Cantor Confusion
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From: Dik T. Winter on 14 Oct 2006 23:18 In article <1160855574.230559.141770(a)e3g2000cwe.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Dik T. Winter schreef: .... > NO with contemporary mathematics. But with my renormalized mathematics: > YES. The gist of renormalized mathematics is to add a tiny bit of error > propagation to the exact outcome. (According to my matra: a little bit > of Physics.) You know: empirical sciences are _never_ free of "errors", > inexactness and uncertainity. I know. But that does not (in my opinion) have repercussions on mathematics. As has been stated by von Neumann: "it is impossible to find solutions of fourty equations with fourty unknowns". Error analysis is the key-point here. Until J. H. Wilkinson it was only forward error analysis, and that leads to nothing. Since his work we also have backward error analysis, and that gives results. But if you, as a scientist, refuses to do anything like error analysis, your results are at most mediocre. > > However, let's have a look at the entier function. > > lim(x from 0 -> 1) entier(x) = 0, but entier(1) = 1. > > It seems that the limit is different from the actual value of the > > function. > > If we convolute the entier function with a Gaussian function with very > small spread sigma, then it becomes continuous (: kind of a smoothened > staircase). And your argument evaporates accordingly. Yes, and in the mathematical sense the function does not work any longer as it should. Something like a function not always returning an integral value. > And your argument evaporates accordingly. Yes, that is the > power of renormalization. Take mathematics "as is" and add that little > bit of Physics. I think you should add a little bit of Mathematics to Physics. > For your eyes only. My "numerical differentiation schemes" work with > the same Gaussians. This doesn't mean that they are better than yours. > It only means that I find it important to have a uniform theory here. Pray come up with a theory that explains why your methods do not work better than my methods. And further explain that when I said your methods would suffer the same problems as my methods (when decreasing step-size), how that could follow from your theory, while I based my opinion onlyon standard theory. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 14 Oct 2006 23:21 In article <1160856043.795135.198610(a)h48g2000cwc.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Dik T. Winter schreef: > > In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl> > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > > > > Set Theory is simply not very useful. > > > > Oh. So you think that Banach spaces are not very useful? You think that > > a book like "The Algebraic Eigenvalue Problem" is not very useful? You > > may note that both are heavily based on set theory. > > Think I could rewrite the relevant stuff in those books without using > any set theory. Think ahead. > I'm not actually going to do it, though. Why not? > (BTW, I find > Banach Spaces not very useful either) They are (as far as I know) used in the design of methods to solve partial differential equations. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 14 Oct 2006 23:26 In article <1160856895.115824.134080(a)b28g2000cwb.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > stephen(a)nomail.com schreef: > > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: .... > > > Pray warn me when 2 has changed sufficiently to be the square of a > > > rational. > > > I would not like to miss that moment. > > > > The day the circle is squared cannot be to far behind. > > Come on, guys! You all know that, in the world of approximations, > 2 _is_ the square of a rational and the circle _is_ squared. I thought you were talking mathematics? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 15 Oct 2006 00:31 In article <45319f23(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > You fellers sure are dense. Han's simply saying that the universe is > consistent, or it doesn't exist at all. Consistent with what? With itself sure, but not necessarily with our view of it.
From: Virgil on 15 Oct 2006 00:44
In article <4531a0fa(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > y=9x is discontinuous? y = 9n is discontinuous! Or does TO claim that the number of iterations changes continuously? >My god, I had no idea how bad it had gotten! We had no idea how far TO had gotten from reality. > > > >> The function f(t) = 9t is continuous, because the function 1/9t is > >> continuous. But f(n) = 9 n, for n in N, is not. > > > > Yes, but that is not the number of balls in the vase. > > After iteration n, there's 9n balls in the vase. Does TO claim that 9n moves continuously up to 9(n+1) as n jumps discontinuously from n to n+1? |