Verità matematiche e forme della natura da Galileo a Newton (review)

In the last two decades, much attention has been paid to the quaestio de certitudine mathematicarum, a Renaissance debate that profoundly influenced seventeenth-century discussions concerning the status and function of mathematics. As Emilio Sergio announces in his preface, the scope of the present book is to analyze how seventeenth-century authors, from Galileo to Newton, thought about the question of the certainty of mathematics and about the possibility of a mathesis universalis.

Sergio's introduction deals with the sixteenth-century quaestio. He first analyzes Alessandro Piccolomini's influential Commentarium de certitudine mathematicarum (1547), which denied the status of demonstrationes potissimae to mathematical proofs, but admitted the possibility of a scientia communis to all mathematical disciplines, and then presents the views of Francesco Barozzi, Petrus Ramus, Giacomo Zabarella, Benito Pereira, Adriaan van Roomen, and Giuseppe Biancani.

The main body of the book is divided into seven chapters, each devoted to a seventeenth-century author. In chapter 1, Sergio shows that although Galileo Galilei did not pay systematic attention to the quaestio, the way in which he practiced science constitutes a clear answer to it. The analysis of accelerated motion reveals Galileo's faith in the heuristic power of mathematics and also his belief that the new mathematical physics could incorporate "the classical relations of geometrical reasoning . . . mathematical demonstrations and sensory experiences could be to one another as compositio to resolutio" (89).

Chapter 2 is devoted to Tommaso Campanella, who denied (pace Galileo) that mathematical signs had a value in themselves, but rather believed that quantities and magnitudes expressed relations describing man's connection with the world. But while underlining the fictitious, noncausal nature of pure mathematics, Campanella identified the metaphysical foundation of the mundus mathematicus in spatial extension.

Chapter 3 deals with Francis Bacon, whose disinterest in the research of a mathesis universalis is described as the natural consequence of the idea that nature "as an organized complex exceeds man's innate tendency towards an ordo cognoscendi" (158). Mathematics was assigned by Bacon the role of an "auxiliary science" in the investigation of physical reality.

René Descartes, to whom chapter 4 is devoted, cultivated the project of unifying all scientific disciplines through the creation of a mathesis universalis, intended not only as the totality of principles common to all mathematical disciplines, but also as a universal science dealing with quantity, order, and measure. Sergio's thesis is that the Cartesian project originated from the mathematical research carried out from 1619, and that Descartes "calibrated his methodological considerations in the light of a broader project of reform of the encyclopedia of sciences, which included philosophy of knowledge, philosophia prima, and natural philosophy" (176). Sergio shows how Descartes's solution to the quaestio was a synthesis between the concept of order resulting from his mathematical research, his theory of intuitus, and his doctrine of eternal truths.

Chapter 5, devoted to Thomas Hobbes, deals with three main themes: his physicalist foundation of geometrical entities; the composition of the continuum; and the notion of thought as a calculus, which is anchored in Hobbes's theory of perception.

Chapter 6 analyzes Robert Boyle's views concerning the relation between experimental procedure and quantitative thought. Sergio explains how Boyle managed to stipulate a programmatic alliance between Bacon's idea of nature as a sylva and Galileo's idea of nature as a book written in the simple language of mathematics.

Chapter 7, finally, is devoted to Newton. Sergio admits that no trace of a reception of the quaestio de certitudine mathematicarum is to be found in the works of Newton, who was rather interested in specific themes such as the continuity between ancient and modern mathematical methods, the relation between algebra and geometry, the nature of number, and the concept of mathesis (323). Sergio shows that the Principia represents the application of the idea of mathesis developed in works such as the Geometria curvilinea. Among the many...