Abstract
In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersen, K.: 1985, ‘Cavalieri's Method of Indivisibles’, Archives for the History of Exact Sciences 31, 291–367.
Arnauld, A.: 1667, Nouveaux Élémens de Géométrie, Paris.
Arnauld, A.: 1775–81, Oeuvres de Messire Antoine Arnauld ..., Du Pac de Bellegarde and Hautefage eds., Paris.
Arnauld, A. and P. Nicole: 1662, Logique de Port Royal, Paris.
Arnauld, A. and P. Nicole, 1872, The Port-Royal Logic, trans. by Thomas Spencer Baynes, 7th ed., Edinburgh.
Barozzi, F.: 1560, Opusculum, in quo una Oratio, & duae Quaestiones: altera de certitudine, & altera de medietate Mathematicarum continentur, Padua.
Barrow, I.: 1734, The Utility of Mathematical Learning being ..., London.
Beeson, M.: 1985, Foundations of Constructive Mathematics, Springer, Berlin.
Belaval, Y.: 1960, Leibniz Critique de Descartes, Gallimard, Paris.
Biancani, G.: 1615, De natura mathematicarum, Bononiae.
Bopp, K.: 1902, ‘Antoine Arnauld, der grosse Arnauld, als Mathematiker’, Abhandlungen zur Geschichte der mathematischer Wissenschaften, Bd. 14, B. G. Teubner, Leipzig.
Brody, B. A.: 1972, ‘Towards an Aristotelean Theory of Scientific Explanation’, Philosophy of Science 39, 20–31.
Cavalieri, B.: 1635, Geometria indivisibilibus continuorum nova quadam ratione promota, Bononiae.
Cavalieri, B.: 1647, Exercitationes Geometricae Sex, Bononiae.
Cavalieri, B.: 1966, Geometria degli indivisibili, translated with Introduction and Notes by L. Lombardo Radice, UTET, Torino.
De Gottignies, G. F.: 1669, Elementa Geometriae Planae, Angeli Bernado, Romae.
Descartes, R.: 1897–1913, Oeuvres, Adam and Tannery eds., Paris.
Dummett, M.: 1977, Elements of Intuitionism, Oxford University Press, Oxford.
Fabry, H.: 1669, Synopsis Geometrica, Lugduni Gallorum.
Giacobbe, G. C.: 1972a, ‘Il commentarium de certitudine mathematicarum disciplinarum di Alessandro Piccolomini’, Physis 14(2), 162–93.
Giacobbe, G. C.: 1972b, ‘Francesco Barozzi e la Quaestio de certitudine mathematicarum’, Physis 14(4), 357–74.
Giacobbe, G. C.: 1976, ‘Epigoni nel seicento della “Quaestio de certitudine mathematicarum”: Giuseppe Biancani’, Physis 18(1), 5–40.
Giacobbe, G. C.: 1977, ‘Un gesuita progressista nella “Quaestio de certitudine mathematicarum” rinascimentale: Benito Pereyra’, Physis 19, 51–86.
Giusti, E.: 1980, Bonaventura Cavalieri and the Theory of Indivisibles, Cremonese, Roma.
Guldin, P.: 1635–41, Centroborycae, G. Gelbhaar, Viennae Austriae.
Hintikka, J. and U. Remes: 1974, The Method of Analysis, D. Reidel, Dordrecht.
Hobbes, T.: 1660: Examinatio et emendatio mathematicae odiernae, qualis explicatur in libris Johannis Wallisii distributa in sex dialogos, in Hobbes, T.: 1865, Vol. IV.
Hobbes, T.: 1865, Opera Philosophica quae Latine scripsit, ed. W. Molesworth, London.
Jardine, N.: 1988, ‘The Epistemology of the Sciences’, in Schmitt and Skinner (eds.), The Cambridge History of Renaissance Philosophy. Cambridge University Press, Cambridge, pp. 658–711.
Lakatos, I. (ed.): 1967, Problems in the Philosophy of Mathematics, North Holland.
Mancosu, P.: forthcoming, ‘Aristotelean Logic and Euclidean Mathematics: Seventeenth Century Developments of the Quaestio de certitudine mathematicarum’. Accepted for publication in Studies in History and Philosophy of Science.
Pardies, G. I.: 1671, Élémens de Géométrie, Paris; 2d ed., 1673.
Pascal, B.: 1963, Oeuvres Complètes, Editions du Seuil, pp. 348–59.
Pererius, B.: 1576, De communibus omnium rerum naturalium principiis et affectionibus libri quindecim, Rome.
Philosophical Transactions: 1670, London, Vol. 5, pp. 2054–57. Reprinted by Johnson Reprint Corporation and Klaus Reprint Corporation, New York, 1963.
Piccolomini, A.: 1547, Commentarium de certitudine mathematicarum disciplinarum, Padua.
Proclus: 1970, A Commentary on the First Book of Euclid's Elements, translated with Introduction and Notes by Glenn R. Morrow, Princeton University Press, Princeton.
Rivaltus, D.: 1615, APXIMHΔOYΣ ПANTA ΣΩZOMENA, Archimedia opera quae extant novis demonstrationibus commentariisque illustrata per Davidem Rivaltum a Flurantia ..., Parisiis.
Scaliger, J.: 1594, Iosephi Scaligeri ... Cyclometrica elementa duo, Lugduni Batavorum.
Szabó, A. K.: 1976, The Beginnings of Greek Mathematics, D. Reidel, Dordrecht.
Torricelli, E.: 1644, ‘De Solido Hyperbolico Acuto’, in Opera Geometrica, Florentiae. Reprinted in G. Loria and G. Vassura (eds.), Opere di Evangelista Torricelli, Stabilimento Tipografico Montanari, Faenza, pp. 1919–44.
Vuillemin, J.: 1962, Philosophie de l'Algebre, Presses Universitaires de France, Paris.
Vuillemin, J.: 1981, ‘Trois Philosophes Intuitionistes: Epicure, Descartes et Kant’, Dialectica 35(1–2), 21–41.
Wallis, J.: 1655, Arithmetica Infinitorum, London. Reprinted in Wallis, 1696–99, Vol. 1.
Wallis, J.: 1665, Mathesis Universalis: sive aritheticum opus integrum, London. Reprinted in Wallis, 1696–99, Vol. 1.
Wallis, J.: 1685, A Treatise of Algebra, London. Reprinted in Wallis, 1696–99, Vol. 2.
Wallis, J.: 1687, Institutio Logicae, London. Reprinted in Wallis, 1696–99, Vol. 3.
Wallis, J.: 1696–99, Opera Mathematica, Oxonii, 3 vols.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mancosu, P. On the status of proofs by contradiction in the seventeenth century. Synthese 88, 15–41 (1991). https://doi.org/10.1007/BF00540091
Issue Date:
DOI: https://doi.org/10.1007/BF00540091