Notre Dame Journal of Formal Logic

On the Inconsistency of Mumma's Eu

Nathaniel Miller

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Abstract

In several articles, Mumma has presented a formal diagrammatic system Eu meant to give an account of one way in which Euclid's use of diagrams in the Elements could be formalized. However, largely because of the way in which it tries to limit case analysis, this system ends up being inconsistent, as shown here. Eu also suffers from several other problems: it is unable to prove several wide classes of correct geometric claims and contains a construction rule that is probably computationally intractable and that may not even be decidable.

Article information

Source
Notre Dame J. Formal Logic Volume 53, Number 1 (2012), 27-52.

Dates
First available: 9 May 2012

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1336586236

Digital Object Identifier
doi:10.1215/00294527-1626509

Zentralblatt MATH identifier
06040393

Mathematical Reviews number (MathSciNet)
MR2925267

Subjects
Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 51M99: None of the above, but in this section

Keywords
diagrams case analysis geometry

Citation

Miller, Nathaniel. On the Inconsistency of Mumma's Eu. Notre Dame Journal of Formal Logic 53 (2012), no. 1, 27--52. doi:10.1215/00294527-1626509. http://projecteuclid.org/euclid.ndjfl/1336586236.


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References

  • Euclid, The Thirteen Books of Euclid's Elements Translated from the Text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix, 2nd edition, edited by T. L. Heath, Dover Publications Inc., New York, 1956.
  • Luengo, I., "A diagrammatic subsystem of Hilbert's geometry", pp. 149–76 in Logical Reasoning with Diagrams, edited by G. Allwein and J. Barwise, vol. 6 of Studies in Logic and Computation, Oxford University Press, New York, 1996.
  • Luengo, I. P., Diagrams in Geometry, ProQuest LLC, Ann Arbor, 1995. Ph.D. Thesis, Indiana University.
  • Manders, K., "The Euclidean diagram", pp. 80–133 in The Philosophy of Mathematical Practice, edited by P. Mancosu, Oxford University Press, Oxford, 2008.
  • Miller, N., "Computational complexity of diagram satisfaction in Euclidean geometry", Journal of Complexity, vol. 22 (2006), pp. 250–74.
  • Miller, N., Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, Studies in the Theory and Applications of Diagrams. CSLI Publications, Stanford, 2007.
  • Mumma, J., Intuition Formalized: Ancient and Modern Methods of Proof in Elementary Geometry, Ph.D. thesis, Carnegie Mellon University, 2006. Available at www.contrib.andrew.cmu.edu/~ jmumma/list.html
  • Mumma, J., "Ensuring generality in Euclid's diagrammatic arguments", pp. 222–35 in Diagrammatic Representation and Inference: 5th International Conference, Diagrams 2008, edited by G. Stapleton, J. Howse, and J. Lee, vol. 5223 of Lecture Notes in Computer Science, Springer, Berlin, 2008.
  • Mumma, J., "Review of Euclid and his twentieth century rivals: Diagrams in the logic of Euclidean geometry", Philosophia Mathematica, vol. 16 (2008), pp. 256–64.
  • Mumma, J., "Proofs, pictures, and Euclid", Synthese, vol. 175 (2010), pp. 255–87.