1. Introduction

The emergence and establishment of classical mechanics has always served as a prime example for debates on the nature of science and its conceptual development. Accordingly, its study has often been a place for a fruitful interaction between history and philosophy of science. Galileo has since long been an emblem for modern experimental science (for a recent example see Boumans Reference Boumans2003). At the same time, Galileo's own use of experiments has been a hotly debated topic during the past century. This paper is intended to shed some light on the role that one set of experiments played within Galileo's science. Although of historical interest, it must primarily be seen as contributing to the discussions on the nature and function of different types of scientific experiments, discussions of which have been gaining momentum in the philosophy of science over the past two decades.

In Section 3, I will discuss Galileo's use of the experiments with a pendulum that he describes in the first day of his Dialogues Concerning Two New Sciences (Galileo Reference Galilei, Crew and Salvio[1638] 1954). I ascribe to these experiments a much more central role in Galileo's science than is usually admitted. In Section 4, I will analyze the light this sheds on the paradox of conceptual novelty, which will be introduced in Section 2.

2. The Paradox of Conceptual Novelty

Alexandre Koyré is famous (in some quarters, infamous) for denying that experiments played any fundamental role in Galileo's establishment of his science (see especially Koyré Reference Koyré[1939] 1966, Reference Koyré1968). He gives two arguments for this claim. They are connected, but the falsity of the first doesn't imply that we can neglect the second (although there has been a tendency to do so). After extensively quoting Galileo's description of his inclined plane experiments in the Two New Sciences, Koyré, in a typical example showing the essentially rhetorical nature of the first argument, remarks: “It is obvious that the Galilean experiments are completely worthless: the very perfection of their results is a rigorous proof of their incorrection” (Koyré Reference Koyré1968, 94). Of course, Koyré is right in remarking that Galileo must have exaggerated the results of his experiments, but this need not imply that Galileo did not perform them. Indeed, extensive research on his manuscripts has revealed that Galileo did perform experiments with inclined planes, and moreover, that at least some of his results were in essential agreement with his theory (for a recent assessment of these experiments, see Hahn Reference Hahn2002). What should be remembered, however, is that Koyré had a philosophical motivation for making provocative claims like the one quoted. This provides a second argument for denying experiments a fundamental role. At several places Koyré admits that Galileo might well have had an experimental program (although he didn't execute it properly), but he stresses that these experiments can only consist in the testing of well-formulated theoretical claims (Koyré Reference Koyré[1939] 1966, 143, 155). For an experimental mathematical science such as classical mechanics to be possible, a theoretical language must first be established. Experimental results cannot be at the foundation of a new mathematical science; they can only come at the end, in the form of nature's answers to theory's questions (Koyré Reference Koyré[1939] 1966, 13, 156; Koyré Reference Koyré1968, 75–76). Before experiments establishing the law of fall were possible, the very idea of motion already had to be re-conceptualized.

Koyré was not the first to make this philosophical point. A famous forerunner was Pierre Duhem (Reference Duhem[1906] 1914). Both Koyré and Duhem oppose a naïve “abstractive empiricism,” which would see the establishment of mathematical science grounded in inductive generalizations from empirically observed regularities. As Duhem incessantly stresses, this goes counter to the symbolic character of any truly mathematical science. Before an experimental fact can become epistemologically relevant for a mathematical physical theory, it must first be translated into a symbolic language. This implies that the conceptual structure of the theory must already be established before an empirical fact can have any epistemic bearing on this theory. Koyré shares this insight, but couples it to a view which sees the historical development of physics as characterized by a discontinuous break during the scientific revolution (a term which he understandably championed). In Galileo's work we see the replacement of one conceptual whole for another one. (The link with Thomas Kuhn's ideas is obvious, and Kuhn stressed his indebtedness to Koyré (Kuhn Reference Kuhn[1962] 1996, 3).)

