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.
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Notes
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.
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.
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.
Hintikka-Remes, Chapter II.
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).
See Kleene, Stephen C., Mathematical Logic, John Wiley & Sons. Inc., New York, 1968, p. 289.
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.
See Gulley, Norman, ‘Greek Geometrical Analysis’, Phronesis 3 (1958), 1–14.
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.
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.
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.
Cf. LLGI, pp. 141 and 178–185.
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.
Tannery, Paul, Mémoires Scientifiques II, p. 1 for the quadratrix.
Cf. Tarski, A., A Decision Method for Elementary Algebra and Geometry, second edition, University of California Press, Berkeley and Los Angeles, 1951, p. 45.
Cf., e.g., Prawitz, Dag, ‘Advances and Problems in Mechanical Proof Procedures’, Machine Intelligence 4, 59–71.
See La Géométrie, p. 299 of the original.
See Prawitz, Dag, op. cit.
Cf. Hintikka-Remes, Chapter VII.
Cf. LLGI, Chapter IX.
Such as Gentzen’s Extended Hauptsatz.
For such a possible dependency, see Hintikka-Remes, Chapter III.
Cf. Chapter VI of Hintikka-Remes.
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Hintikka, J., Remes, U. (1976). Ancient Geometrical Analysis and Modern Logic. In: Cohen, R.S., Feyerabend, P.K., Wartofsky, M.W. (eds) Essays in Memory of Imre Lakatos. Boston Studies in the Philosophy of Science, vol 39. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1451-9_17
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