An uncountable countable set
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From: Virgil on 26 Sep 2006 14:45 In article <84b61$4518e269$82a1e228$10258(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > The history of science is strewn with scientists getting their theories > > wrong and then having to change their minds about how things work. > > Yes. That is called learning from your mistakes. > > > The only major time that happened in mathematics was the discovery of > > non-Eucidean geometry, and Euclidean geometry was based mostly on the > > conlusins of science, not pure math. > > It's a bit off-topic but I have the following question. The mathematical > discipline called 'analytical geometry' - as teached in our schools - is > the algebraic equivalent of Euclidian geometry. OK ? Parallellism of two > straight lines can be detected algebraically. And moreover, two straight > lines CAN be parallel. How then can 'analytical geometry' translate into > an algebraic equivalent for non-Euclidian geometries? > There are models of all the non-Euclidean geometries which can be embedded in Euclidean geometry, but in them the "lines" are not the straight lines of Cartesian geometry. For an example, geometry on the surface of a sphere. Take the great circles on a sphere as the "straight" lines (the shortest distance between two points on the sphere along its surface necessarily being part of a great circle). This can all be done very nicely within Cartesian geometry. There is even a specialty called spherical geometry, which is an embedding of a non-Euclidean geometry in Euclidean geometry.
From: Virgil on 26 Sep 2006 15:08 In article <45193afa(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45189e62(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45187864(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Han de Bruijn wrote: > >>>>> Tony Orlow wrote: > >>>>> > >>>>>> Mike Kelly wrote: > >>>>>> > >>>>>>> Han de Bruijn wrote: > >>>>>>> > >>>>>>>> Mike Kelly wrote: > >>>>>>>> > >>>>>>>>> Han de Bruijn wrote: > >>>>>>>>> > >>>>>>>>>> Mike Kelly wrote: > >>>>>>>>>> > >>>>>>>>>>> What the hell are you talking about? Arguing with someone who > >>>>>>>>>>> can't > >>>>>>>>>>> speak English is getting aggravating. > >>>>>>>>>> My English is much better than your Dutch. > >>>>>>>>> So what? Your English is still too poor for this discussion to be > >>>>>>>>> fruitful. > >>>>>>>> Still don't get the point, huh? > >>>>>>>> > >>>>>>>> You are lacking even the most elementary form of politeness. It's > >>>>>>>> very > >>>>>>>> impolite to cut of a discussion with somebody from a foureign > >>>>>>>> country - > >>>>>>>> somebody who is doing his best to communicate with you - only > >>>>>>>> because > >>>>>>>> you are obviously superior in expressing your thoughts within your > >>>>>>>> own > >>>>>>>> mother's tongue. > >>>>>>> You're a very rude person yourself, Han. I generally don't feel the > >>>>>>> need to be civil to those who won't reciprocate. > >>>>>> I don't think I have ever found Han to be rude, except when he > >>>>>> referred to my "babbling" recently. Ahem. But anyway, while we > >>>>>> disagree on the actuality of any infinity, we have the open mind of > >>>>>> spirited debate, and feel no need to get nasty. > >>>>> I may be rude sometimes, but I never get _personal_ by calling somebody > >>>>> an "idiot" or a "crank". Tony's "babbling" translates with Euroglot as > >>>>> "babbelen" in Dutch, which is a word I can use here in the conversation > >>>>> with my collegues without making them very angry (if I say "volgens mij > >>>>> babbel je maar wat"). But, of course, I cannot judge the precise impact > >>>>> of the word in English. Apologies if it is heavier than I thought. > >>>> Do you have the saying, "Shallow brooks babble, and still waters run > >>>> deep"? I figured you picked up the usage from this forum, actually. It's > >>>> meant, in English, to mean you aren't making any sense. :) > >>>> > >>>>>> Furthermore, I have never had any trouble understanding what Han is > >>>>>> saying, except where he is using some mathematical construct with > >>>>>> which I am not familiar. His English is not bad, and blaming your > >>>>>> disagreement on his inability to communicate is kind of low. > >>>>> Thank you very much, Tony, for this sort of defense. > >>>> My pleasure. It seemed like a vacuous excuse. I get pretty sick of those > >>>> diversionary tactics. > >>>> > >>>>>> So, let's engage in lively debate, and maintain our civility, while > >>>>>> chopping each other's arguments to pieces. Of course, this can only > >>>>>> happen if we don't consider our arguments to be part of our anatomy. > >>>>>> Otherwise, it gets personal. > >>>>>>>>> You are misinterpreting virtually all my posts. You claim that > >>>>>>>>> you're > >>>>>>>>> not dishonest so I have to conclude you're simply incapable of > >>>>>>>>> comprehending written English. This makes this whole subthread > >>>>>>>>> pointless. > >>>>>>>> I have only this kind of trouble with _you_ and nobody else on the > >>>>>>>> web. > >>>>>>> Really? You've never had anybody else other than me complain that you > >>>>>>> misinterpret their posts? I suppose I must have hallucinated dozens > >>>>>>> of > >>>>>>> posts I've seen of just that, then. > >>>>>>> > >>>>>>> You've never had anyone other than me struggling to understand what > >>>>>>> the > >>>>>>> devil you mean by your broken English? I must have hallucinated, for > >>>>>>> example, "A little physics would be no idleness in mathematics", then > >>>>>>> :)? > >>>>>> Well, that's a difficult type of quote. Han - I wouldn't mind working > >>>>>> on exactly how you want to say that in English, if you like. :) > >>>>> Uhm, since litteraly everybody is complaining ... Let it be an > >>>>> encrypted > >>>>> message then :-) > >>>> Well, it seems to me that perhaps you're saying