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A FORMAL SYSTEM FOR EUCLID’S ELEMENTS

Published online by Cambridge University Press:  01 December 2009

JEREMY AVIGAD ,
EDWARD DEAN  and
JOHN MUMMA
JEREMY AVIGAD*
Affiliation:
Department of Philosophy and Department of Mathematical Sciences, Carnegie Mellon University
EDWARD DEAN*
Affiliation:
Department of Philosophy, Carnegie Mellon University
JOHN MUMMA*
Affiliation:
Division of Logic, Methodology, and Philosophy of Science, Suppes Center for History and Philosophy of Science
*
*JEREMY AVIGAD, DEPARTMENT OF PHILOSOPHY, CARNEGIE MELLON UNIVERSITY, PITTSBURGH, PA 15213 E-mail:avigad@cmu.edu
†EDWARD DEAN, DEPARTMENT OF PHILOSOPHY, CARNEGIE MELLON UNIVERSITY, PITTSBURGH, PA 15213 E-mail:edean@andrew.cmu.edu
‡JOHN MUMMA, DIVISION OF LOGIC METHODOLOGY, AND PHILOSOPHY OF SCIENCE AT THE SUPPES, CENTER FOR HISTORY AND PHILOSOPHY OF SCIENCE, BUILDING 200, STANFORD, CA 94305–2024 E-mail:john.mumma@gmail.com
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Abstract

We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.

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Type
Research Article
Information
The Review of Symbolic Logic , Volume 2 , Issue 4 , December 2009 , pp. 700 - 768
Copyright
Copyright © Association for Symbolic Logic 2009

