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**Philosophy of mathematics and mathematical practice in the seventeenth century.**
*(English)*
Zbl 0939.01004

New York, NY: Oxford University Press. viii, 275 p. (1996).

This book endeavours to bridge the gap between studies on the philosophy of mathematics and studies on the mathematical research (here somewhat unfortunately called mathematical practice) in the 17th century. In order to show how profound the interaction between these fields was, the author (mainly following the historical order) “point(s) out which theories, concepts, and theorems were at the center of foundational reflections.”

Chapter 1 provides the background by outlining the main arguments put forward in the Renaissance debate about the certainty of mathematics (Quaestio de certitudine mathematicarum). Its starting point was Aristotle’s theory of scientific knowledge, as formulated in the Posterior Analytics: only if we know the cause of the results of which we possess knowledge, is it scientific knowledge. (Aristotle accepted four causes: formal, material, efficient, and final.) Can mathematical demonstrations be causal and thereby establish the same kind of certainty as scientific knowledge? While this was flatly denied by some on ontological grounds, others analyzed the mathematical proof structure, especially in Euclid and Archimedes. Proofs by contradiction obviously are not causal, but also proofs by superposition are problematic in so far as they involve motion. Thus various attempts were made to circumvent them.

Chapter 2 is devoted to Cavalieri’s geometry of indivisibles (a kind of forerunner of the theory of integration), Guldin’s criticism thereof, and also his method of determining centers of gravity. Guldin, wary of the risk of ending up with an atomistic theory of the continuum, insisted on explicitly geometrical constructions. He rejected Cavalieri’s idea of considering plane figures as a collection of infinitely many lines and solids as a collection of infinitely many planes. Guldin was also able to determine a considerable number of surfaces and volumes of rotation by applying “Guldin’s rule” although he had no valid proof for this rule, as Cavalieri rightly criticized. Despite their attempts to develop mathematics only by ostensive proofs, neither Cavalieri nor Guldin were able to develop a program for a complete replacement of proofs by contradiction.

In Chapter 3, the author takes a fresh look at Descartes’ Géométrie of 1637, investigating its foundational assumptions, the exclusion of mechanical curves, and the algebraization of mathematics. A novel interpretation in this much-researched field is the relation of Descartes’ foundational choice to the attempted quadrature of the circle by Clavius in 1591. Moreover, Mancosu emphasizes that Descartes’ appeal to a priori or extensive proofs places his project in the same category as those of Cavalieri and Guldin: they agreed on the metamathematical preference for direct proofs over proofs by contradiction. With the algebraization of mathematics a new type of foundational problems appeared: how is it possible to reason about symbols, as opposed to concrete geometrical objects? In addition, the classical theory of ratios could now be interpreted in terms of fractions of numbers, which (when negative numbers were included) gave rise to much-discussed paradoxes.

Chapter 4 deals with the problem whether there is historical continuity in the development of seventeenth century mathematics, as opposed to Renaissance mathematics. The author is able to establish that many mathematicians of the time, when discussing the nature of mathematics, put forward arguments that can only be understood on the basis of the Renaissance dispute about the Quaestio or, in other words, that the Aristotelian conception of science still strongly influenced mathematical practice. This is illustrated in sections entitled Motion and Genetic Definitions and The “Causal” Theories of Arnauld and Bolzano. (Here the boundary of the 17th century is surpassed, as it is in the next section Proofs by Contradiction from Kant to the Present).

Paradoxes of the Infinite is the theme of Chapter 5. Problems connected with infinitely small and infinitely large quantities that startled mathematicians (such as Galileo, Cavalieri, Tacquet, Torricelli, Leibniz, Barrow, Hobbes, and Wallis) in the 17th century are discussed. It was, in particular, Torricelli’s determination of the (finite) volume of an infinitely long solid that provoked a discussion on how to account epistemologically for infinitistic theorems in geometry.

In Chapter 6, the contemporary debate about the foundations of the Leibnizian differential calculus is studied as a revealing example of the interaction between mathematical practice and the philosophy of mathematics. (Newton is excluded since, due to the delayed publication of his results, the foundational debate about his method of fluxions started only around the turn of the century.) After an exposition of the essence of the calculus following l’Hospital, the debates with Clüver and Nieuwentijt and the debates in the Paris Academy of Sciences (Rolle, Varignon, Saurin) around 1700 are scrutinized by the author.

