Journal of Automated Reasoning

, Volume 62, Issue 1, pp 1–68

Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq

• Pierre Boutry
• Charly Gries
• Julien Narboux
• Pascal Schreck
Article

Abstract

In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid’s 5th postulate. Our study is performed in the context of Tarski’s neutral geometry, or equivalently in Hilbert’s geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for point equality. Our formalization provides a clarification of the conditions under which different versions of the postulates are equivalent. Following Beeson, we study which versions of the postulate are equivalent, constructively or not. We distinguish four groups of parallel postulates. In each group, the proof of their equivalence is mechanized using intuitionistic logic without continuity assumptions. For the equivalence between the groups additional assumptions are required. The equivalence between the 34 postulates is formalized in Archimedean planar neutral geometry. We also formalize a variant of a theorem due to Szmielew. This variant states that, assuming Aristotle’s axiom, any statement which hold in the Euclidean plane and does not hold in the Hyperbolic plane is equivalent to Euclid’s 5th postulate. To obtain all these results, we have developed a large library in planar neutral geometry, including the formalization of the concept of sum of angles and the proof of the Saccheri–Legendre theorem, which states that assuming Archimedes’ axiom, the sum of the angles in a triangle is at most two right angles.

Keywords

Euclid Parallel postulate Formalization Neutral geometry Coq Classification Foundations of geometry Decidability of intersection Aristotle’s axiom Archimedes’ axiom Saccheri–Legendre theorem Sum of angles