We seem to be presented with a paradox: where do Galileo's new concepts come from, if they cannot be abstracted from experimentally established regularities? Whence conceptual novelty? Koyré had an answer: Galileo's metaphysical inclinations, his “Archimedean Platonism,” guided and structured his conceptual revolution—metaphysics is what makes measurement possible. Undoubtedly, this solution sounds rather naive, and Koyré's answer doesn't stand close scrutiny, but the problem that he raised is genuine.

In their brilliant study, Renn et al. (Reference Renn, Damerow and Rieger2000) tackle the problem of Galileo's conceptual novelty and propose an ingenious solution, based on a careful study of all the relevant material. By focussing on Galileo's proof techniques, they show in compelling detail how he was struggling to derive theorems which are valid in classical mechanics by means of a conceptual apparatus which was still “preclassical.” Nevertheless, the generation of scientists immediately following Galileo could take these theorems as their starting point, and let them dictate the appropriate conceptual framework, which then became “classical.”

To properly solve the riddle of Galileo's conceptual novelty, it is necessary to push back the borderline. Galileo's science at the same time was continuous with a preclassical conceptual system, and did constitute a break with it—for those who came after him. I believe that this will prove to be a lasting insight. Nevertheless, I also believe that Damerow et al. overlook something that is crucial to fully characterize how this break was made possible by Galileo's work. But let me first expand a little bit on the mechanism for conceptual transformation that they propose.

The results achieved by Galileo implicitly define a new conceptual structure. These results were reached by applying given conceptual means to peculiar problems. The nature of these problems, although defined in terms of the given conceptual apparatus, forced Galileo to stretch the limits of application of these conceptual means, showing the need to transcend this conceptual apparatus. Through his exploration of these limits, Galileo's work already implicitly started the necessary conceptual restructuring. Thus, the nature and origin of these problems becomes an important question.

Damerow et al. locate two possible sources that can force someone like Galileo to explore the limits of his conceptual apparatus: the introduction of new objects of investigation, and the combination of conceptual means that stood unconnected before. It is on the first of these that I want to focus. The most important new object with which Galileo was confronted was naturally accelerated motion, and more specifically the knowledge that the space traversed by such a motion was as the square of time spent in this motion.Footnote ¹ His attempts at trying to incorporate this knowledge into a theoretical framework forced him to stretch his conceptual apparatus, which was not suited to handle accelerated motion in an unproblematic way. Hence, at the basis of Galileo's implicit conceptual restructuring there lies empirically given information. As I will show in Section 4, by conceiving of this information as “given,” an essential element of the conceptual transformation brought about by Galileo remains hidden.Footnote ² But first I will have to go back to the first day of Galileo's Two New Sciences.

3. Galileo's Use of Experiments

The task Galileo set himself in constructing a mathematical science of motion was to bring the diverse elements involved in the phenomena of local motion into a well-determined relation, capable of being manipulated with his geometry of proportional magnitudes. If a stable model for the free fall of bodies could be constructed, a host of properties surrounding the motion of bodies could be analysed and investigated, thanks to the powerful and secure ways of mathematical reasoning. As is well known, Galileo succeeded—within the limits set by his conceptual apparatus—by considering the fall of bodies in a void, a limit situation in which all bodies undergo the same uniform acceleration.

4. Galileo's Experiments and Conceptual Novelty

The main flaw of Koyré's treatment of the problem of conceptual novelty is that he isn't sufficiently aware of the way new concepts can be implicitly constructed by putting older concepts to new uses. This neglect is what makes conceptual novelty appear paradoxical. However, as shown by Damerow et al., this construction is far from straightforward. I will now argue that they also overlook an essential element entering into this construction.

As was explained in Section 2, central to the argument of Damerow et al. is the idea that Galileo's exploration of the limits of preclassical mechanics implicitly defined the concepts of classical mechanics. This exploration was triggered and constrained by the attempt to apply the preclassical concepts to the problem of accelerated motion. True, but I want to stress that Galileo's conceptual restructuring was not only constrained but also positively controlled by experimental means. He was not only trying to incorporate empirical information into his conceptual framework, but also experimentally reworking the problem situation. Galileo was not only stretching the limits of application of his preclassical concepts, but also experimentally investigating the proper domain of application for a true science of falling bodies. The stabilization of concepts is not independent of the stabilization of the empirical situation. The implicit definition of the classical concepts is co-constituted by the application of the preclassical concepts to carefully selected, and not as such given, situations. By establishing that the case of fall in a void was the most general, and hence could be called the natural behavior of bodies, Galileo did pave the way for what would become the central domain of application of classical mechanics. The transition from an Aristotelian dynamical framework to the Newtonian is unthinkable without this intervention.

