Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Automated Deduction — CADE-15 (CADE 1998)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1421))

Included in the following conference series:

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.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.

    MathSciNet  MATH  Google Scholar 

  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. 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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  6. F. De Gandt. Force and Geometry in Newton's Principia. Princeton University Press, Princeton, New Jersey, 1995.

    MATH  Google Scholar 

  7. D. Kapur. Geometry Theorem Proving using Hilbert's Nullstellensatz. Proceedings of SYMSAC'86, Waterloo, 1986, 202–208.

    Google Scholar 

  8. H. J. Keisler. Foundations of Infinitesimal Calculus. Prindle, Weber & Schmidt, 20 Newbury Street, Boston, Massachusetts, 1976.

    MATH  Google Scholar 

  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. 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. L. C. Paulson. Isabelle: A Generic Theorem Prover. Springer, 1994. LNCS 828.

    Google Scholar 

  12. A. Robinson. Non-Standard Analysis. North-Holland Publishing Company, 1980. 1966, first edition.

    Google Scholar 

  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. 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. D. T. Whiteside. The Mathematical Principles Underlying Newton's Principia Mathematica. Glasgow University Publication 138, 1970.

    Google Scholar 

  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.

    MATH  Google Scholar 

  17. W.-t. Wu. Mechanical Theorem Proving of Differential Geometries and Some of its Applications in Mechanics. J. Automated Reasoning 7 (1991), 171–191.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Laboratory, University of Cambridge, New Museums Site, Pembroke Street, CB2 3QG, Cambridge

    Jacques D. Fleuriot & Lawrence C. Paulson

Authors
  1. Jacques D. Fleuriot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lawrence C. Paulson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Claude Kirchner Hélène Kirchner

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fleuriot, J.D., Paulson, L.C. (1998). A combination of nonstandard analysis and geometry theorem proving, with application to Newton's Principia. In: Kirchner, C., Kirchner, H. (eds) Automated Deduction — CADE-15. CADE 1998. Lecture Notes in Computer Science, vol 1421. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054241

Download citation

  • DOI: https://doi.org/10.1007/BFb0054241

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64675-4

  • Online ISBN: 978-3-540-69110-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation