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The Analytical Method in Descartes' Geometrie

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The Analytical Method in Descartes' Geometrie THE ANALYTICAL METHOD IN DESCARTES' GEOMETRIE GIORGIO ISRAEL Dipartimento di Matematica Università degli Studi di Roma 'La Sapienza' P.le A. Moro, 5 00185 - ROMA To describe La Géométrie as an essai of the Cartesian method, or as the application of the rules given in Discours de la méthode, has paradoxically contributed to an undervaluation of existing connections between this brilliant and famous essai and Descartes' philosophical work. In a way this is a paradox, considering the fact that this description of La Géométrie underlines the dependency of Descartes' only complete mathematical treatment on the method to follow «pour bien conduire sa raison et chercher la vérité dans les sciences » and on the metaphysical principles on which it is based. Nevertheless the connection between La Géométrie and the Cartesian method thus established appears weak. Because of this unsatisfactory situation, the essays dedicated to the study of this text appear to be split into 'philosophical-humanistic' analyses and 'scientific' analyses. Let us try to clarify the previous statement, beginning with the reasons why the connection between La Géométrie and the rules of Discours de la méthode appears weak.The fundamental reason lies in the vagueness of the methodological rules expressed in the Discours and summarized in the four famous rules governing the scientific thought, even if Leibniz' severity seems excessive when he compares them with common recipes and sums them up in the almost obvious rule: «sume quod debes, operare ut debes et habebis quod optas ».1 Nevertheless it is difficult to deny that those who aim at establishing a tight connection between the rules of the Discours and the contents of La Géométrie, by trying to demonstrate in some way 1 G.W.Leibniz, Die philosophischen Schriften von Leibniz , 7 voll., herausgeg. von C.J. Gerhardt, Berlin, 1875-90, vol.IV, p. 329. that the latter represent an application of the first, as if Descartes had endeavoured to obtain the results of La Géométrie as a direct application of his methodological rules, would be disappointed, and achieve little more than the impression of a vague link . The situation appears different, however, when the whole of Descartes' work is considered and not only the Discours. Then, particularly when referring to the Regulæ ad directionem ingenii, it is possible to trace a much tighter connection between Descartes' method and the contents of La Géométrie, and at the same time to examine some historiographical questions on viewing Descartes' mathematical work from a different angle. The aim of this article is to attempt to highlight these connections and to briefly consider the historiographical questions mentioned above. As an introduction we will use some observations by E.J. Dijksterhuis, which, even though rather general, emphasize the existing link between La Géométrie and the Regulæ .2 Dijksterhuis observes that « […] if you really want to get to know Descartes' method, you should not read the enchanting Discours, which is more a causerie than a treatise, but rather the Regulæ ad directionem ingenii [_]. As a matter of fact, the Regulæ contain an exposition of the so-called Mathesis universalis, which Descartes always considered one of his major methodological discoveries and which he hoped to see applied in all natural sciences.»3 Further on he continues: «The essay La Géométrie, in which Descartes presents his new discovery, fully deserves [_] to be described as a demonstration of the Cartesian method; yet it does not contain an application of the four rules of the Discours, to which this essay constitutes an appendix. In fact, the true Discours de la méthode is set up by the Regulæ ad directionem ingenii ».4 Dijksterhuis identifies this methodology in the Mathesis universalis and consequently the Cartesian ideal in the process of making science 2 The following quotations are taken from E.J. Dijksterhuis, The Mechanization of the World Picture , Oxford University Press, 1961. 3 Ibidem, p. xxx. 4 Ibidem , p. xxx. 2 mathematical, which establishes a central role for La Géométrie as the first step of this process and as a model for its realization. Nevertheless, the way in which he characterizes the Mathesis universalis and the methodology he derives from it is not only vague but also misleading, in a way typical of many ambiguities in historiography dealing with these topics. First of all Dijksterhuis completely identifies the Mathesis universalis with the algebra speciosa of Viète: Consequently, Descartes' ideal would be nothing but the systematic Òapplication of algebraic methods Ó5 to all science. In this way La Géométrie is nothing but the application of algebraic methods to geometry 6, which, in part, is true, but in our opinion insufficient to describe the characteristic features of Cartesian geometry. Secondly, Dijksterhuis identifies the deductive Cartesian method with the logical deductive method of modern mathematics, explicitly referring to the axiomatic method, which