Abstract
Proofs contain important mathematical knowledge and for mathematical knowledge management it is important to represent them adequately. They can be given at different levels of abstraction and writing a proof is typically a compromise between two extremes. On the one hand it should be in full detail so that it can be checked without using any intelligence, on the other hand it should be concise and informative. Making everything fully explicit is not adequate for most mathematical fields since easy parts do not need any communication. In particular in traditional proofs, computations are typically not made explicit, but a reader is expected to check them for him- or herself. Barendregt formulated a principle, the Poincaré Principle, which allows to separate reasoning and computation. However, should any computation be hidden? Or only easy computations? What is easy? How can we be sure that computations are correct? In this contribution, relevant notions are discussed and a principle is introduced which allows for checkable proofs which give a choice to see on request two different types of argument. The first type of argument states why any computation of this kind is correct. The second type states a (typically lengthy) detailed low-level proof of a trace of the computation.
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References
Ayer, A.J.: Language, Truth and Logic. Victor Gollancz Ltd., 2nd edn., London, UK 1951 edn.(1936)
Barendregt, H.: The impact of the lambda calculus in logic and computer science. Bull. Symbolic Logic 3(2), 181–215 (1997)
Barendregt, H., Cohen, A.M.: Electronic communication of mathematics and the interaction of computer algebra systems and proof assistants. Journal of Symbolic Computation 32(5), 3–22 (2001)
Boole, G.: An Investigation of The Laws of Thought, Cambridge, UK., Macmillan, Barclay, & Macmillan, New York (reprinted by Dover Publication 1854) (1958)
Constable, R.L., et al.: Implementing Mathematics with the Nuprl Proof Development System. Prentice Hall, Englewood Cliffs (1986)
Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle, reprint in: Begriffsschrift und andere Aufsätze, J.A., (ed.), Hildesheim. Also in Logiktexte, Berka, K., Kreiser, L. (eds.), pp. 82–112 (1879)
Hardy, G.: A Mathematician’s Apology. Cambridge University Press, London (1940)
van Heijenoort, J. (ed.): From Frege to Gödel – A Source Book in Mathematical Logic, pp. 1879–1931. Havard University Press, Cambridge, Massachusett (1967)
Huet, G., et al.: The Coq Theorem Prover (Version 8.0). http://coq.inria.fr/doc-eng.html
Kerber, M.: Why is the Lucas-Penrose argument invalid? In: Furbach, U. (ed.) KI 2005. LNCS (LNAI), vol. 3698, pp. 380–393. Springer, Heidelberg (2005)
Kerber, M., Kohlhase, M., Sorge, V.: Integrating computer algebra into proof planning. Journal of Automated Reasoning 21(3), 327–355 (1998)
Leibniz, G.W.: Projet et essais pour arriver à quelque certitude pour finir une bonne partie des disputes et pour avancer l’art d’inventer. In: Berka, K., Kreiser, L. (eds.). Logiktexte, ch. I.2, pp. 16–18. Akademie-Verlag, german translation, Berlin, Germany (1686) (1983)
Nederpelt, R., Geuvers, H., de Vrijer, R. (eds.): Selected Papers on Automath. Studies in Logic and the Foundations of Mathematics, vol. 133. North-Holland, Amsterdam (1994)
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Kerber, M. (2006). A Dynamic Poincaré Principle. In: Borwein, J.M., Farmer, W.M. (eds) Mathematical Knowledge Management. MKM 2006. Lecture Notes in Computer Science(), vol 4108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11812289_5
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DOI: https://doi.org/10.1007/11812289_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37104-5
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