something like, "Those > >>>> with their heads in the abstract should keep their feet in the > >>>> concrete", though that sounds a little funny. > >>>> > >>>> Math=Science? > >>> > >>> Scientists, particularly those in the sciences most dependent on > >>> mathematics, tend to think that all mathematics is, or should be, a > >>> subservient to their particular fragment of science. > >>> > >>> Mathematicians know better. > >> Define "better". Those that work in various areas of science share a > >> notion which defines science. Theories which have no means of > >> verification are not science, but philosophy. In mathematics, > >> verification really consists of corroboration by other means, agreement > >> between different approaches. In science, where you find a contradiction > >> with your theory, it needs revision. So, the scientific approach to > >> mathematics requires some criterion for universal consistency, as > >> measured by the predictions of the various theories that comprise it. > >> Where two theories collide, one or both is in error. I think that's > >> better. > > > > TO mistakes misapplication of mathematics, which is an error by > > scientists, as an error of the mathematics. > > > > That's not what I said, and you know it. When an application of mathematics to physics, or some other science, does not predict what actually is observed in that science, why does TO insist the problem is inherent in the mathematics alone and not in the improper application of mathematics to science? > > > > > The history of science is strewn with scientists getting their theories > > wrong and then having to change their minds about how things work. > > The same holds true for mathematics. How long did it take to resolve the > inconsistencies in set theory to the satisfaction of the mathematical > community? It hasn't been done yet to the satisfaction of all mathematicians, and probably never will be.
From: Virgil on 26 Sep 2006 15:17 > Han de Bruijn wrote: > > Tony Orlow wrote: > > > >> Han de Bruijn wrote: > >>> > >>> I may be rude sometimes, but I never get _personal_ by calling somebody > >>> an "idiot" or a "crank". Tony's "babbling" translates with Euroglot as > >>> "babbelen" in Dutch, which is a word I can use here in the conversation > >>> with my collegues without making them very angry (if I say "volgens mij > >>> babbel je maar wat"). But, of course, I cannot judge the precise impact > >>> of the word in English. Apologies if it is heavier than I thought. > >> > >> Do you have the saying, "Shallow brooks babble, and still waters run > >> deep"? I figured you picked up the usage from this forum, actually. > >> It's meant, in English, to mean you aren't making any sense. :) > > > > I have talked to my collegues and they have told me that "babbling" in > > English indeed has a meaning which is somewhat different from "babbelen" > > in Dutch. It is more like our "lallen", to be translated in English as > > "talking while you are drunk". Is that correct ? > > Yes, or just with an inherently incoherent mind. > > > > > "Babbelen" in Dutch is more like "having a nice chat". It's not that it > > doesn't make any sense, but it's not very deep either. It's more social > > than intellectual. In the biblical story of the Tower of Babel, the tower builder's speech becomes incomprehensible to each other, and, at least in English, much of that meaning remains in the word "babble". The sounds babies make before they have learned to talk is called 'babbling'.
From: Virgil on 26 Sep 2006 15:24 In article <45193e6f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Han de Bruijn wrote: > > Tony Orlow wrote: > > > >> Virgil wrote: > >>> > >>> Mathematicians know better. > >> > >> Define "better". Those that work in various areas of science share a > >> notion which defines science. Theories which have no means of > >> verification are not science, but philosophy. In mathematics, > >> verification really consists of corroboration by other means, > >> agreement between different approaches. In science, where you find a > >> contradiction with your theory, it needs revision. So, the scientific > >> approach to mathematics requires some criterion for universal > >> consistency, as measured by the predictions of the various theories > >> that comprise it. Where two theories collide, one or both is in error. > >> I think that's better. > > > > Precisely ! In mathematics, there are contradictory approaches, such as > > constructivism (Brouwer) against axiomatism (Hilbert). Its practicioners > > are asked to be "nice" to each other and to "reconciliate" the different > > points of view, which turns out to be a hopeless task. Such a situation > > would be unthinkable if mathematics aimed to be a science. > > > > Han de Bruijn > > > > Well, Han, I'm not sure I agree with the statement that reconciliation > is hopeless. Is it hopeless to reconcile the wave nature of elementary > entities with their particle nature? It is close to hopeless to expect those who reject the law of the excluded middle (constructionists) and those who insist on it (formalists) to agree. > There is confusion about my "definition" of infinitesimals, because I > can see the validity both in nilpotent infinitesimals and in those that > are further infinitely divisible. Until TO can come up with an axiom system which simultaneously allows his infinitesimals to be both nilpotent and not, he is in trouble. > Constructivism and Axiomatism are two sides of a coin. They can be > reconciled in larger framework, I think. Better men that TO will ever be have tried.
From: MoeBlee on 26 Sep 2006 19:01
Tony Orlow wrote: > Constructivism and Axiomatism are two sides of a coin. They can be > reconciled in larger framework, I think. I don't know what your definition of 'axiomatism' is, but there are axiomatic systems for constructive mathematics. MoeBlee |