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IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS : A study in early modern conceptions of impossibility

Abstract : This dissertation deals with impossibility results in the context of early modern geometry (XVIIth century). The main problems and questions I shall address in my study are the following. How did early modern geometers prove (or argued for) the impossibilities of solving construction problems by prescribed means? Can we identify similar structures and similar roles in different instances of these impossibility arguments? My starting point is one of the first examples of algebraic thinking in geometry, namely, Descartes’ epoch-making « La Géométrie » (1637). My examination of « La Géométrie » mainly concerns the methodological points of this treatise: the foundations of the distinction between geometrical and mechanical curves, and the classification of curves and problems. A general thesis I advance in my work is that conditional impossibility claims exerted a twofold methodological, or metatheoretical role. Firstly, they contribute to frame the demarcation between acceptable and non acceptable curves. Secondly, conditional impossibility claims enter in the classification of problems on the ground of the curves which construct them, sketched in the third Book of « La Géométrie » and commented by Van Schooten in his latin editions from 1649 and 1659. The presence of impossibility claims in a treatise, like Descartes’ « Géométrie », devoted to lay down the fundamentals of a method to solve all problems of geometry, is not surprising, in so far such a method should provide the guidelines in order to solve each problem according to the most adequate means. An interesting sketch of a classification into possible and impossible problems can be found in Descartes’ correspondence with Mersenne. In particular, the circle-squaring problem is considered by Descartes an « impossible problem ». More generally, the circle-squaring problem stood as an intriguing problem in the context of XVIth and XVIIth century research: it was not only a difficult mathematical question, but it had an important metatheoretical role, I surmise. Indeed decisions about its solvability in principle would contribute to frame the subject matter of geometry, by demarcating legitimate from illegitimate solving methods, as in the outstanding attempt led by Descartes in « La Géométrie ». Furthermore, in the second half of XVIIth century, arguments asserting that the quadrature problem could not be solved by algebraic method would be invoked in order to demarcate finite from infinitesimal analysis. I investigate in this work some fragments of two mathematical works in detail: James Gregory’s work « Vera circuli et hyperbolae quadratura » (1667), and G. W. Leibniz’s « De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis » (1676). In this second part of my work, I argue the general thesis that impossibility claims concerning the circle-squaring problem acquires a new status in the second half of XVIIth century. In order to explain this claim, I detail a critical examination of James Gregory’s work « Vera circuli et hyperbolae quadratura ». In this text, in fact, Gregory sets out to search for a way to reduce the problem of squaring any sector of a central conic (the circle, the ellipse and the hyperbola), to an algebraic equation, and comes up with an argument in order to prove that the impossibility of this endeavor. Gregory’s argument is faulty and was heavily criticized by his contemporaries, but it shows an uncommon insight, for his time, into impossibility results. Moreover, Gregory’s thesis on the impossibility of finding an algebraic quadrature of the central conic sections are historically relevant because they exerted, through a subsequent controversy with Christiaan Huygens, a deep influence on Leibniz’s mathematics. The chapter 8 of my dissertation is indeed dedicated to Leibniz’s lenghty treatise « De quadratura arithmetica circuli ellipseos et hyperbolae », composed and ultimated during Leibniz’s stay in Paris. Although the treatise circulated, under different versions, among Leibniz friends and colleagues mathematicians from 1674, and historical evidence shows that Leibniz had a manuscript ready for publication in the year 1676, this document got lost, and the treatise never saw the publication in Leibniz’s lifetime. My interest for the « De quadratura arithmetica » will be mainly directed towards the the concluding proposition LI, considered by Leibniz as the ‘crowning’ of his treatise: a theorem on the impossibility of squaring the circle, the ellipse and the hyperbola. Leibniz allegedly proves, by an indirect argument, that there is no quadrature of the central conic sections (namely, the circle, the ellipse and the hyperbola) that is more geometrical than his own. Since the solution presented in the « De quadratura arithmetica » is obtained through an infinite series, the above claim amounts to saying that the solution to the quadrature of the circle, the ellipse and the hyperbola cannot be obtained by a finite algebraic equation. In chapter 8 I examine in detail the influence of the controversy between Gregory and Huygens over the genesis, the conception and certain results presented in Leibniz’s « De Quadratura Arithmetica ». I then discuss the mathematical and methodological meaning of Leibniz’s impossibility result, and I argue that Gregory’s « Vera Circuli et hyperbolae quadratura » played a dependable rôle concerning the function of Leibniz’s impossibility argument within the organization of the treatise on the arithmetical quadrature of the circle and the conic sections. Finally, in a concluding chapter, I respond to the questions raised in the beginning by assessing the function of impossibility results in the context of XVIIth century mathematics and, more particularly, with respect to the case studies discussed in this dissertation. On this concern, I assess these case studies both with respect to today extrathereotical impossibility theorems, and with respect to the metatheoretical claims of antiquity. 
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Davide Crippa. IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS : A study in early modern conceptions of impossibility. History, Philosophy and Sociology of Sciences. Univeristé Paris Diderot Paris 7, 2014. English. ⟨tel-01098493⟩



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