In an appendix of well over 30 pages, an English translation of Biancani’s De Mathematicarum Natura Dissertatio of 1615 is given. Another 35 pages contain a wealth of notes and references. A bibliography of well over 300 titles and a nine-page index round off this thoughtful and well-written account, that illustrates how the attachment to the Aristotelean conception of science caused seventeenth century mathematicians to develop more direct approaches to mathematics in order to ensure unquestionable certainty.

Chapter 1 provides the background by outlining the main arguments put forward in the Renaissance debate about the certainty of mathematics (Quaestio de certitudine mathematicarum). Its starting point was Aristotle’s theory of scientific knowledge, as formulated in the Posterior Analytics: only if we know the cause of the results of which we possess knowledge, is it scientific knowledge. (Aristotle accepted four causes: formal, material, efficient, and final.) Can mathematical demonstrations be causal and thereby establish the same kind of certainty as scientific knowledge? While this was flatly denied by some on ontological grounds, others analyzed the mathematical proof structure, especially in Euclid and Archimedes. Proofs by contradiction obviously are not causal, but also proofs by superposition are problematic in so far as they involve motion. Thus various attempts were made to circumvent them.

Chapter 2 is devoted to Cavalieri’s geometry of indivisibles (a kind of forerunner of the theory of integration), Guldin’s criticism thereof, and also his method of determining centers of gravity. Guldin, wary of the risk of ending up with an atomistic theory of the continuum, insisted on explicitly geometrical constructions. He rejected Cavalieri’s idea of considering plane figures as a collection of infinitely many lines and solids as a collection of infinitely many planes. Guldin was also able to determine a considerable number of surfaces and volumes of rotation by applying “Guldin’s rule” although he had no valid proof for this rule, as Cavalieri rightly criticized. Despite their attempts to develop mathematics only by ostensive proofs, neither Cavalieri nor Guldin were able to develop a program for a complete replacement of proofs by contradiction.

In Chapter 3, the author takes a fresh look at Descartes’ Géométrie of 1637, investigating its foundational assumptions, the exclusion of mechanical curves, and the algebraization of mathematics. A novel interpretation in this much-researched field is the relation of Descartes’ foundational choice to the attempted quadrature of the circle by Clavius in 1591. Moreover, Mancosu emphasizes that Descartes’ appeal to a priori or extensive proofs places his project in the same category as those of Cavalieri and Guldin: they agreed on the metamathematical preference for direct proofs over proofs by contradiction. With the algebraization of mathematics a new type of foundational problems appeared: how is it possible to reason about symbols, as opposed to concrete geometrical objects? In addition, the classical theory of ratios could now be interpreted in terms of fractions of numbers, which (when negative numbers were included) gave rise to much-discussed paradoxes.

Chapter 4 deals with the problem whether there is historical continuity in the development of seventeenth century mathematics, as opposed to Renaissance mathematics. The author is able to establish that many mathematicians of the time, when discussing the nature of mathematics, put forward arguments that can only be understood on the basis of the Renaissance dispute about the Quaestio or, in other words, that the Aristotelian conception of science still strongly influenced mathematical practice. This is illustrated in sections entitled Motion and Genetic Definitions and The “Causal” Theories of Arnauld and Bolzano. (Here the boundary of the 17th century is surpassed, as it is in the next section Proofs by Contradiction from Kant to the Present).

Paradoxes of the Infinite is the theme of Chapter 5. Problems connected with infinitely small and infinitely large quantities that startled mathematicians (such as Galileo, Cavalieri, Tacquet, Torricelli, Leibniz, Barrow, Hobbes, and Wallis) in the 17th century are discussed. It was, in particular, Torricelli’s determination of the (finite) volume of an infinitely long solid that provoked a discussion on how to account epistemologically for infinitistic theorems in geometry.

In Chapter 6, the contemporary debate about the foundations of the Leibnizian differential calculus is studied as a revealing example of the interaction between mathematical practice and the philosophy of mathematics. (Newton is excluded since, due to the delayed publication of his results, the foundational debate about his method of fluxions started only around the turn of the century.) After an exposition of the essence of the calculus following l’Hospital, the debates with Clüver and Nieuwentijt and the debates in the Paris Academy of Sciences (Rolle, Varignon, Saurin) around 1700 are scrutinized by the author.

In an appendix of well over 30 pages, an English translation of Biancani’s De Mathematicarum Natura Dissertatio of 1615 is given. Another 35 pages contain a wealth of notes and references. A bibliography of well over 300 titles and a nine-page index round off this thoughtful and well-written account, that illustrates how the attachment to the Aristotelean conception of science caused seventeenth century mathematicians to develop more direct approaches to mathematics in order to ensure unquestionable certainty.

Reviewer: C.J.Scriba (Hamburg)