References

1. 1.
Avigad, J., Dean, E., Mumma, J.: A formal system for Euclid’s elements. Rev. Symb. Log. 2, 700–768 (2009)
2. 2.
Amiot, A.: Eléments de géométrie [texte imprimé]: rédigés d’après le nouveau programme de l’enseignement scientifique des lycées; suivis d’un complément à l’usage des élèves de mathématiques spéciales (1870)Google Scholar
3. 3.
Amira, D.: Sur l’axiome de droites parallèles. L’Enseign. Math. 32, 52–57 (1933)
4. 4.
Alama, J., Pambuccian, V.: From absolute to affine geometry in terms of point-reflections, midpoints, and collinearity. Note di Mat. 36(1), 11–24 (2016)
5. 5.
Bachmann, F.: Zur Parallelenfrage. In: Cortés, V., Richter, B. (eds.) Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 27, pp. 173–192. Springer, Berlin (1964)Google Scholar
6. 6.
Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1973)
7. 7.
Beeson, M., Boutry, P., Narboux, J.: Herbrand’s theorem and non-Euclidean geometry. Bull. Symb. Log. 21(1), 111–122 (2015)
8. 8.
Boutry, P., Braun, G., Narboux, J.: From Tarski to Descartes: formalization of the arithmetization of Euclidean geometry. In: The 7th International Symposium on Symbolic Computation in Software (SCSS 2016), EasyChair Proceedings in Computing, Tokyo, Japan, pp. 15 (March 2016)Google Scholar
9. 9.
Braun, G., Boutry, P., Narboux, J.: From Hilbert to Tarski. In: Eleventh International Workshop on Automated Deduction in Geometry, Proceedings of ADG 2016, Strasbourg, France, p. 19 (2016)Google Scholar
10. 10.
Beeson, M.: A constructive version of Tarski’s geometry. Ann. Pure Appl. Log. 166(11), 1199–1273 (2015)
11. 11.
Beeson, M.: Constructive geometry and the parallel postulate. Bull. Symb. Log. 22(1), 1–104 (2016)
12. 12.
Beeson, M.: Brouwer and Euclid. Indagationes Mathematicae. To appear in a special issue devoted to Brouwer (2017)Google Scholar
13. 13.
Bell, J.L.: Hilbert’s $$\epsilon$$-operator in intuitionistic type theories. Math. Log. Q. 39(1), 323–337 (1993)
14. 14.
Birkhoff, G.D.: A set of postulates for plane geometry (based on scale and protractors). Ann. Math. 33, 329–345 (1932)
15. 15.
Braun, D., Magaud, N.: Des preuves formelles en Coq du théorème de Thalès pour les cercles. In: Baelde, D., Alglave, J. (eds.) Vingt-sixièmes Journées Francophones des Langages Applicatifs (JFLA 2015), Le Val d’Ajol, France (2015)Google Scholar
16. 16.
Braun, G., Narboux, J.: From Tarski to Hilbert. In: Ida, T., Fleuriot, J. (eds.) Post-proceedings of Automated Deduction in Geometry 2012. LNCS, vol. 7993, pp. 89–109. Springer, Edinburgh (2012)Google Scholar
17. 17.
Braun, G., Narboux, J.: A synthetic proof of Pappus’ theorem in Tarski’s geometry. J. Autom. Reason. 58(2), 23 (2017)
18. 18.
Boutry, P., Narboux, J., Schreck, P., Braun, G.: A short note about case distinctions in Tarski’s geometry. In: Botana, F., Quaresma, P. (eds.) Proceedings of the 10th International Workshop on Automated Deduction in Geometry, Volume TR 2014/01 of Proceedings of ADG 2014, pp. 51–65. University of Coimbra, Coimbra (2014)Google Scholar
19. 19.
Boutry, P., Narboux, J., Schreck, P., Braun, G.: Using small scale automation to improve both accessibility and readability of formal proofs in geometry. In: Botana, F., Quaresma, P. (eds.) Proceedings of the 10th International Workshop on Automated Deduction in Geometry, Volume TR 2014/01 of Proceedings of ADG 2014, pp. 31–49. University of Coimbra, Coimbra (2014)Google Scholar
20. 20.
Bonola, R.: Non-Euclidean Geometry: A Critical and Historical Study of Its Development. Courier Corporation, North Chelmsford (1955)
21. 21.
Borsuk, K., Szmielew, W.: Foundations of Geometry. North-Holland, Amsterdam (1960)
22. 22.
Cajori, F.: A History of Elementary Mathematics. Macmillan, London (1898)
23. 23.
Cohen, C., Mahboubi, A.: Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination. Log. Methods Comput. Sci. 8(1:02), 1–40 (2012)
24. 24.
Coxeter, H.S.M.: Non-Euclidean Geometry. Cambridge University Press, Cambridge (1998)
25. 25.
Dehlinger, C., Dufourd, J.-F., Schreck, P.: Higher-order intuitionistic formalization and proofs in Hilbert’s elementary geometry. In: Richter-Gebe, J., Wang, D. (eds.) Automated Deduction in Geometry, Lectures Notes in Computer Science, vol. 2061, pp. 306–324 (2001)Google Scholar
26. 26.
Dehn, M.: Die Legendre’schen Sätze über die Winkelsumme im Dreieck. Math. Ann. 53(3), 404–439 (1900)
27. 27.
Duprat, J.: Fondements de géométrie euclidienne. https://hal.inria.fr/hal-00661537 (2010)
28. 28.
Euclid, Heath, T.L., Densmore, D.: Euclid’s Elements: All Thirteen Books Complete in One Volume: The Thomas L. Heath Translation. Green Lion Press, Santa Fe (2002)
29. 29.
Friedrich, C., Bolyai, F.: Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai. BG Teubner, Leipzig (1899)Google Scholar
30. 30.