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References

BIBLIOGRAPHY

Avigad, J. (2009). Review of Marcus Giaquinto, Visual thinking in mathematics: an epistemological study. Philosophia Mathematica, 17:95–108.CrossRefGoogle Scholar
Barrett, C., & Tinelli, C. (2007). CVC3. In Damm, W., and Hermanns, H., editors. Computer Aided Verification (CAV) 2007. Berlin: Springer, pp. 298–302.Google Scholar
Beeson, M. (to appear). Constructive Euclidean geometry. In Arai, T., Chi Tat, C., Downey, R., Brendle, J., Feng, Q., Kikyo, H., and Ono, H., editors. Proceedings of 10th Asian Logic Conference. Kobe, Japan.Google Scholar
Berkeley, G. (1734). A Treaties Concerning the Principles of Human Knowledge. Reprinted in Clarke, D. M., editor. Berkeley, CA: Philosophical writings and Cambridge, UK: Cambridge University Press, pp. 67–150, 2008.Google Scholar
Bockmayr, A., & Weispfenning, V. (2001). Solving Numerical Constraints. In Robinson, A., and Voronkov, A., editors. Handbook of Automated Reasoning. Amsterdam, The Netherlands: Elsevier Science, pp. 751–842.CrossRefGoogle Scholar
Bos, H. J. M. (2001). Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. New York, NY: Springer.CrossRefGoogle Scholar
Buss, S. R. (1998). An introduction to proof theory. In Buss, Samuel R., editor. The Handbook of Proof Theory. Amsterdam, The Netherlands: North-Holland, pp. 1–78.Google Scholar
Chou, S. C., & Gao, X. S. (2001). Automated reasoning in geometry. In Robinson, A., and Voronkov, A., editors. Handbook of Automated Reasoning. Amsterdam, The Netherlands: Elsevier Science, pp. 707–750.CrossRefGoogle Scholar
Chou, S. C., Gao, X. S., & Zhang, J. Z. (1994). Machine Proofs in Geometry. Singapore: World Scientific.CrossRefGoogle Scholar
Collins, G. E. (1975). Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Brakhage, H., editor. Automata Theory and Formal Languages. Berlin: Springer, pp. 134–183.Google Scholar
Coxeter, H. S. M. (1969). Introduction to Geometry (second edition). New York, NY: John Wiley & Sons Inc.Google Scholar
Dean, E. (2008). In defense of Euclidean proof. Master’s Thesis, Carnegie Mellon University.Google Scholar
Dershowitz, N., & Plaisted, D. A. (2001). Rewriting. In Robinson, A., and Voronkov, A., editors. Handbook of Automated Reasoning. Amsterdam, The Netherlands: Elsevier Science, pp. 535–607.CrossRefGoogle Scholar
Euclid, . (1956). The Thirteen Books of the Elements (second edition, Vols. I–III). New York, NY: Dover Publications. Translated with introduction and commentary by Sir Thomas L. Heath, from the text of Heiberg. The Heath translation has also been issued as Euclid’s Elements: All Thirteen Books Complete in One Volume Green Lion Press, Santa Fe, 2002.Google Scholar
Friedman, M. (1985). Kant’s theory of geometry. Philosophical Review, 94, 455–506. Revised version in Michael Friedman, Kant and the Exact Sciences. Cambridge, MA: Cambridge: Harvard University Press, 1992.CrossRefGoogle Scholar
Gebert, J. R., & Kortenkap, U. H. (1999). The Interactive Geometry Software Cinderella. Berlin: Springer.Google Scholar
Giaquinto, M. (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Goodwin, W. M. (2003). Kant’s philosophy of geometry. PhD Thesis, University of California, Berkeley.Google Scholar
Hartshorne, R. (2005). Geometry: Euclid and Beyond. New York, NY: Springer.Google Scholar
Hilbert, D. (1899). Grundlagen der Geometrie. In Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen. Leipzig, Germany: Teubner. Translated by Leo Unger as Foundations of Geometry, Open Court, La Salle, 1971. Ninth printing, 1997.Google Scholar
Hungerford, T. W. (1974). Algebra. New York, NY: Springer.Google Scholar
Kant, I. (1998). Kritik der reinen Vernunft (first edition 1781, second edition 1787). Translated and edited by Paul Guyer and Allen W. Wood as Critique of Pure Reason. Cambridge, UK: Cambridge University Press.Google Scholar
Key Curriculum Press, editor. (2002). The Geometer’s Sketchpad: Student Edition. Emeryville, CA: Key Curriculum.Google Scholar
Knorr, W. R. (1985). The Ancient Tradition of Geometric Problems. Boston, MA: Birkhäuser.Google Scholar
Krantz, D., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of Measurement, Vols. I and II. New York, NY: Academic Press.Google Scholar
Leibniz, G. (1949). New Essays Concerning Human Understanding. La Salle, IL: Open Court Publishing.Google Scholar
Leitgeb, H. (to appear). On formal and informal provability. In Linnebo, O., and Bueno, O., editors. New Waves in Philosophy of Mathematics. Palgrave Macmillan.Google Scholar
Macbeth, D. (preprint). Diagrammatic reasoning in Euclid’s Elements. Preprint.Google Scholar
Mancosu, P. (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. New York, NY: Oxford University Press.CrossRefGoogle Scholar
Manders, K. (2008a) Diagram-based geometric practice. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 65–79.CrossRefGoogle Scholar
Manders, K. (2008b). The Euclidean diagram. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 80–133. MS first circulated in 1995.CrossRefGoogle Scholar
Manna, Z., & Zarba, C. G. (2003). Combining decision procedures. In Aichernig, B., and Maibaum, T., editors. Formal Methods at the Crossroads. Berlin: Springer, pp. 381–422.Google Scholar
