Advertisement

A combination of nonstandard analysis and geometry theorem proving, with application to Newton's Principia

  • Jacques D. Fleuriot
  • Lawrence C. Paulson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1421)

Abstract

The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains “infinitesimal” elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton's proofs from being mechanised using only the existing geometry theorem proving (GTP) techniques.

Using concepts from Robinson's Nonstandard Analysis (NSA) and a powerful geometric theory, we introduce the concept of an infinitesimal geometry in which quantities can be infinitely small or infinitesimal. We reveal and prove new properties of this geometry that only hold because infinitesimal elements are allowed and use them to prove lemmas and theorems from the Principia.

Keywords

Nonstandard Analysis Readable Proof Geometry Theory Diagrammatic Reasoning Automatic Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. M. Ballantyne and W. W. Bledsoe. Automatic Proofs of Theorems in Analysis Using Nonstandard Analysis. J. of the Association of Computing Machinery, Vol. 24, No. 3, July 1977, 353–374.MathSciNetzbMATHGoogle Scholar
  2. 2.
    T. Bedrax. Infmal: Prototype of an Interactive Theorem Prover based on Infinitesimal Analysis. Liciendo en Mathematica con Mencion en Computation Thesis. Pontifica Universidad Catolica de Chile, Santiago, Chile, 1993.Google Scholar
  3. 3.
    G. Berkeley. The Analyst: A Discourse Addressed to an Infidel Mathematician. The World of Mathematics, Vol. 1, London. Allen and Unwin, 1956, 288–293.Google Scholar
  4. 4.
    S. C. Chou, X. S. Gao, and J. Z. Zhang. Automated Generation of Readable Proofs with Geometric Invariants, I. Multiple and Shortest Proof Generation. J. Automated Reasoning 17 (1996), 325–347.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. C. Chou, X. S. Gao, and J. Z. Zhang. Automated Generation of Readable Proofs with Geometric Invariants, II. Theorem Proving with Full-angles. J. Automated Reasoning 17 (1996), 349–370.MathSciNetzbMATHGoogle Scholar
  6. 6.
    F. De Gandt. Force and Geometry in Newton's Principia. Princeton University Press, Princeton, New Jersey, 1995.zbMATHGoogle Scholar
  7. 7.
    D. Kapur. Geometry Theorem Proving using Hilbert's Nullstellensatz. Proceedings of SYMSAC'86, Waterloo, 1986, 202–208.Google Scholar
  8. 8.
    H. J. Keisler. Foundations of Infinitesimal Calculus. Prindle, Weber & Schmidt, 20 Newbury Street, Boston, Massachusetts, 1976.zbMATHGoogle Scholar
  9. 9.
    I. Newton. The Mathematical Principles of Natural Philosophy. Third edition, 1726. Translation by A. Motte (1729). Revised by F. Cajory 1934. University of California Press.Google Scholar
  10. 10.
    S Novak Jr. Diagrams for Solving Physical Problems. Diagrammatic Reasoning: Cognitive and Computational Perspectives, AAAI Press/MIT Press, 753–774, 1995. (Eds. Janice Glasgow, N. Hari Narayana, and B. Chandrasekaram).Google Scholar
  11. 11.
    L. C. Paulson. Isabelle: A Generic Theorem Prover. Springer, 1994. LNCS 828.Google Scholar
  12. 12.
    A. Robinson. Non-Standard Analysis. North-Holland Publishing Company, 1980. 1966, first edition.Google Scholar
  13. 13.
    R. Chuaqui and P. Suppes. Free-Variable Axiomatic Foundations of Infinitesimal Analysis: A Fragment with Finitary Consistency Proof. J. Symbolic Logic, Vol. 60, No. 1, March 1995.Google Scholar
  14. 14.
    D. Wang. Geometry Machines: From AI to SMC. 3rd Internationa Conference on Artificial Intelligence and Symbolic Mathematical Computation. (Stey, Austria, September 1996), LNCS 1138, 213–239.Google Scholar
  15. 15.
    D. T. Whiteside. The Mathematical Principles Underlying Newton's Principia Mathematica. Glasgow University Publication 138, 1970.Google Scholar
  16. 16.
    W.-t. Wu. On the Decision Problem and the Mechanization of Theorem in Elementary Geometry. Automated Theorem Proving: After 25 years. A.M.S., Contemporary Mathematics, 29 (1984), 213–234.zbMATHGoogle Scholar
  17. 17.
    W.-t. Wu. Mechanical Theorem Proving of Differential Geometries and Some of its Applications in Mechanics. J. Automated Reasoning 7 (1991), 171–191.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jacques D. Fleuriot
    • 1
  • Lawrence C. Paulson
    • 1
  1. 1.Computer LaboratoryUniversity of CambridgeCambridge

Personalised recommendations