Abstract
In a celebrated contribution on “mathematical problems”, at the Second International Mathematicians Congress (Paris, 8 August 1900), David Hilbert speaks about the conviction that our mind is capable of answering all questions it asks. Hilbert raises the question of whether this axiom is characteristic only of mathematical thinking, or whether it is a general and essential law of our mind. He shares the conviction that there is an answer to every mathematical question, and that we can find it through pure thinking: “in Mathematics there is no Ignorabimus”.1
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Notes
D. Hilbert, Gesammelte Abhandlungen, III. Band, pp. 297–298, Springer-Verlag, Berlin, 1935.
D. Hilbert and W. Ackermann, Grundzüge der theoretischen Logik, 3.§1, Springer-Verlag, Berlin, 1928.
All these basic papers are collected in M. Davis (ed.), The Undecidable, Raven Press, New York, 1965.
A. Wiles, “Modular elliptic curves and Fermat’s Last Theorem”, Annals of Mathematics 141(3), pp. 443–552, 1995 and R. Taylor and A. Wiles, “Ring-theoretic properties of certain Hecke algebras”, Annals of Mathematics 141(3), p. 553, 1995.
I. Kant, Kritik der reinen Vernunft (Vorrede zur zweiten Auflage 1787, B X-XVI), pp. 16–20, Felix Meiner, Hamburg, 1956.
D. Hilbert, Gesammelte Abhandlungen, III. Band, pp. 383–385, Springer-Verlag, Berlin, 1935.
Certainly the problem of the complexity of mathematical proofs (see the contributions of H.-Ch. Reichel-this volume, pp. 3-14-and F.T. Arecchi — this volume, pp. 61-82) also challenges Kant’s view: indeed it is hard to accept that some mathematical result is a priori in my mind, even if I know the method of reaching it, if the proof is so complex that I cannot survey it; a “simple” decomposition into prime factors can become so intractable when the number is large, that one cannot go beyond results that are very probable rather than true. Nevertheless intractability of mathematical problems does not reveal any incompleteness in principle.
Such numbers are referred to as computable (A.M. Turing, “On Computable Numbers”, Proceedings of The London Mathematical Society 42, p. 230, 1937). Notice that irrational numbers as π and e are computable, for they are real numbers whose expressions as a decimal are generated by a finite number of instructions or a computer program. The set of the computable numbers is enumerable. A set of real numbers containing elements which are not computable, as for instance the continuum, is not enumerable. We are restricting our considerations to the computable numbers.
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Suarez, A. (1997). The Limits of Mathematical Reasoning: In Arithmetic there will always be Unsolved Solvable Problems. In: Driessen, A., Suarez, A. (eds) Mathematical Undecidability, Quantum Nonlocality and the Question of the Existence of God. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5428-4_4
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