Meng, J., & Paulson, L. C. (2008). Translating higher-order clauses to first-order clauses. Journal of Automated Reasoning, 40, 35–60.CrossRefGoogle Scholar
Miller, N. (2008). Euclid and his Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry. Stanford, CA: CSLI. Based on Miller’s thesis (2001). A diagrammatic formal system for Euclidean geometry. PhD Thesis, Cornell University.Google Scholar
Morrow, G., editor. (1970). Proclus: A Commentary on the First Book of Euclid’s Elements. Princeton, NJ: Princeton University Press.Google Scholar
de Moura, L. M., & Bjørner, N. (2008). Z3: An efficient SMT solver. In Ramakrishnan, C. R., and Rehof, J., editors. Tools and Algorithms for the Construction and Analysis of Systems (TACAS) 2008. Berlin: Springer, pp. 337–340.Google Scholar
Mueller, I. (1981). Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Cambridge, MA: MIT Press.Google Scholar
Mumma, J. (2006). Intuition formalized: Ancient and modern methods of proof in elementary geometry. PhD Thesis, Carnegie Mellon University.Google Scholar
Mumma, J. (2008). Review of Euclid and his twentieth century rivals, by Nathaniel Miller. Philosophia Mathematica, 16, 256–264.CrossRefGoogle Scholar
Mumma, J. (to appear). Proofs, pictures, and Euclid. Synthese.Google Scholar
Narboux, J. (2007). A graphical user interface for formal proofs in geometry. Journal of Automated Reasoning, 39, 161–180.CrossRefGoogle Scholar
Negri, S. (2003). Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem. Archive for Mathematical Logic, 42, 389–401.CrossRefGoogle Scholar
Negri, S., & von Plato, J. (1998). Cut elimination in the presence of axioms. Bulletin of Symbolic Logic, 4, 418–435.CrossRefGoogle Scholar
Netz, R. (1999). The Shaping of Deduction in Greek mathematics: A Study of Cognitive History. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Panza, M. (preprint). The twofold role of diagrams in Euclid’s plane geometry. Preprint.Google Scholar
Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig, Germany: Teubner.Google Scholar
Peano, G. (1889). I principii di Geometria, logicamente esposti. Turin, Italy: Bocca.Google Scholar
von Plato, J. (1995). The axioms of constructive geometry. Annals of Pure and Applied Logic, 76, 169–200.CrossRefGoogle Scholar
von Plato, J. (1998). A constructive theory of ordered affine geometry. Indagationes Mathematicae, 9, 549–562.CrossRefGoogle Scholar
Prevosto, V., & Waldmann, U. (2006). Spass+T. In Sutcliffe, G., Schmidt, R., and Schulz, S., editors. Empirically Successful Computerized Reasoning (ESCoR) 2006. pp. 18–33.Google Scholar
Saito, K. (2006). A preliminary study in the critical assessment of diagrams in Greek mathematical works. SCIAMVS, 7, 81–44.Google Scholar
Schulz, S. (2002). E—A brainiac theorem prover. Journal of AI Communications, 15, 111–126.Google Scholar
Shabel, L. (2003). Mathematics in Kant’s Critical Philosophy: Reflections on Mathematical Practice. New York, NY: Routledge.Google Scholar
Shabel, L. (2004). Kant’s “argument from geometry.” Journal of the History of Philosophy, 42, 195–215.CrossRefGoogle Scholar
Shabel, L. (2006). Kant’s philosophy of mathematics. In Guyer, P., editor. The Cambridge Companion to Kant (second edition). Cambridge, UK: Cambridge University Press.Google Scholar
Stein, H. (1990). Eudoxus and Dedekind: On the ancient Greek theory of ratios and its relation to modern mathematics. Synthese, 84, 163–211.CrossRefGoogle Scholar
Tappenden, J. (2005). Proof style and understanding in mathematics I: Visualization, unification, and axiom choice. In Mancosu, P., Jorgensen, K. F., and Pedersen, S. A., editors. Visualization, Explanation and Reasoning Styles in Mathematics. Berlin: Springer, pp. 147–214.CrossRefGoogle Scholar
Tarski, A. (1959). What is elementary geometry? In Henkin, L., Suppes, P., and Tarski, A., editors. The Axiomatic Method: With Special Reference to Geometry and Physics (first edition). Amsterdam, the Netherlands: North-Holland, pp. 16–29.CrossRefGoogle Scholar
Tarski, A., & Givant, S. (1999). Tarski’s system of geometry. Bulletin of Symbolic Logic, 5, 175–214.CrossRefGoogle Scholar
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic Proof Theory (second edition). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Veroff, R. (2001). Solving open questions and other challenge problems using proof sketches. Journal of Automated Reasoning, 27, 157–174.CrossRefGoogle Scholar
Weidenbach, C., Schmidt, R., Hillenbrand, T., Rusev, R., & Topic, D. (2007). System description: Spass version 3.0. In Pfenning, F., editor. CADE-21: 21st International Conference on Automated Deduction. Berlin: Springer, pp. 514–520.CrossRefGoogle Scholar
Wilgus, W. (1998). Exploring the Basics of Geometry with Cabri. Dallas, TX: Texas Instruments.Google Scholar
Wu, W. T. (1994). Mechanical Theorem Proving in Geometries. Vienna, Austria: Springer. Translated from the 1984 Chinese original by Xiao Fan Jin and Dong Ming Wang.CrossRefGoogle Scholar
Ziegler, M. (1982). Einige unentscheidbare Körpertheorien. L’enseignement mathématique, 28, 269–280. Unpublished translation by Michael Beeson.Google Scholar