Skip to main content
SpringerLink
Log in

A Dynamic Poincaré Principle

  • Conference paper
Mathematical Knowledge Management (MKM 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4108))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ayer, A.J.: Language, Truth and Logic. Victor Gollancz Ltd., 2nd edn., London, UK 1951 edn.(1936)

    Google Scholar 

  2. Barendregt, H.: The impact of the lambda calculus in logic and computer science. Bull. Symbolic Logic 3(2), 181–215 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boole, G.: An Investigation of The Laws of Thought, Cambridge, UK., Macmillan, Barclay, & Macmillan, New York (reprinted by Dover Publication 1854) (1958)

    Google Scholar 

  5. Constable, R.L., et al.: Implementing Mathematics with the Nuprl Proof Development System. Prentice Hall, Englewood Cliffs (1986)

    Google Scholar 

  6. 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)

    Google Scholar 

  7. Hardy, G.: A Mathematician’s Apology. Cambridge University Press, London (1940)

    Google Scholar 

  8. van Heijenoort, J. (ed.): From Frege to Gödel – A Source Book in Mathematical Logic, pp. 1879–1931. Havard University Press, Cambridge, Massachusett (1967)

    MATH  Google Scholar 

  9. Huet, G., et al.: The Coq Theorem Prover (Version 8.0). http://coq.inria.fr/doc-eng.html

  10. 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)

    Chapter  Google Scholar 

  11. Kerber, M., Kohlhase, M., Sorge, V.: Integrating computer algebra into proof planning. Journal of Automated Reasoning 21(3), 327–355 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Google Scholar 

  13. 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)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, The University of Birmingham, Birmingham, B15 2TT, England

    Manfred Kerber

Authors
  1. Manfred Kerber
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Computer Science, Dalhousie University, 6050 University Avenue, Halifax, B3H 1W5, Nova Scotia, Canada

    Jonathan M. Borwein

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    William M. Farmer

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

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

Download citation

  • DOI: https://doi.org/10.1007/11812289_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37104-5

  • Online ISBN: 978-3-540-37106-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation