Analogy and diagonal argument

Zbigniew Tworak



In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme.


analogy; diagonal argument; antinomy; limitative theorems; provability; refutability; undecidability; truth

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Cantor, G., “Über eine elementare Frege der Mannigfaltigkeitslehre”, Jahresbericht der Mathematiker-Vereinigung 1 (1890–91), 75–78, reprinted in: E. Zermelo (ed.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Berlin, J. Springer 1932, pp. 278–281.

Church, A., “An unsolvable problem of elementary number theory”, American Journal of Mathematics 58 (1936), 345–363, reprinted in: M. Davis (ed.), The Undecidable, Raven Press, New York 1965, pp. 89–107.

Gödel, K., “Über formal unnentscheidbare Sätze der ‘Principia mathematica’ und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931), 173–198; translated in: J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge,

MA, 1967, pp. 596–616.

Gumański, L., “Remarks on Cantor’s Diagonal Method and Some Related Topics”, Ruch Filozoficzny LVII, 3/4 (2000), 437–448.

Hallett, M., Cantorian Set Theory and Limitation of Size, Clarendon Press, Oxford, 1984.

Krajewski, S., Twierdzenie Gödla i jego interpretacje filozoficzne. Od mechanicyzmu do postmodernizmu, Wydawnictwo IFiS PAN, Warszawa, 2003.

Martin, R. L., “On a Puzzling Classical Validity”, Philosophical Review LXXXVI, 4, (1977), 454–473.

Murawski, R., Filozofia matematyki. Zarys dziejów, PWN, Warszawa, 1995.

Murawski, R., Recursive Functions and Metamathematics, Synthese Library vol. 286, Kluwer Academic Publishers, Dordrecht – Boston – London, 1999.

Russell, B., The Principles of Mathematics, Cambridge University Press, Cambridge, 1903.

Simmons, K., “The diagonal argument and the Liar”, Journal of Philosophical Logic 19 (1990), 277–303.

Smullyan, R., Diagonalization and Self-Reference, Clarendon Press, Oxford, 1994.

Tarski, A., Pojęcie prawdy w językach nauk dedukcyjnych, Warszawa, 1933; translated in: Logic, semantics, metamathematics. Papers from 1923 to 1938, Clarendon Press, Oxford 1956, pp. 152–277.

Turing, A., “On computable numbers, with an application to Entscheidungsproblem”, Proceedings of the London Mathematical Society Ser. 2, Vol. 42 (1937), 230–265; reprinted in: M. Davis (ed.), The Undecidable, Raven Press, New York 1965, pp. 116–154.

Woleński, J., Metamatematyka a epistemologia, PWN, Warszawa, 1993.

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ISSN: 2300-9802 (electronic version)

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