Rose-Mary Sargent has aptly remarked that experiment constitutes a third methodological category that mixes elements from classical empiricism and rationalism (Sargent Reference Sargent1995, 231, n. 50). This also holds true with respect to the problem of conceptual meaning. The meaning of modern scientific concepts is fully determined neither by the conceptual structure of which they are a part, nor by the empirical objects, or properties, or whatever, to which they are supposed to refer. It is only the way that these aspects are put together by experimental means that gives these concepts their full meaning. At the same time, the character of the situations thus described takes on a new dimension.

As a result of Galileo's experimental analysis, it becomes possible for him to attribute the presence of the pure phenomenon to actually occurring instances of free fall, transforming the character of the latter through this attribution. From now on, it will become possible to speak meaningfully about the velocity and the acceleration of the falling objects, and especially about the (mathematical) relations obtaining between them, as defined and analyzed at the theoretical level of the new science. At the same time, the meaning of the abstract concepts of velocity and acceleration will be co-constituted through this attribution. The experiments with the pendulum are essential to all this, because they secure the reference of the pure phenomenon in non-pure situations. In contradistinction to Koyré's view, the abstractly floating concepts are tied to empirical reality from the beginning.

What I called the paradox of conceptual novelty is related to the well known riddle of the applicability of mathematics to nature. That new concepts cannot be abstracted from empirical knowledge—this would not be paradoxical if these concepts were not supposed to constitute a physical science, i.e., be applicable to nature. Galileo famously claimed that the book of nature is written in a mathematical language, but the preceding discussion reveals how the reading of that book can only take place via the mediation of experimentally constructed situations. Experimental activity is what makes possible a mathematical science of nature, by securing the reference of theoretically defined concepts.

5. Concluding Remarks

It is clear that the perspective I have taken here is informed by the so-called “new experimentalism” that found a very influential and early expression in Ian Hacking's Representing and Intervening (Hacking Reference Hacking1983). Hacking was very effective in drawing attention to the multiple functions that experiments can have, other than confirming or falsifying theoretical claims. Experiments not only “give” empirical content to theories, but also have a structure that shapes this content. My critique of Damerow et al. has exactly been that they focus too exclusively on the way Galileo represented problems. By intervening experimentally he also reconstituted these problems, transforming their nature in a way that was essential for the establishment of the new conceptual structure of classical mechanics.

Both Galileo's experiments leading up to the extrapolation to the situation of fall in a void (Section 3.1), and his experiments with the pendulum to exhibit the presence of the pure case of free fall in actually occurring instances (Section 3.3), show many of the characteristics of what Friedrich Steinle has called “exploratory experiments”: “Exploratory experimentation … is driven by the elementary desire to obtain empirical regularities and to find out proper concepts and classifications by means of which those regularities can be formulated” (Steinle Reference Steinle1997, S70). He also stresses that, in this respect, “of particular importance is the idea of elaborating ‘pure’ or ‘simple’ cases” (S73). I am in complete agreement with Steinle's insistence on the fact that these kinds of experiments can be epistemologically very relevant and should not be relegated to some kind of mystical “context of discovery.” One important difference between the cases he discusses, and the case of Galileo discussed here, is that Galileo was not merely exploring experimentally, but exploring the limits of his conceptual apparatus at the same time (as described by Damerow et al.). A truly satisfactory picture can be gained only by focussing both on how he was representing and on how he was intervening.

Let me fittingly end by linking Galileo's work with his pendulum, trying to retract a meaningful signal from the noisy actual behavior of falling bodies, to the work of that other Florentine giant.