A combination of nonstandard analysis and geometry theorem proving, with application to Newton's Principia
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.
KeywordsNonstandard Analysis Readable Proof Geometry Theory Diagrammatic Reasoning Automatic Proof
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- 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
- 7.D. Kapur. Geometry Theorem Proving using Hilbert's Nullstellensatz. Proceedings of SYMSAC'86, Waterloo, 1986, 202–208.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