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. In his next letter, Cantor claimed that this was indeed his point: where Riemann andothers had casually spoken of a space that requires n coordinates as if that number wasknown to be invariant, he felt that this invariance required proof. Far from wishing toturn my result against the article of faith of the theory of manifolds, I rather wish touse it to secure its theorems, he wrote.The required theorem turned out to be true,indeed, but proving it took much longer than either Cantor or Dedekind could haveguessed: it was finally proved by Brouwer in 1910.The long and convoluted story ofthat proof can be found in [3], [11], and [12].Finally, one should point out that it was only some three months later that Cantorfound what most modern mathematicians consider the obvious way to prove thatthere is a bijection between the interval minus a countable set and the whole interval.In a letter dated October 23, 1877, he took an enumeration ƽ of the rationals and let"·½ = 2/2½.Then he constructed a map from [0, 1] sending ·½ to ·2½-1, ƽ to ·2½, andevery other point h to itself, thus getting a bijection between [0, 1] and the irrationalnumbers between 0 and 1.7.MATHEMATICS AS CONVERSATION.Is the real story more interesting thanthe story of Cantor s surprise? Perhaps it is, since it highlights the social dynamic thatunderlies mathematical work.It does not render the theorem any less surprising, butshifts the focus from the result itself to its proof.The record of the extended mathematical conversation between Cantor and Dede-kind reminds us of the importance of such interaction in the development of mathe-matics.A mathematical proof is, after all, a kind of challenge thrown at an idealizedopponent, a skeptical adversary that is reluctant to be convinced.Often, this adversaryis actually a colleague or collaborator, the first reader and first critic.A proof is not a proof until some reader, preferably a competent one, says it is.Untilthen we may see, but we should not believe.REFERENCES1.K.-R.Biermann, Dedekind, in Dictionary of Scientific Biography, C.C.Gillispie, ed., Scribners, NewYork, 1970 1981.208 © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118 2.W.Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathe-matics, Princeton University Press, Princeton, 2007.3.J.W.Dauben, The invariance of dimension: Problems in the early development of set theory and topology,Historia Math.2 (1975) 273 288.doi:10.1016/0315-0860(75)90066-X4., Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press,Princeton, 1990.5.R.Dedekind, Essays in the Theory of Numbers (trans.W.W.Beman), Dover, Mineola, NY, 1963.6., Gesammelte Mathematische Werke, R.Fricke, E.Noether, and O.Ore, eds., Chelsea, New York,1969.7., Stetigkeit und Irrationalzahlen, 1872, in Gesammelte Mathematische Werke, vol.3, item L,R.Fricke, E.Noether, and O.Ore, eds., Chelsea, New York, 1969.8.W.Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford UniversityPress, Oxford, 1996.9.R.Gray, Georg Cantor and transcendental numbers, Amer.Math.Monthly 101 (1994) 819 832.doi:10.2307/297512910.J.Hadamard, An Essay on the Psychology of Invention in the Mathematical Field, Princeton UniversityPress, Princeton, 1945.11.D.M.Johnson, The problem of the invariance of dimension in the growth of modern topology I, Arch.Hist.Exact Sci.20 (1979) 97 188.doi:10.1007/BF0032762712., The problem of the invariance of dimension in the growth of modern topology II, Arch.Hist.Exact Sci.25 (1981) 85 267.doi:10.1007/BF0211624213.H.Meschkowski.Cantor, in Dictionary of Scientific Biography, C.C.Gillispie, ed., Scribners, New York,1970 1981.14.E.Noether und J.Cavaillès, Briefwechsel Cantor Dedekind, Hermann, Paris, 1937.FERNANDO Q.GOUVÊA is Carter Professor of Mathematics at Colby College in Waterville, ME.He isthe author, with William P.Berlinghoff, of Math through the Ages: A Gentle History for Teachers and Others.This article was born when he was writing the chapter in that book called Beyond Counting. So it s Bill sfault.Department of Mathematics and Statistics, Colby College, Waterville, ME 04901fqgouvea@colby.eduMarch 2011] WAS CANTOR SURPRISED? 209 [ Pobierz caÅ‚ość w formacie PDF ]
