Galileo's Paradox
in [Math]
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From: Lester Zick on 10 Dec 2006 12:27 On Sat, 09 Dec 2006 19:28:44 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >David Marcus 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. >> > >Yes, but I'm not sure why he wants that. Like I said in another post, I >think it has to do with his infinitesimal change in radius as dr/dt or >something, uh, nonstandard like that. Not really, Tony. That was just a collateral observation. The point is that finite numbers of infinitesimals don't alter finite magnitudes any more than odd powers of i alter finite magnitudes. This effort was simply an attempt to analyze the problem in terms analogous to the treatment of complex numbers. If you have a better way to analyze the problem by all means have at it. But you need to have something better than constituent points and infinitesimals where there don't appear to be any. ~v~~
From: Tony Orlow on 10 Dec 2006 12:39 Lester Zick wrote: > 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. > It means that one can get within any finite difference from the target quantity within a finite number of iterations of your specification process. For instance, you can specify any decimal fraction to within 10^-x with x digits to the right of the decimal point. >>>>> 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. > Not surprising, since straight lines are infinitesimally curved, and therefore a figment of your imagination. ;) >>>>> 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. > Oh, his is an interesting approach, all about extending the language. But, never mind. >>>>> 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. > Sure you can, in the end. The slope is a constant 0. That's what the derivative says. >>>>> 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. > Then you might want to flesh it out a little. Assumptions of clarity are little better than assumptions of truth. >>>>> 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. > Generally "ni" would be taken to mean "0 increased by i n times", not "1 multiplied by i n times". >>>>> 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. > Uh, okay. Still don't understand what you're saying, but very well. >>>>> 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. > I don't see how it indicates any such thing. As usual, I think you are using all kinds of nonstandard terminology that makes it hard to tell what you're saying. >>>>> 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. > The number line is not necessarily straight, but may be viewed as a circumference of an infinite circle. Depends what you're trying to do. >>>>> 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. > Can a finite egg change the finite amount of flour in your pancake batter? It all adds up to a finite number of finite pancakes. :) The number i is a finite distance from the origin, 0. That's why it's a finite quantity. It cannot be compared with 1, such that i<1 or 1<i, but that doesn't mean it can't be determined to be a finite quantity. After all, who in their right mind thinks that an infinity quantity, squared, can equal a finite quantity like -1? You have to admit, that's a pretty strange conclusion to have to draw. And if it's not infinite or finite, is it infinitesimal? It's not real, but it's finite. >>>>> 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~~ The trick is done formulaically, and using the assumption of a unit infinity equal to the number of reals per unit interval. Where we allow allow such a beast, it's little germ-like friend, the unit infinitesimal, follows, which allows additivity over an infinite sample space, and infinite sets with easily integrated measure, and other nice things like that. Where they all exist together on the hyperreal line/circle, that kind of thing gets possible, and I don't see why infinitesimals should be orthogonal to reals, and what direction you would want infinite quantities to go, or whether you are considering any kind of infinite hierarchy of extra dimensions for extra infinities and infinitesimals or not. So, how is YOUR trick "done"? What happens, in mechanical terms, when you, say, multiply an infinitesimal by an infinite quantity? Is it always finite? :) Tony
From: Lester Zick on 10 Dec 2006 12:39 On Sun, 10 Dec 2006 05:01:28 +0000 (UTC), stephen(a)nomail.com wrote: >Tony Orlow <tony(a)lightlink.com> wrote: >> Tonico wrote: > >>> So the above, and very specially the despise and offensive tone many >>> idiotic trolls/cranks use to refer to PROFESSIONAL mathematicians just >>> because they don't abide by their whims is what makes me call you >>> people what you are: trolls/cranks. > >> There is nothing wrong with expecting science to satisfy intuition. > >Other than the fact that science again and again has proven >to not satisfy intuition. > >It was intuitive that heavier objects fall faster than lighter objects. > >It was intuitive that heat was a fluid. > >It was intuitive that an object in motion must have a force acting on it. > >It was intuitive that light was a wave in a physical medium. > >It was intuitive that objects obey Gallilean transformations. Ah, Stephen, now the master of intuition as well. What is intuitive to one is not necessarily intuitive to another.So how you can intuitively claim what is necessarily intuitive to all remains a mystery. Now if you were arguing truth that would be another matter. Then you could claim that truth need not conform to intuition. But you don't argue truth. All you argue is that your counter intuition is superior to the intuition of another. A truly remarkable and highly problematic intuition at best. ~v~~
From: Tony Orlow on 10 Dec 2006 12:40 Lester Zick wrote: > 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~~ Do you lump infinites and infinitesimals together, or are they mutually orthogonal too? 01oo
From: Tony Orlow on 10 Dec 2006 12:50
Lester Zick wrote: > On Sat, 09 Dec 2006 19:28:44 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> David Marcus 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. >>> >> Yes, but I'm not sure why he wants that. Like I said in another post, I >> think it has to do with his infinitesimal change in radius as dr/dt or >> something, uh, nonstandard like that. > > Not really, Tony. That was just a collateral observation. The point is > that finite numbers of infinitesimals don't alter finite magnitudes > any more than odd powers of i alter finite magnitudes. Finite numbers of infinitesimals don't change finite magnitudes in any finite way, but infinite numbers of infinitesimals can sum to finite or infinite values. Odd powers of i equal i, and all powers of 1 equal 1. That's not particularly related, as I see it. This effort was > simply an attempt to analyze the problem in terms analogous to the > treatment of complex numbers. If you have a better way to analyze the > problem by all means have at it. But you need to have something better > than constituent points and infinitesimals where there don't appear to > be any. > > ~v~~ There appear to be infinitesimals on the real line. Where you have some infinite number of reals in a unit interval, and assume a constant density in the continuum, you can handily declare such a constant unit infinity as a count per unit, and tie measure to infinite count of infinitesimal intervals. I see no reason to place them somewhere else. It's just that, in calculus, the derivative is the measure of change, perpendicular to the x direction along which we are taking the derivative, or the slope. The change in that change is the curvature of the function at a given point, or the change perpendicular to the graph at that point. But, just because those changes are "perpendicular" doesn't indicate to me that infinitesimals values are "perpendicular" to real numbers. Infinitesimals were a tool in deriving the calculus. I don't see that this application of them makes other interpretations wrong. 01oo |