constitutes its complete codification.7 In reality, these two comparisons are strictly correlated so that the discussion of one leads directly to the discussion of the other. We will start by commenting on the second comparison, recognizing that it is misleading, which a brief reading of La Géométrie demonstrates . As will be clarified later, the Cartesian deductivism clearly has 'constructivistic' character: the only kind of reasoning allowed is that which will give an explicit construction of the entity under investigation or the result being demonstrated. Consequently any form of reasoning ab absurdo is excluded in Cartesian mathematics; moreover the entities all have to be constructible , which makes it impossible to define them in a conventional or axiomatic way. Furthermore, the admissible deductive chains must be finite; 5 Ibidem, p. xxx. 6 This is the meaning that Dijksterhuis attributes to the term 'analytical geometry'; Descartes is considered his creator: «With the introduction of the new symbolic algebra in geometry he actually became the creator of analytical geometry and consequently the author of one of the most fundamental reforms in mathematics.» (Ibidem , p. 543). 7 The «possibility of setting out propositions in deductive chains» which Cartesius speaks of is identified with the possibility of «expressing the acquired knowledge in axioms». He continues: « The intention of the cartesian method is [_] to make scientific thought occur [_] through deduction, starting out with axioms, and through algebra». (Ibidem , p. 542). 3 consequently, also the rudimentary forms of inductive reasoning in Descartes' work differentiate from modern mathematical inductive reasoning which, by means of a finite number of steps, makes it possible to pass from the finite to the infinite. Thus Cartesian deductivism is 'constructivistic' and 'finitistic', i.e. far from, if not the opposite of, the 'logical-formal' deductivism of modern mathematics. Descartes seems conscious of the particular nature of his method and its position in comparison with past traditions in mathematics. When Descartes criticizes the "vulgar mathematics"8 of his time, he does not only refer to a sort of intuitive-experimental knowledge, in which the validity of the discoveries is particularly uncertain because of the frailty of the method used to obtain them;9 he also criticizes the deductivism of classical mathematics, in particular that of the ancients' and the synthetic method on which it is based .10 Therefore the 'analytical' method he proposes is neither an intuitive procedure, which relies on the senses, nor an abstract formal deductive procedure, which is unable to account for the way in which the 8 R. Descartes, Regulæ ad directionem ingenii, in Opuscula Posthuma , Amsterdam, 1701. See: R. Descartes, Regulæ ad directionem ingenii, A. T., X; as well as G. Le Roy ed.: R. Descartes, Regulæ ad directionem ingenii, (texte revu et traduit par G. Le Roy), Boivin, Paris, 1933. In the following references for both editions will be given; in this case s. A. T., X a p. 376 e l'ed. Le Roy a p.34. 9 Descartes talks about these demonstrations «quae casu saepius quam arte inveniuntur, et magis ad oculos et imaginationem pertinent quam ad intellectum», Ibidem , A.T. X, p.376; ed.Le Roy, pp. 34-36. 10 Descartes remembers to have read nearly everything from the beginning that is usually taught in Arithmetics and Geometry and comments: « Sed in neutra Scriptores, qui mihi abunde satisfecerint, tunc forte incidebant in manus: nam plurima quidem in iisdem legebam circa numeros, quae subductis rationibus vera esse experiebar; circa figuras vero, multa ispismet oculis quondammodo exhibebant et ex quibusdam consequentibus concludebant; sed quare haec ita se habeant, et quomodo invenirent, menti ipsi non satis videbantur ostendere.» (Ibidem , A.T. X, p.375; ed.Le Roy, pp. 32). 4 discovery was reached - similar to the one characterizing the forms of reasoning of ancient mathematics.11 The difference between the analytical and synthetic method and Descartes' evaluations of them are shown in an extremely clear manner in a passage of the Risposte to the Seconde Obiezioni to the Meditationes.12 Here Descartes points out that in the works of the geometer the methods of demonstration are twofold: Òl'une se fait par l'analyse ou résolution, et l'autre par la synthèse ou compositionÓ13 and he continues: «L'analyse montre la vraie voie par laquelle une chose a été méthodiquement inventée, et fait voir comment les effets dépendent des causes; en sorte que, si le lecteur la veut suivre, et jeter les yeux soigneusement sur tout ce qu'elle contient, il n'entendra pas moins parfaitement la chose ainsi démontrée, et ne la rendra pas moins sienne, que si lui-même l'avait inventée. Mais cette sorte de démonstration n'est pas propre à convaincre les lecteurs opiniâtres ou peu attentifs: car si on laisse échapper, sans y prendre garde, la moindre des choses qu'elle propose, la nécessité de ses conclusions ne paraîtra point; et on n'a pas coutume d'y exprimer fort amplement les choses qui sont 11 Modern axiomatic mathematicians consider it the basis of their method.
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