeps and clicks, like Morse code. 2) While Western languages have a small alphabet, and words with an averga of a large number of characters, one can have a large alphabet, the size of the entire vocabulary, where each word is a single character. So, Asian pictographic languages is not excluded by that definition. 3) I have specifically been talking about digital number systems, which are a special case of languages, complete, and with a unique quantitative interpretation for any finite string. It doesn't help to wander off into sociological and psychological realms when discussing digital number system.
From: Tony Orlow on 10 Dec 2006 09:52 Virgil wrote: > In article <457b8a2f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Language is a set of strings from an alphabet. > > Like Chinese and Japanese? > > Languages are acoustic at first, and are still learnt that way. > Ideogramatics comes later. An acoustic sound IS a symbol, which we string together in time, as opposed to doing it on paper.
From: Tony Orlow on 10 Dec 2006 09:54 Virgil wrote: > In article <457b8ccf(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <457b5606(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> There is nothing wrong with saying E and N have the same cardinality. >>>>>>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse >>>>>>>> Function Rule, which indeed can parametrically compare sets mapped >>>>>>>> onto >>>>>>>> the real line, using real valued functions, as a generalization to set >>>>>>>> density. It works for finite and infinite sets. So, what is wrong with >>>>>>>> trying to form a more cohesive theory of infinite set size, which >>>>>>>> distinguishes set sizes that cardinality cannot? There certainly seems >>>>>>>> to be intuitive impetus for such a theory. >>>>>>> Fine. Please state your rule. Let's take a look. >>>>>> I've been over this a lot, but hey, what's one more time. Practice makes >>>>>> perfect. >>>>>> >>>>>> We start with the notion of infinite-case induction, such that a >>>>>> equation proven true for all n greater than some finite k holds also for >>>>>> any positive infinite n. >>>>> What is a "positive infinite n"? >>>>> >>>> A value greater than any finite value. >>> But if "n", is, as usual, reserved for indicating natural numbers, those >>> which are members of every inductive set, there is no such thing. >>> >> What, now mathematics has declared what the single letter n means? It's >> a variable. > > Every variable has a domain of definition. The variable 'n' is quite > commonly required to have the set of finite naturals as its domain. > > I merely remarked that when that is the case, there are no values for n > other than finite ones. > >>>> Why do you have a problem with the mere suggestion of an infinite value? >>> It is arithmetical operations with infinite values absent any >>> definition of what those operations mean or what properties they have, >>> to which we make legitimate objections. >>> >> Not really. > > Really. > No. Reread the following: >> If the expressions used can themselves be ordered using >> infinite-case induction, then we can say that one is greater than the >> other, even if we may not be able to add or multiply them. Of course, >> most such arithmetic expressions can be very easily added or multiplied >> with most others. Can you think of two expressions on n which cannot be >> added or multiplied? > > I can think of legitimate operations for integer operations that cannot > be performed for infinites, such as omega - 1. Omega is illegitimate schlock. Read Robinson and see what happens when omega-1<omega. >>>> Surely you must have guessed enough what I meant to follow the >>>> paragraph? (sigh) >>> Guessing is not a reliable way of finding things out. >> Of course, reading is a little more reliable, but when one gets stuck on >> such words as "positive" and "infinite", maybe one needs to do a little >> guessing. > > If TO's descriptive powers are so insubstantial as to leave us > perpetually guessing what he means, he will never be able to convey > anything of mathematical interest or significance. If you don't know what "positive" or "infinite" mean, that's not my fault.
From: Lester Zick on 10 Dec 2006 12:14 On Sat, 09 Dec 2006 16:23:36 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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. :) Not really. "Arbitrarily close" doesn't mean anything that I can tell. It could mean adjacent; it could mean far away. >>>> 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? Sure. Same for planes etc. They can't even derive straight lines. >>>> 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. :) Well I'll skip it if you don't mind. I've had quite enough assumptions of truth for one lifetime. >>>> 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 derivative of a flat line, r, is 0dr not zero, Tony. You can't just forget about the dr. >>>> 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? Not that I know of. I don't know if the idea has even been suggested before. >>>> 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. Actually infinitesimals are produced by the calculus, Tony. That's how we derive straight lines through derivative tangents to curves. I use the term "ni" to indicate the number of i's in this context. >>>> 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? Result of a definite integral. >>>> 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. No we've been over this before, Tony. The derivative of a flat line such as a finite number line, r, is 0dr indicating to me at least that you don't have any infinite number of infinitesimals within that flat line. That's what I'm trying to show with the finite kdr term in the infinitesimal phase. Infinite numbers of infinitesimals only show up in the context of sloped lines with respect to a flat number line. >>>> 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? Regardless as long as the line is flat. And the finite number line you're suggesting is certainly flat. Infinitesimals only show up in relation to other lines which are not flat with respect to that line. >>>> 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. How do you figure i is finite, Tony? Can it be added to and change the finite magnitude of 6? Or does 6 remain 6 in finite terms regardless? i just exists in another non finite phase until it is compounded evenly in terms of itself. >>>> 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. We've had polyphase numbers around for what, over a century now, Tony? I'm just trying to explain the mechanics of infinitesimal-finite transitions in analogous terms. It looks to me like you're just determined to do finite arithmetic with infinitesimals without being able to show how the trick is done. I haven't explored every implication of what I'm suggesting here but as far as I can see at the moment it looks reasonable. ~v~~
From: Lester Zick on 10 Dec 2006 12:16
On Sat, 9 Dec 2006 17:09:54 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Tony Orlow wrote: >> Lester Zick wrote: >> > On Fri, 08 Dec 2006 14:20:32 -0500, Tony Orlow <tony(a)lightlink.com> >> > wrote: >> > >> >>> 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. > >I think Lester would like to avoid having a real number plus an >infinitesimal be a real number. So, he wants to keep them separate, like >we do when we add a real number and an imaginary number. True except I'd like to avoid having finite and non finite numbers lumped together. ~v~~ |