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. In his next letter, Cantor claimed that this was indeed his point: where Riemann andothers had casually spoken of a space that requires n coordinates as if that number wasknown to be invariant, he felt that this invariance required proof. Far from wishing toturn my result against the article of faith of the theory of manifolds, I rather wish touse it to secure its theorems, he wrote.The required theorem turned out to be true,indeed, but proving it took much longer than either Cantor or Dedekind could haveguessed: it was finally proved by Brouwer in 1910.The long and convoluted story ofthat proof can be found in [3], [11], and [12].Finally, one should point out that it was only some three months later that Cantorfound what most modern mathematicians consider the obvious way to prove thatthere is a bijection between the interval minus a countable set and the whole interval.In a letter dated October 23, 1877, he took an enumeration ƽ of the rationals and let"·½ = 2/2½.Then he constructed a map from [0, 1] sending ·½ to ·2½-1, ƽ to ·2½, andevery other point h to itself, thus getting a bijection between [0, 1] and the irrationalnumbers between 0 and 1.7.MATHEMATICS AS CONVERSATION.Is the real story more interesting thanthe story of Cantor s surprise? Perhaps it is, since it highlights the social dynamic thatunderlies mathematical work.It does not render the theorem any less surprising, butshifts the focus from the result itself to its proof.The record of the extended mathematical conversation between Cantor and Dede-kind reminds us of the importance of such interaction in the development of mathe-matics.A mathematical proof is, after all, a kind of challenge thrown at an idealizedopponent, a skeptical adversary that is reluctant to be convinced.Often, this adversaryis actually a colleague or collaborator, the first reader and first critic.A proof is not a proof until some reader, preferably a competent one, says it is.Untilthen we may see, but we should not believe.REFERENCES1.K.-R.Biermann, Dedekind, in Dictionary of Scientific Biography, C.C.Gillispie, ed., Scribners, NewYork, 1970 1981.208 © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118 2.W.Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathe-matics, Princeton University Press, Princeton, 2007.3.J.W.Dauben, The invariance of dimension: Problems in the early development of set theory and topology,Historia Math.2 (1975) 273 288.doi:10.1016/0315-0860(75)90066-X4., Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press,Princeton, 1990.5.R.Dedekind, Essays in the Theory of Numbers (trans.W.W.Beman), Dover, Mineola, NY, 1963.6., Gesammelte Mathematische Werke, R.Fricke, E.Noether, and O.Ore, eds., Chelsea, New York,1969.7., Stetigkeit und Irrationalzahlen, 1872, in Gesammelte Mathematische Werke, vol.3, item L,R.Fricke, E.Noether, and O.Ore, eds., Chelsea, New York, 1969.8.W.Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford UniversityPress, Oxford, 1996.9.R.Gray, Georg Cantor and transcendental numbers, Amer.Math.Monthly 101 (1994) 819 832.doi:10.2307/297512910.J.Hadamard, An Essay on the Psychology of Invention in the Mathematical Field, Princeton UniversityPress, Princeton, 1945.11.D.M.Johnson, The problem of the invariance of dimension in the growth of modern topology I, Arch.Hist.Exact Sci.20 (1979) 97 188.doi:10.1007/BF0032762712., The problem of the invariance of dimension in the growth of modern topology II, Arch.Hist.Exact Sci.25 (1981) 85 267.doi:10.1007/BF0211624213.H.Meschkowski.Cantor, in Dictionary of Scientific Biography, C.C.Gillispie, ed., Scribners, New York,1970 1981.14.E.Noether und J.Cavaillès, Briefwechsel Cantor Dedekind, Hermann, Paris, 1937.FERNANDO Q.GOUVÊA is Carter Professor of Mathematics at Colby College in Waterville, ME.He isthe author, with William P.Berlinghoff, of Math through the Ages: A Gentle History for Teachers and Others.This article was born when he was writing the chapter in that book called Beyond Counting. So it s Bill sfault.Department of Mathematics and Statistics, Colby College, Waterville, ME 04901fqgouvea@colby.eduMarch 2011] WAS CANTOR SURPRISED? 209 [ Pobierz caÅ‚ość w formacie PDF ]