From: renfrod@central.edu (Dave L. Renfro) Subject: Re: What is the point of rigor? Date: 24 Oct 2000 00:47:08 -0400 Newsgroups: sci.math Summary: [missing] Martin Green [sci.math Sun, 22 Oct 2000 22:28:49 GMT] wrote > Martin Green wrote: > >> > Are there any examples of....a mathematical result which >> > appeared to be right, until the Bertrand Russell types had >> > a go at it and found out that it was wrong? > > Dennis Feldman evaded answering: > >> Many; and major ones, at that. Have a look at the history of >> analysis. Or the invention of fractals. Or Godel theorems... > > Did you even read what I wrote? I didn't ask for examples of > new branches of mathematics....I asked if there were any cases > of old results which were proven wrong by the application of > "rigorous" methods. The failure to distinguish between uniform and non-uniform convergence of series of functions. [Early (before approx. 1820) work in Fourier series.] Assuming term-by-term integration of infinite series of functions. [Many mathematicians, middle 1800's.] The failure to distinguish between nowhere dense sets and sets whose Jordan measure equals zero. [Du Bois-Reymond's, Hankel's, etc. (approx. 1870 to 1885) work in integration theory.] [From Thomas Hawkins' book, pp. 43-44] "Between the appearance of Ampere's paper [in 1806] and 1870, the proposition that any (continuous) function is differentiable in general [i.e. except for points isolated from each other] was stated and proved in most of the leading texts on the calculus." [Ten references to incorrect proofs are given.] Lebesgue's incorrect assertion (made without proof) in a very highly regarded 1905 paper that the projection onto a line of a G_delta set in the plane is a Borel set. Here are some things I doubt would have been discovered without the application of a fair amount of rigor: The existence of transcendental numbers, nowhere differentiable continuous functions, Banach-Tarski decompositions, Peano curves, Godel incompleteness results, various independence results in ZFC, real numbers whose decimal expansions cannot be coded up by the use of a computer program, free ultrafilters on the natural numbers, the additive group of real numbers is isomorphic to the additive group of complex numbers [this one is for Pertti Lounesto], Sierpinski-Zygmund functions, C^infinity functions that are nowhere analytic, the fact that for each positive integer n at least one of exp(e^n), exp(e^(2n)), exp(e^(3n)) is transcendental, the prime number theorem, 2^(Aleph_0) pairwise disjoint subsets of [0,1] that each have outer Lebesgue measure 1, the set of non-continuity points of a function from the reals to the reals can be exactly the set of transcendental numbers but it cannot be the set of rational numbers, the subset (rationals)^2 union (irrationals)^2 is a pathwise connected subset of R^2, there exist non-measurable functions that satisfy the intermediate value property, . . . . . . . Dave L. Renfro ============================================================================== From: wrameyxiii@home.remove13.com (Wade Ramey) Subject: Re: What is the point of rigor? Date: Tue, 24 Oct 2000 19:43:48 GMT Newsgroups: sci.math Summary: [missing] In article , "Martin Green" wrote: > Looking at it intuitively, I'd > say there are lots of zigzags there, and the sum of the series should be > well-defined for any value of x. My intuition disagrees. What do we do now, take a vote? > I notice, by the way, that he did not see fit to provide an example > of how you can get in trouble by integrating before summing, the way > I gave an example of how you can get in trouble by differentiating. That's on p. 15 of your local real analysis text. Here's a simple example. On R, set f_n = 1 on [n, n+1], 0 elsewhere. Then f_n -> 0 pointwise everywhere on R, yet the integral of f_n over R is 1 for all n. So the integral of the limit is not the limit of the integrals. Because every sequence can be viewed as a series, you have your example. Wade PS: Martin, I am enjoying your thread ============================================================================== From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: What is the point of rigor? Date: 23 Oct 2000 21:06:51 -0500 Newsgroups: sci.math In article <44OI5.24606$6O5.2255352@news1.rdc1.mb.home.com>, Martin Green wrote: >Martin Green wrote: >> > Are there any examples of ...a mathematical result which appeared >> > to be right, until the Bertrand Russell types had a go at it and found >out >> > that it was wrong?? Here is one in statistics. There are many problems which are invariant under quite reasonable transformations. Objectivity would say that invariant solutions are good. One can often find out quite easily what the best invariant solution is. However, there are reasonable problems for which this is very definitely not the case. One may argue that the first one of these found as not reasonable, but this led to the quite natural ones which are known. The obtaining of a rigorous treatment led to the type of previously known mathematical paradox which were the natural counterexamples. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 ============================================================================== From: Pertti Lounesto Subject: Re: Q: An example of a function Date: Thu, 26 Oct 2000 07:35:44 +0300 Newsgroups: sci.math Gerry Myerson wrote: > If you're into books - remember books? - Gelbaum & Olmsted, > Counterexamples in Analysis, is full of stuff like this. There are other books on counterexamples. See M. Capobianco, J. Molluzo: Examples and Counterexamples in Graph Theory. North Holland, Amsterdam, 1978. Ya.S. Dubnov: Mistakes in Geometric Proofs. Heath, Boston, 1963. J.E. Forn\ae ss, B. Stens\o nes: Lectures on Counterexamples in Several Complex Variables. Princeton UP, Princeton, NJ, 1987. B.R. Gelbaum, J.H.M. Olmsted: Counterexamples in Analysis. Holden-Day, San Francisco, 1964. B.R. Gelbaum, J.H.M. Olmsted: Theorems and Counterexamples in Mathematics. Springer, New York, 1990. B. Hauchecorne: Les contre-exemples en math\'ematiques. Ellipses, Paris, 1988. S.M. Khaleelulla: Counterexamples in Topological Vector Spaces. Springer, Berlin, 1982. I. Lakatos: Proofs and Refutations. The Logic of Mathematical Discovery. Cambridge University Press, Cambridge, 1976. P. Lounesto: Counterexamples in Clifford algebras with CLICAL, pp. 3-30 in R. Ablamowicz et al. (eds.): Clifford Algebras with Numeric and Symbolic Computations. Birkh\"auser, Boston, 1996. J.P. Romano, A.F. Siegel: Counterexamples in Probability and Statistics. Wadsworth & Brooks, Monterey, CA, 1986. L.A. Steen, J.A. Seebach: Counterexamples in Topology. Holt, Rinehart and Winston, New York, 1970. Springer, New York, 1978. J. Stoyanov: Counterexamples in Probability. John Wiley, Chichester, 1987, 1997. G.L. Wise, E.B. Hall: Counterexamples in Probability and Real Analysis. Oxford University Press, New York, 1993.