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An Editor Recalls Some Hopeless Papers

  • Wilfrid Hodges (a1)

§1. Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument (I mean the one proving that the set of real numbers and the set of natural numbers have different cardinalities) have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.

These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.

In 1995 an engineer named William Dilworth, who had published a refutation of Cantor's argument in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, sued for libel a mathematician named Underwood Dudley who had called him a crank ([9] pp. 44f, 354).

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[1] Aristotle, Metaphysics X–XIV, Loeb Classical Library, Harvard University Press, Cambridge, Mass., 1935.
[2] BarrowJohn D., Theories of everything: the quest for ultimate explanation, Oxford University Press, Oxford, 1990.
[3] BarwiseJon and EtchemendyJohn, The liar, Oxford University Press, New York, 1987.
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[5] BethE.W., Semantic entailment and formal derivability, The philosophy of mathematics (HintikkaJaakko, editor), Oxford University Press, 1969, pp. 941.
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[7] CantorGeorg, Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. I (1891), pp. 7578.
[8] CarrollLewis, What the tortoise said to Achilles, Mind, vol. 4 (1895), pp. 278280.
[9] DudleyUnderwood, Mathematical cranks, Mathematical Association of America, Washington, DC, 1992.
[10] DummettMichael, Frege, philosophy of language, Duckworth, London, 1973.
[11] FisherAlec, The logic of real arguments, Cambridge University Press, Cambridge, 1988.
[12] FraenkelAbraham A., Abstract set theory, 4th ed., North-Holland, Amsterdam, 1976, revised by Azriel Lévy.
[13] GelernterH., Realization of a geometry theorem-proving machine, Proceedings of international conference on information processing, UNESCO House, Paris, 1959, pp. 273282.
[14] GödelKurt, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Collected works (FefermanS. et al., editors), vol. II, Oxford University Press, Oxford, 1990, pp. 240251.
[15] HodgesWilfrid, Critical commentary on P. Johnson-Laird and R. Byrne, ‘Deduction’, Behavioral and Brain Sciences, vol. 16 (1993), p. 353f.
[16] Johnson-LairdP. N. and ByrneRuth M. J., Deduction, Lawrence Erlbaum, Hove, 1991.
[17] KampHans and ReyleUwe, From discourse to logic, Kluwer, Dordrecht, 1993.
[18] KleeneStephen Cole, Introduction to metamathematics, North-Holland, Amsterdam, 1952.
[19] RipsLance J., The psychology of proof, MIT Press, Cambridge, Mass., 1994.
[20] RussellBertrand, The principles of mathematics, George Allen and Unwin, London, 1903.
[21] SmithD. E., History of mathematics, vol. II, Dover, New York, 1958.
[22] WittgensteinLudwig, Remarks on the foundations of mathematics, Blackwell, Oxford, 1956.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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