An uncountable countable set
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From: Han de Bruijn on 26 Sep 2006 04:01 Dik T. Winter wrote: > In article <1159211074.494116.142040(a)e3g2000cwe.googlegroups.com> > Han.deBruijn(a)DTO.TUDelft.NL writes: > ... > > I'm still flabbergasted why those difficult proofs as for Fermat's Last > > Theorem or the Poincare Conjecture are not proved then with the full > > power of modern computers. > > Perhaps because computers can only be used to prove finitely many cases > and that FLT and Poincare are not tangible to be reduced to finitely > many cases (like 4CT)? Has Pythagoras ever been proved automatically? Han de Bruijn
From: Han de Bruijn on 26 Sep 2006 04:13 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
From: Han de Bruijn on 26 Sep 2006 04:18 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? (I've asked this question before, but got no answer) Han de Bruijn
From: Tony Orlow on 26 Sep 2006 10:33 Virgil wrote: > In article <45189d2a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45187409(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <451149ef(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>> >>>>>> Consider the equally spaced staircase from (0,0) to (1,1), as the number >>>>>> of steps increases from 1 without bound. Is it the same as the diagonal >>>>>> line? Inductively we can prove that the length of the staircase is 2 at >>>>>> every step. Does it really suddenly become sqrt(2) in the infinite case? >>>>> There is no "infinite case", there is only a limit case. >>>>> >>>> Then noon never comes and the vase is never empty, >>> >>> Both are limit cases, which are what actually occurs, given the models. >> Is this state actually achieved? > > > Those limits actually exist, according to the apropriate definitions of > limits, without ever any infinite state being "achieved", whatever that > might mean. > That invalidates the entire ball-and-vase problem, if noon can never be reached and your infinite state of affairs realized. So, you are conjecturing about what occurs when something impossible happens. That's very useful. > >> If so, then why do you say "There is no >> infinite case"? > > > Because there isn't any. There is no noon in the Zeno machine? > >> If not, then why do you say the limit case "actually >> occurs"? > > I don't. > > The existence of a limit does not require the existence of a "limit > case". > See above: "Both are limit cases, which are what actually occurs, given the models." >>>> Each such pair denotes, through these relative >>>> coordinates, a length and a direction. >>> So what length and direction are indicated by the pair {(0,0),(0,0)} ? >> That's a pair of pairs, first of all. >> >> {0,0} would represent a null segment, really a point, without size or >> direction. >> >> For all x, {1/x,1/x} always has a slope of 1 and a length of sqrt(2)/x. >> The corresponding section of the staircase, though, is two segments, >> {0,1/x} and {1/x,0}, one with infinite slope, and one with zero slope. >> They are clearly differently defined segments, direction-wise, and >> length-wise. > > Not in the "limit case" which TO keeps insisting must exist. > Nor even in the limit. Why not? Hand waving does not suffice. As x increases without bound, the ration between 1/x and 1/x remains precisely at 1, whereas the ratios between 0 and 1/x, and 1/x and 0, remain constant at 0 and oo, respectively. I suppose if you consider 1/x=0 for any infinite x, you can claim that the staircase degrades into a sequence of directionless points, but that's clearly not the case with the corresponding segments in the diagonal.
From: Tony Orlow on 26 Sep 2006 10:36
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. > > 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? What ever happened to infinitesimals? > > 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. If you say so. Somehow I thought theories about numbers starting at 1 and all being countable, or always positive, or always finite, or not having a square root if negative, were all theories that were overturned given later advances. Where calculus was invented, the theory took a long time to formally state. So, you're claim is simply wrong. |