An uncountable countable set
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From: Virgil on 25 Sep 2006 23:47 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. > If so, then why do you say "There is no > infinite case"? Because there isn't any. > 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". > >> 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.
From: Virgil on 25 Sep 2006 23:54 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. The history of science is strewn with scientists getting their theories wrong and then having to change their minds about how things work. 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.
From: Han de Bruijn on 26 Sep 2006 03:20 Virgil wrote: > In article <2OudnWd259sao4XYnZ2dnUVZ_u-dnZ2d(a)onvoy.com>, > "Gerard Schildberger" <Gerard46(a)rrt.net> wrote: > >>| Virgil wrote: >>|> Han.deBruijn wrote: >>|>> Virgil wrote: >>|> > Def: 0.333... = lim_{n -> oo} Sum_{k = 1..n} 1/3^n >> >>|> Oh? >>|> >>|> 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729 + ... > 4.993141289 > 0.333 >>|> >>|> Han de Bruijn >> >>| My typo. My (1/3)^n should have been (3/10)^n. >>| >>| But your typo says >>| "1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729 + ... > 4.993141289" >>| Which is off by a factor of almost 10. >> >>for (3/10)^n, the sum for 100 terms is around 0.42857142857+ >>________________________________________________________Gerard S. > > I'm really being sloppy today, (3/10)^n should be 3/ (10^n). Virgil, shall we both go to bed (separately, huh ;-) instead of trying to post_before_we_think, so late in the evening? >>for (3/10)^n, the sum for 100 terms is around 0.42857142857+ >>> ________________________________________________________Gerard S. > > And the sum of the geometric series as a whole is 1/2 . Heheh ... No. The sum of _that_ series is 1/(1 - 3/10) - 1 = 3/7 = (see above) Han de Bruijn
From: Han de Bruijn on 26 Sep 2006 03:43 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 ? "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. > Math=Science? Definitely, yes! Han de Bruijn
From: Han de Bruijn on 26 Sep 2006 03:49
Virgil wrote: > 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. I have suggested a proper place for mathematics in: http://hdebruijn.soo.dto.tudelft.nl/QED/klassiek.htm What do you think, Virgil? Han de Bruijn |