Discourse, Mathematics, Demonstration, and Science in Galileo's Discourses Concerning Two New Sciences

Galileo's Discourses Concerning Two New Sciences is, indisputably, one of the founding documents of the modern, mathematical sciences of nature. ¹ As such, it is not surprising that it is more discussed (or, more accurately, referred to) than read. Galileo is accepted as the founder of a mathematized physics--one in which time, space, and motion are treated as physically measurable and, theoretically, mathematical quantities, and in which laws of motion, depending only on these quantities, are abstracted from the other properties that material bodies possess. He is also known as the founder of an experimental physics in which such physical objects, suitably idealized (or, more specifically, measurable quantities associated with them, or with their idealized and mathematically formulated laws), are subject to verification and detailed examination. The Two New Sciences is the work in which all of this is set forth as the culmination of Galileo's lifelong study of the problems of motion and force, or weight ("force" is not exactly Galileo's term, but it is used by Stillman Drake in his translation).

However, even a cursory reading produces a number of seemingly perplexing and unexpected features, which significantly complicate this view. There is, for example, enough mathematics in the text to persuade both mathematically inclined and nonmathematically inclined readers of the central role of mathematics in Galileo's science; indeed, the quantity and level of mathematical argument is sufficient to dissuade many nonmathematically inclined readers from penetrating very deeply into the text. On the other hand, the text is by no means purely mathematical in nature, and the nonmathematical aspects may, in a perverse manner, dissuade the mathematically inclined from taking the text as a whole seriously enough to give it more than a selective reading. This combination, although found elsewhere in Galileo's works, presents particular complexities here, and this may help to account for the relatively low level of readership of the Two New Sciences and the prevailing, somewhat stereotyped views of the book. ²

To address these and other issues that arise in confronting this text, we present here a reading of The Two New Sciences that attempts to account for both its content and form, for both its mathematical/ scientific aspects and some of the more literary aspects of its structure. Our aim is to point out and explore some key aspects of Galileo's conception of the mathematical sciences of nature, and the ways in which this conception differs from certain accepted views of his project as well as from certain views of the nature of mathematical physics (as we would call it now) that arose in the wake of his work. Accordingly, the emphasis of our analysis is more "critical" or "philosophical" than "historical," arising from a textual reading of the Two New Sciences and the exploration of the architecture of its key concepts, accompanied by a somewhat closer-than-customary analysis of Galileo's mathematics. We will discuss the consequences of our readingat the end of this article. We would like, however, to state at the outset four key points that arise from our analysis, since, on these points in particular, Galileo's approach to science and his views differ notably from those of many practitioners, philosophers, and apologists of modern science, who often try to identify Galileo as a principal precursor. Our four points are as follows:

(1) While mathematics is a crucial and defining dimension of Galilean sciences, mathematical arguments enter these sciences in more complex and diverse ways than is commonly thought.

(2) Similarly, the plural term sciences must be taken quite seriously, for Galileo's concept of science entails, and his scientific practice enacts, a plurality of sciences.

(3) Galileo's concept and practice of science instantiate radically distinct views of certain notions traditionally associated with the project of modern science, such as mathematical and...