Galileo's Paradox
in [Math]
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From: Lester Zick on 10 Dec 2006 19:18 On Sun, 10 Dec 2006 12:40:17 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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? Reciprocals as far as I can tell. I don't know what the modern math doctrince entails. Doesn't really matter very much if it isn't true. ~v~~
From: Lester Zick on 10 Dec 2006 19:21 On Sun, 10 Dec 2006 12:50:43 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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. But that's not what I remember you saying previously, Tony. I seem to recall you indicating r+dr>r. > Odd powers of i equal i, and all powers of 1 equal 1. >That's not particularly related, as I see it. It's related if you maintain r+dr>r. >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. So flat lines don't have a derivative 0dr? ~v~~
From: Lester Zick on 10 Dec 2006 19:23 On Sun, 10 Dec 2006 15:47:12 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Mike Kelly wrote: >> David Marcus wrote: >> > Tony Orlow wrote: >> > > David Marcus wrote: >> > > > Tony Orlow wrote: >> > > >> Well, I have been through much of that regarding such specific language >> > > >> approaches as the T-riffic digital numbers, but that's not necessary for >> > > >> this purpose. It suffices to say that, if a statement is proved true for >> > > >> all n greater than some finite k, that that also includes any postulated >> > > >> infinite values of n, since they are greater than any finite k. I don't >> > > >> need to construct these numbers. Consider them axiomatically declared. >> > > > >> > > > Then list the axioms for them. >> > > >> > > (sigh) >> > > >> > > infinite(x) <-> A yeR x>y >> > >> > Is that your only axiom? If so, then state your first theorem about them >> > and give the proof. >> >> The only thing I can come up with is "There are no infinite numbers". >> Am I close? > >You are about as close as I am. I can prove "A xeR (not infinite(x))". >Doesn't seem like a very interesting theory, so far. Unlike truth interest lies in the eye of the beholder. Just ask Bob. ~v~~
From: cbrown on 10 Dec 2006 19:25 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> Why do you have a problem with the mere suggestion of an infinite value? > >>>>> I don't have a problem with infinite values. However, you have to do > >>>>> more than merely say "positive infinite n". Assuming your positive > >>>>> infinite things aren't the same as something that we already know about > >>>>> (in which case, you should just say so), you either need to give a > >>>>> construction of these positive infinite things or you need to specify > >>>>> their properties. You haven't done either. For example, how many of > >>>>> these things are there? How do they relate to each other? How do they > >>>>> interact with the natural numbers? Are the operations of addition and > >>>>> multiplication defined for them? > >>>> Well, I have been through much of that regarding such specific language > >>>> approaches as the T-riffic digital numbers, but that's not necessary for > >>>> this purpose. It suffices to say that, if a statement is proved true for > >>>> all n greater than some finite k, that that also includes any postulated > >>>> infinite values of n, since they are greater than any finite k. I don't > >>>> need to construct these numbers. Consider them axiomatically declared. > >>> Then list the axioms for them. > >> (sigh) > >> (T1) infinite(x) <-> A yeR x>y > >> infinite(x) <-> A yeR x>y > > > > Is that your only axiom? If so, then state your first theorem about them > > and give the proof. > > > > That's the only one necessary for what defining a positive infinite n. A > whole array of theorems pop forth... Before going there, you might want to start by adding the axiom: (T2) exists B such that infinite(B) Otherwise, who cares if you can prove a whole bunch of theorems about something that doesn't exist? Cheers - Chas
From: Virgil on 10 Dec 2006 19:43
In article <457c12f6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Well, I have been through much of that regarding such specific language > >> approaches as the T-riffic digital numbers, but that's not necessary for > >> this purpose. It suffices to say that, if a statement is proved true for > >> all n greater than some finite k, that that also includes any postulated > >> infinite values of n, since they are greater than any finite k. I don't > >> need to construct these numbers. Consider them axiomatically declared. > > > > Then list the axioms for them. > > > > (sigh) > > infinite(x) <-> A yeR x>y This may serve in some sense as a definition, but does not imply that any such x exists, so it is totally unsatisfactory as an axiom declaring or requiring such x's existence. One can equally well say "x is Santa Claus if and only if ...", but no matter how one fills in that blank, it doesn't mean any Santa exists. |