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Ancient Geometrical Analysis and Modern Logic

  • Jaakko Hintikka
  • Unto Remes
Chapter
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 39)

Abstract

The old heuristic method known as analysis (geometrical analysis) was famous in Antiquity, and in the course of the history of Western thought its generalizations have played an important and varied role. It is nevertheless far from obvious what this renowned method of the ancient geometers really was. One reason for this difficulty of understanding the method is the scarcity of ancient descriptions of the procedure of analysis. Another is the relative failure of these descriptions to do justice to the practice of analysis among ancient mathematicians.

Keywords

Sequent Calculus Modern Logic Ancient Method Reidel Publishing Company Tableau Method 
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Notes

  1. 1.
    Hintikka, Jaakko and Remes, Unto, The Method of Analysis: Its Geometrical Origin and Its General Significance (Boston Studies in the Philosophy of Science, Vol. 25), D. Reidel Publishing Company, Dordrecht, 1974.Google Scholar
  2. 2.
    Beth, E. W., ‘Semantic Entailment and Formal Derivability’, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R., 18, n: of 13, Amsterdam 1955, reprinted in Jaakko Hintikka (ed.), The Philosophy of Mathematics, Oxford University Press, Oxford, 1969. — Beth refers to Plato’s Philebus 18 B-D, and to Aristotle’s Metaphysics IV, 3, 1005b2, and to Leibniz. Cf. The Philosophy of Mathematics, p. 19, note 8.Google Scholar
  3. 3.
    Cf. Pappi Alexandrini Collections Quae Supersunt, Fr. Hultsch (ed.), Weidmann, Berlin, Vols. I-III, 1876–77, 634–36. Cf. Hintikka, Jaakko, Logic, Language-Games, and Information (= LLGI), Clarendon Press, Oxford, 1973, Chapter IX, and Hintikka-Remes, Chapter II, for secondary literature on Pappus’ description.Google Scholar
  4. 4.
    Hintikka-Remes, Chapter II.Google Scholar
  5. 5.
    Hultsch, pp. 830–32. The translation is from Heath, T. L., The Thirteen Books of Euclid’s Elements, Cambridge University Press, Cambridge, 1926, pp. 141–142. Pace Heath, we speak of construction in analysis as a part of the transformation (Part I of analysis).Google Scholar
  6. 6.
    See Kleene, Stephen C., Mathematical Logic, John Wiley & Sons. Inc., New York, 1968, p. 289.Google Scholar
  7. 9.
    Cf. Hintikka-Remes, Appendix 1. This Appendix is by Prof. Arpad Szabó. For the two types of reductive arguments, see Lakatos, I. (ed.), Problems in the Philosophy of Mathematics, North-Holland Publishing Company, Amsterdam, 1967, p. 10 (comment by Prof. Kalmar), and p. 25 (reply by Prof Szabó). For the early history of the reductive arguments, see Knorr, W., The Evolution of the Euclidean Elements, D. Reidel Publishing Company, Dordrecht, 1975, Chapter II.Google Scholar
  8. 10.
    See Gulley, Norman, ‘Greek Geometrical Analysis’, Phronesis 3 (1958), 1–14.CrossRefGoogle Scholar
  9. 11.
    For the original treatment of the cut formula in the sequent calculus, see Gentzen, G., ‘Untersuchungen über das logische Schliessen’, Mathematische Zeitschrift 39 (1934), 176–210 and 405–431; for an earlier idea of eliminating modus ponens, see Herbrand, J., in Warren D. Goldfarb and J. van Heijenoort (eds.), Logical Writings, D. Reidel Publishing Company, Dordrecht-Holland, 1971, pp. 40ff.CrossRefGoogle Scholar
  10. 12.
    For the observations concerning the necessity of dealing with auxiliary constructions in geometry, see Proclus In. Pr. Eucl. Comm. (ed. Friedlein), p. 78, line 12ff., and Euclides: Suppl. Anaritii Comm. (ed. by Curtze), pp. 88 and 106.Google Scholar
  11. 13.
    Cf. Beth’s article in The Philosophy of Mathematics, p. 37, and Beth, Aspects of Modern Logic, D. Reidel Publishing Company, Dordrecht-Holland, 1970, p. 44; Hintikka, LLGI, p. 215.Google Scholar
  12. 14.
    Cf. LLGI, pp. 141 and 178–185.Google Scholar
  13. 17.
    Cf. Tarski, A., ‘What Is Elementary Geometry?’ in The Philosophy of Mathematics, pp. 165–175, reprinted from The Axiomatic Method (ed. by L. Henkin, P. Suppes, and A. Tarski), North-Holland Publishing Company, Amsterdam, 1959, pp. 16–29. We refer the reader to Tarski’s paper also for the sense of ‘elementary’ which we are presupposing here.Google Scholar
  14. 18.
    Tannery, Paul, Mémoires Scientifiques II, p. 1 for the quadratrix.Google Scholar
  15. 19.
    Cf. Tarski, A., A Decision Method for Elementary Algebra and Geometry, second edition, University of California Press, Berkeley and Los Angeles, 1951, p. 45.Google Scholar
  16. 20.
    Cf., e.g., Prawitz, Dag, ‘Advances and Problems in Mechanical Proof Procedures’, Machine Intelligence 4, 59–71.Google Scholar
  17. 21.
    See La Géométrie, p. 299 of the original.Google Scholar
  18. 22.
    See Prawitz, Dag, op. cit. Google Scholar
  19. 23.
    Cf. Hintikka-Remes, Chapter VII.Google Scholar
  20. 24.
    Cf. LLGI, Chapter IX.Google Scholar
  21. 25.
    Such as Gentzen’s Extended Hauptsatz.Google Scholar
  22. 26.
    For such a possible dependency, see Hintikka-Remes, Chapter III.Google Scholar
  23. 27.
    Cf. Chapter VI of Hintikka-Remes.Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht-Holland 1976

Authors and Affiliations

  • Jaakko Hintikka
    • 1
  • Unto Remes
    • 1
  1. 1.Academy of FinlandFinland

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