Galileo's Paradox
in [Math]
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From: Virgil on 9 Dec 2006 15:50 In article <457af56a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > But, > there are serious questions regarding the measure of infinite sets, as > Six Letters points out. Questions about mathematics considered serious by non-mathematicians most often merely demonstrate that they are non-mathematicians. > It's not unreasonable to consider the proper > subset as always smaller than the superset, though that is clearly not > how cardinality works for infinite sets. It is not unreasonable to have different measures of set size depending on what properties of those sets are being measured. And it is not unreasonable to note that different measures will behave differently, that when for one measure of one property, two sets are the same but for a measure of a different property the same two sets differ, that the two measures will disagree as well. > So, there is an ongoing attempt > in various directions to develop a better theory, or to reject it as an > absurdity. The absurdity is to suppose that sets have only one measurable quality. > I prefer the first route, where IFR and N=S^L grow out of > infinite-case induction and provide a means for finely ordering all > sorts of infinite sets, countable and uncountable. IFR only applies, if at all, only for order preserving functions from ordered sets to ordered sets, which leaves out all unordered sets and multiply ordered sets. It cannot even deal with subsets of the Cartesian plane. N = S^L is only demonstrably valid for finite sets, for which there are no problems with cardinality anyway, so is entirely redundant.
From: Virgil on 9 Dec 2006 16:02 In article <457af7ba$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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. Finite strings are alleged to have TWO ends, a lef-hand end AND a right-hand end, so merely specifying that a string has a "left-hand end" does not indicate that it must be finite. > 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. What does "potentially infinite" mean to TO? I do not know of any mathematical system in which any of the pseudo-definitions of "potentially infinite" so far presented here make any sense.
From: Tony Orlow on 9 Dec 2006 16:06 David Marcus wrote: > Tony Orlow 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. :) > > Trolls are smarter than cranks. > Do you think? Trolls hide under bridges and try to scare you. Cranks get your car started. Trolls just try to get attention. Cranks are hard at work, even if you think their machine is useless. Trolls might be more socially "effective", but then, a lot of stupid people get lots of attention. At least I'm not a statue.
From: Virgil on 9 Dec 2006 16:11 In article <457af8dd(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > > > >>>> 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. > > > > You have a very limited view of what mathematics is. A better > > description is, Mathematics is the study of patterns. I believe Saunders > > MacLane presented this view in his book "Mathematics, Form and > > Function". You might like this book. MacLane believed set theory is not > > the best foundation for mathematics. > > > > I have a simplistic view of it. There is measure, the language of > measure, and the operations within the language. To me, logic is a form > of mathematics where all values lie within the first unit interval. I > try to stick to the basics - Occam's Razor, eh? Thanks for the > reference, by the way. > > > TOny Saunders Mac Lane preferred to have his family name spelled with a space between "Mac" and "Lane". E.g., See http://en.wikipedia.org/wiki/Saunders_Mac_Lane It is a preference often ignored. When TO says "there is measure", does he have a mathematically satisfactory definition of what he means by a "measure"? TO might take as an example, the definition in: http://en.wikipedia.org/wiki/Measure_theory
From: Virgil on 9 Dec 2006 16:12
In article <457afa7e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <MPG.1fe42308b91485ee989a0b(a)news.rcn.com>, > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > >> Virgil wrote: > >>> In article <4579d1c7(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> I chose to work within computer science, after having planned to become > >>>> a mathematician, for the obvious reasons.... > >>> Couldn't cut it as a mathematician? > >> Probably couldn't understand any of his math courses. > > > > I think TO once said that he disagreed with one of his math teachers so > > strongly that he decided to give up on math as a major. > > It was never my major. > > > > > I suppose pat of it might have been that the teacher was a graduate > > assistant with a bad attitude, but I rather think that a large part of > > it was the student's bad attitude. > > I understood my math courses and aced them, thanks. I did fine with > transfinitology, even though I didn't believe it. I disagreed with the > teacher on it, but we didn't have a conflict over it. It certainly made > me think that pure mathematics had a great potential for nonsense, and > that going into an applied math area, like the math machines we call > computers, would be more rewarding and productive. Part of it had to do > with the binary code upon which they are built, which I had been > developing myself in conjunction with a dualistic model of the universe, > and part had to do with the computer being a simple brain, and the model > describing the structure of mind, which might be implemented on a > computer. I never ran away from math. I mixed it with reality. :) To the detriment of both. |