Galileo's Paradox
in [Math]
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From: Tony Orlow on 8 Dec 2006 14:20 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. :) > > 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. > > 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. > > 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. > > 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) > > 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. > > 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. > > 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? > > 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? > > 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? > 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. 01oo
From: Tony Orlow on 8 Dec 2006 14:23 cbrown(a)cbrownsystems.com wrote: > Lester Zick wrote: >> On 4 Dec 2006 17:57:54 -0800, cbrown(a)cbrownsystems.com wrote: >> >>> Lester Zick wrote: >>>> On 4 Dec 2006 11:29:33 -0800, cbrown(a)cbrownsystems.com wrote: >>>> >>>>> Lester Zick wrote: >>>>>> On 3 Dec 2006 11:22:56 -0800, cbrown(a)cbrownsystems.com wrote: >>>>>> > >>>>> Of course it doesn't; "all of mathematics" is an extremely broad range >>>>> of discourse. >>>> So when mathematikers conflate mathematical ignorance with set >>>> "theory" ignorance they are being extremely overly broad? >>>> >>> Not all of Italian cooking involves sauteeing things in olive oil; >>> however it is somewhat bizzare for someone to claim to be a >>> knowledgeable Italian cook without knowing how to sautee things in >>> olive oil. >> Then you undoubtedly qualify as an Italian cook and set mathematiker >> in your spare time. I'm just trying to ascertain the basis for your >> disdain of Italian cooks who don't choose to practice what you preach. >> > > Why do you assume that I disdain cooks (Italian or otherwise) who don't > know how to sautee things in olive oil? I simply think that it's > bizzare to claim to be a knowledgable Italian cook, when one cannot > sautee things in olive oil. I doubt such a person get a job at an > Italian restaurant. > > Cheers - Chas > If you can't sautee reasonably well, in olive oil or axle grease, then you can't cook.
From: Tony Orlow on 8 Dec 2006 14:27 David Marcus wrote: > Tony Orlow wrote: >> Eckard Blumschein wrote: >>> On 11/24/2006 1:20 PM, Richard Tobin wrote: >>>>> * If you have two sets of infinite size, is the union of these sets >>>>> than larger than infinity? What would larger than infinity mean? >>>> That this is not a stumbling block should be clear of you replace >>>> "infinite" with "big". The union of two big sets can be bigger than >>>> either of the original big sets. Like "big", "infinity" is not a >>>> number; it's a description of certain numbers. >>> According to Spinoza, infinity is something that cannot be enlarged >>> (and also not exhausted). It is a quality, not a quantity. >>> >>> There is only one such ideal concept, not different levels of infinity. >> Isn't the purpose of math be to quantify? > > No. 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? Measure=maths. > >> That description by Spinoza >> doesn't lead to mathematics, does it? > > Nothing EB has quoted leads to mathematics. I just agreed that that doesn't, but that statement is unjustified. > >> But, infinite is a term often >> applied to quantities, whether of space, or time, or knowledge, or power... >> >> We have a unit interval, consisting of some infinity of distinct points >> (more than any finite number). We have the unit square, consisting of an >> equally infinite number of distinct parallel unit line segments. The >> unit cube, again, has this infinite number of parallel unit squares >> within it. Now, does it stand to reason that each added dimension to >> this figure multiplies the number of points by this infinite number? >> When we divide the line by this number, we get one point, the 0D square. >> There's really no way you can convince me that the cube does not have >> infinitely more points than the square, or the square than the segment, >> or the segment than the point. > > I think we can all agree that there is really no way to convince you of > anything. > Hmmm....perhaps that's because you're not right. >> These are different levels of infinity. >> >> As far as sequences go, we can also distinguish between different >> infinities, certainly where one is a subset of the other, but also where >> quantitative elements are mapped by formulas referencing their values. >> Those mapping formulas describe the relative sizes of the sets, >> parametrically. >> >> So, Spinoza might be reconsidering a little, were he still around. > And you might, too. TOny
From: Tony Orlow on 8 Dec 2006 14:28 David Marcus wrote: > Tonico wrote: >> Tony Orlow ha escrito: >>> Eckard Blumschein wrote: >>>> On 12/4/2006 9:56 PM, Bob Kolker wrote: >>>>> Eckard Blumschein wrote: >>>>> >>>>>> 2*oo is not larger than oo. Infinity is not a quantum but a quality. >>>>> But aleph-0 is a quantity. >>>> To those who belive in the usefulness of that illusion. >>>> >>> Aleph_0 is a phantom. The aleph_0th natural starting from 1 would be >>> aleph_0. It's not a count of the naturals. There is no smallest infinity >>> but, sorry to have to tell you, Eckard, a whole spectrum of infinities >>> that extend above and below any given infinite expression. Sure, >>> transfinitology is quasi-religious. Actual infinity can be quite >>> sensible, though. :) >>> >>> Tony >> *********************************************************** >> Just like good'ol Mad Journal with Spy vs Spy, but here it is "Troll vs >> Troll"...fascinating. > > A good analogy. > Good for your ego, but not very illustrative. Tony
From: Tony Orlow on 8 Dec 2006 14:33
David Marcus wrote: > Virgil wrote: >> In article <456f34bc(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >>> Virgil wrote: >>>> In article <456e475e(a)news2.lightlink.com>, >>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> Given a >>>>> set density, value range determines count. >>>> Compare the "set densities" of the set of naturals, the set of >>>> rationals, the set of algebraics, the set of transcendentals, the set of >>>> constructibles, and the set of reals. >>> Rather difficultt o formulate relations between those in standard >>> theory. In the name of IST, I'll avoid any criteria including the notion >>> of "standard" and state the following. The size of the set of >>> hypernaturals is the square root of the size of the set of hyperreals. >>> The set of hyperrationals corresponds to the square of the set of >>> hypernaturals, minus all those pairs that are redundant, such as 2/4 or >>> 6/18. That number of the hyperreals are the hyperirrationals. I am not >>> sure how to relatively quantify transcendentals, constrictibles, or >>> algebraics. Those are probably considered all "countable" by you, which >>> doesn't say much about their relative sizes. >> When challenged to support his fool theories, TO resorts to nonsense. > > At least he admitted that his fool theory has no relevance to any of the > number systems commonly used in Mathematics. > Of course I did. It depends on some substantially different assumptions. |