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Implementing Euclid’s straightedge and compass constructions in type theory

  • Ariel Kellison
  • Mark Bickford
  • Robert Constable
Article
  • 18 Downloads

Abstract

Constructions are central to the methodology of geometry presented in the Elements. This theory therefore poses a unique challenge to those concerned with the practice of constructive mathematics: can the Elements be faithfully captured in a modern constructive framework? In this paper, we outline our implementation of Euclidean geometry based on straightedge and compass constructions in the intuitionistic type theory of the Nuprl proof assistant. A result of our intuitionistic treatment of Euclidean geometry is a proof of the second proposition from Book I of the Elements in its full generality; a result that differs from other formally constructive accounts of Euclidean geometry. Our formalization of the straightedge and compass utilizes a predicate for orientation, which enables a concise and intuitive expression of Euclid’s constructions.

Keywords

Constructive type theory Constructive geometry Foundations of geometry 

Mathematics Subject Classification (2010)

68T15 03B35 03F55 

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References

  1. 1.
    Mäenpää, P., von Plato, J.: The logic of Euclidean construction procedures. Acta Philos. Fenn 49, 275–293 (1990)zbMATHGoogle Scholar
  2. 2.
    Heyting, A.: Axioms for intuitionistic plane affine geometry. Studies in Logic and the Foundations of Mathematics 27, 160–173 (1959). [Online]. Available: http://linkinghub.elsevier.com/retrieve/pii/S0049237X09700266 MathSciNetCrossRefGoogle Scholar
  3. 3.
    van Dalen, D.: Outside as a primitive notion in constructive projective geometry. Geom. Dedicata. 60(1), 107–111 (1996). [Online]. Available: http://link.springer.com/10.1007/BF00150870 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dalen, D.V.: Extension problems in intuitionistic plane projective geometry. [Online]. Available: https://www.illc.uva.nl/Research/Publications/Dissertations/HDS-15-Dirk-van-Dalen.text.pdf
  5. 5.
    Mandelkern, M.: A constructive real projective plane. J. Geom. 107(1), 19–60 (2016). [Online]. Available:  https://doi.org/10.1007/s00022-015-0272-4 MathSciNetCrossRefGoogle Scholar
  6. 6.
    von Plato, J.: The axioms of constructive geometry. Ann. Pure Appl. Logic 76(2), 169–200 (1995). [Online]. Available: https://www.sciencedirect.com/science/article/pii/0168007295000052 MathSciNetCrossRefGoogle Scholar
  7. 7.
    von Plato, J.: A constructive theory of ordered affine geometry. Indag. Math. 9(4), 549–562 (1998). [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0019357798800347 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Beeson, M.: Constructive geometry. In: Proceedings of the 10th Asian Logic Conference, pp. 19–84 (2009). [Online]. Available: http://www.worldscientific.com/doi/abs/10.1142/9789814293020_0002
  9. 9.
    Beeson, M.: Brouwer and Euclid. Indag. Math. 29(1), 483–533 (2018). [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0019357717300447 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Constable, R.L., Allen, S.F., Bromley, H.M., Cleaveland, W.R., Cremer, J.F., Harper, R.W., Howe, D.J., Knoblock, T.B., Mendler, N.P., Panangaden, P., Sasaki, J.T., Smith, S.F.: Implementing mathematics with the nuprl proof development system. [Online]. Available: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.4216 (1985)
  11. 11.
    Constable, R.L.: Programs as proofs: a synopsis. Inf. Process. Lett. 16(3), 105–112 (1983). [Online]. Available: https://www.sciencedirect.com/science/article/pii/0020019083900601 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Vesley, R.: Constructivity in geometry. History and Philosophy of Logic 20 (3-4), 291–294 (1999). [Online]. Available: http://www.tandfonline.com/doi/abs/10.1080/01445349950044206 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Heyting, A.: Zur intuitionistischen axiomatik der projektiven geometrie. Math. Ann. 98(1), 491–538 (1928). [Online]. Available: http://link.springer.com/10.1007/BF01451605 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  15. 15.