Gries, C., Boutry, P., Narboux, J.: Somme des angles d’un triangle et unicité de la parallèle: une preuve d’équivalence formalisée en Coq. In: Les vingt-septièmes Journées Francophones des Langages Applicatifs (JFLA 2016)Google Scholar
31. 31.
Grégoire, B, Pottier, L., Théry, L.: Proof certificates for algebra and their application to automatic geometry theorem proving. In: Sturm, T., Zengler, C. (eds.) Post-proceedings of Automated Deduction in Geometry (ADG 2008). Number 6701 in Lecture Notes in Artificial Intelligence (2011)Google Scholar
32. 32.
Greenberg, M.J.: Aristotle’s axiom in the foundations of geometry. J. Geom. 33(1), 53–57 (1988)
33. 33.
Greenberg, M.J.: Euclidean and Non-Euclidean Geometries—Development and History. Macmillan, London (1993)
34. 34.
Greenberg, M.J.: Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries. Am. Math. Mon. 117(3), 198–219 (2010)
35. 35.
Gupta, H.N.: Contributions to the axiomatic foundations of geometry. PhD thesis, University of California, Berkley (1965)Google Scholar
36. 36.
Hartshorne, R.: Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics. Springer, Berlin (2000)
37. 37.
Hilbert, D.: Foundations of Geometry (Grundlagen der Geometrie). Open Court, La Salle, Illinois, 1960. Second English edition, translated from the tenth German edition by Leo Unger. Original publication date (1899)Google Scholar
38. 38.
Kline, M.: Mathematical Thought from Ancient to Modern Times, vol. 3. Oxford University Press, Oxford (1990)
39. 39.
Klugel, G.S.: Conatuum praecipuorum theoriam parallelarum demonstrandi recensio. PhD thesis, Schultz, Göttingen (1763). German translation available http://www2.math.uni-wuppertal.de/~volkert/versuch.html
40. 40.
Legendre, A.M.: Réflexions sur différentes manières de démontrer la théorie des parallèles ou le théorème sur la somme des trois angles du triangle. Mémoires de l’Académie royale des sciences de l’Institut de France, XII: pp. 367–410 (1833)Google Scholar
41. 41.
Lewis, F.P.: History of the parallel postulate. Am. Math. Mon. 27(1), 16–23 (1920)
42. 42.
Laubenbacher, R., Pengelley, D.: Mathematical Expeditions: Chronicles by the Explorers. Springer, Berlin (2013)
43. 43.
Makarios, T.J.M.: A mechanical verification of the independence of Tarski’s Euclidean axiom. Master’s thesis, Victoria University of Wellington (2012)Google Scholar
44. 44.
Martin, G.E.: The Foundations of Geometry and the Non-Euclidean Plane. Undergraduate Texts in Mathematics. Springer, Berlin (1998)Google Scholar
45. 45.
Millman, R.S., Parker, G.D.: Geometry, A Metric Approach with Models. Springer, Berlin (1981)
46. 46.
Marić, F., Petrović, D.: Formalizing complex plane geometry. Ann. Math. Artif. Intell. 74, 271–308 (2014)
47. 47.
Narboux, J.: Mechanical theorem proving in Tarski’s geometry. In: Botana, F., Lozano, E.R. (eds.) Post-Proceedings of Automated Deduction in Geometry 2006. LNCS, vol. 4869, pp. 139–156. Springer, Pontevedra (2007)Google Scholar
48. 48.
Pambuccian, V.: Zum Stufenaufbau des Parallelenaxioms. J. Geom. 51(1–2), 79–88 (1994)
49. 49.
Pambuccian, V.: Axiomatizations of hyperbolic and absolute geometries. In: Prékopa, A., Molnár, E. (eds.) Non-Euclidean Geometries, vol. 581, pp. 119–153. Springer, Berlin (2006). doi:
50. 50.
Pambuccian, V.: On the equivalence of Lagrange’s axiom to the Lotschnittaxiom. J. Geom. 95(1–2), 165–171 (2009)
51. 51.
Pambuccian, V.: Another equivalent of the Lotschnittaxiom. Beiträge zur Algebra und Geometrie/Contrib. Algebra Geom. 58(1), 167–170 (2017)
52. 52.
Papadopoulos, A.: Master Class in Geometry. Notes on Non-Euclidean Geometry, Chapter 1. European Mathematical Society, Zurich (2012)Google Scholar
53. 53.
Pasch, M.: Vorlesungen über neuere Geometrie. Springer, Berlin (1976)
54. 54.
Pejas, W.: Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie. Math. Ann. 143(3), 212–235 (1961)
55. 55.
Piesyk, Z.: The existential and universal statements on parallels. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9, 761–764 (1961)
56. 56.
Rothe, F.: Several Topics from Geometry (2014). http://math2.uncc.edu/~frothe/3181all.pdf
57. 57.
Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie. Springer, Berlin (1983)
58. 58.
Szmielew, W.: Some metamathematical problems concerning elementary hyperbolic geometry. Stud. Log. Found. Math. 27, 30–52 (1959)
59. 59.
Szmielew, W.: A new analytic approach to hyperbolic geometry. Fundam. Math. 50(2), 129–158 (1961)
60. 60.
Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley (1951)
61. 61.
Tarski, A., Givant, S.: Tarski’s system of geometry. Bull. Symb. Log. 5(2), 175–214 (1999)
62. 62.
Trudeau, R.J.: The Non-Euclidean Revolution. Springer, Berlin (1986)

Authors and Affiliations

• Pierre Boutry
• 1
Email author
• Charly Gries
• 1
• Julien Narboux
• 1
• Pascal Schreck
• 1
1. 1.ICube, UMR 7357 CNRSUniversity of StrasbourgIllkirchFrance