    Knuth, D.E.: Axioms and Hulls. Springer, Berlin (1992). [Online]. Available: https://books.google.com/books/about/Axioms_and_hulls.html?id=vghRAAAAMAAJ CrossRefGoogle Scholar
  16. 16.
    Lombard M., Vesley, R.: A common axiom set for classical and intuitionistic plane geometry. Ann. Pure Appl. Logic 95(1-3), 229–255 (1998). [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0168007298000177 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Beeson, M.: Logic of Ruler and Compass Constructions, pp. 46–55. Springer, Heidelberg (2012). [Online]. Available: http://link.springer.com/10.1007/978-3-642-30870-3_6 zbMATHGoogle Scholar
  18. 18.
    Beeson, M.: A constructive version of Tarski’s geometry. Ann. Pure Appl. Logic 166 (11), 1199–1273 (2015). [Online]. Available: http://linkinghub.elsevier.com/retrieve/pii/S0168007215000718MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sernaker, S., Constable, R.L.: Formal exploration of geometry. [Online]. Available: http://www.nuprl.org/MathLibrary/geometry/ (2016)
  20. 20.
    Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie. Springer, Berlin (1983). [Online]. Available: http://link.springer.com/10.1007/978-3-642-69418-9 CrossRefGoogle Scholar
  21. 21.
    Boutry, P., Gries, C., Narboux, J., Schreck, P.: Parallel postulates and continuity axioms: a mechanized study in intuitionistic logic using Coq. Journal of Automated Reasoning, pp. 1–68. [Online]. Available: http://link.springer.com/10.1007/s10817-017-9422-8 (2017)
  22. 22.
    Narboux, J.: Mechanical theorem proving in Tarski’s geometry. In: Automated Deduction in Geometry, pp. 139–156. Springer, Berlin (2006). [Online]. Available: http://link.springer.com/10.1007/978-3-540-77356-6_9
  23. 23.
    Beeson M., Wos, L.: OTTER proofs in Tarskian geometry. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) Automated Reasoning, pp. 495–510. Springer International Publishing, Cham (2014)Google Scholar
  24. 24.
    Meikle L.I., Fleuriot, J.D.: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. [Online]. Available: http://link.springer.com/10.1007/10930755_21 (2003)
  25. 25.
    Calderón, G.: Formalizing constructive projective geometry in Agda. In: LSFA 2017: the 12th Workshop on Logical and Semantic Frameworks, with Applications, Brasília, pp. 150–165 (2017). [Online]. Available: http://lsfa2017.cic.unb.br/LSFA2017.pdf
  26. 26.
    Kahn, G.: Constructive geometry according to Jan von Plato. V5,10 (1995)Google Scholar
  27. 27.
    Constable, R.L.: The semantics of evidence. Cornell University, Ithaca, NY, Tech Rep. (1985)Google Scholar
  28. 28.
    Wadler, P.: Propositions as types. Commun. ACM 58(12), 75–84 (2015). [Online]. Available: http://doi.acm.org/10.1145/2699407 CrossRefGoogle Scholar
  29. 29.
    Avigad, J., Dean, E., Mumma, J.: A formal system for Euclid’ s elements. The Review of Symbolic Logic 2(4). [Online]. Available: http://repository.cmu.edu/philosophy (2009)
  30. 30.
    Heath, T.: The Thirteen Books of Euclid’s Elements. Dover, New York (1956)zbMATHGoogle Scholar
  31. 31.
    Allen, S., Bickford, M., Constable, R., Eaton, R., Kreitz, C., Lorigo, L., Moran, E.: Innovations in computational type theory using Nuprl. J. Appl. Log. 4(4), 428–469 (2006). [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1570868305000704 MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tarski, A., Givant, S.: Tarski’s system of geometry. Bull. Symb. Log. 5(2), 175–214 (1999). [Online]. Available: https://www.cambridge.org/core/product/identifier/S1079898600007010/type/journal_article MathSciNetCrossRefGoogle Scholar
  33. 33.
    Bickford, M.: Constructive analysis and experimental mathematics using the Nuprl proof assistant. [Online]. Available: http://www.nuprl.org/documents/Bickford/reals.pdf (2016)

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ariel Kellison
    • 1
  • Mark Bickford
    • 1
  • Robert Constable
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA

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