The question of the relationships between physics and mathematics is as old as philosophy. In his Physics and his Posterior analytics Aristotle fixed the framework for discussions on the relations and distinctions between physics and mathematics as well as the nature of the mixed sciences right up to the Renaissance and well into the seventeenth century. It could even be argued that the basic impulse toward the mathematization of natural philosophy in the seventeenth century owes less to the “platonism” of Galileo and others invoked by Koyré than to the long tradition of discussions on the mixed sciences. These discussions tended to extend the use of mathematics to domains beyond those considered by Aristotle as the “more physical among the mathematical sciences”, that is, Astronomy, Optics and Harmonics. Mechanics was to be added following the Renaissance recovery of the Pseudo-Aristotelian Mechanical problems. Without concern for what Aristotle ‘really meant’, or for the fact that his physics was not mathematical, philosophers offered different interpretations of his views on mixed sciences, some stressing the incompatibility of physics and mathematics, others pointing to their compatibility using examples taken from the mixed sciences.
Renaissance discussions of the mixed sciences thus contributed to extending the latter’s domain beyond the three canonical fields. John Dee, for example, in his “Mathematical Preface” to the English translation of Euclid’s Elements of geometry of 1570, insisted on the usefulness of mathematics for just about every domain of knowledge. And the seventeenth century saw the publication of many essays on the “usefulness of mathematics” running from the empirical Robert Boyle to the mathematical Isaac Barrow, followed at the turn of the century with essays by Fontenelle and John Arbuthnot. Boyle’s subtitle explicitly suggested that “the Empire of Man may be promoted by the naturalist’s skill in Mathematicks (as well pure, as mixed)”, while for Barrow, the usual distinction between mathematics and the mixed sciences was artificial because mathematical objects “are at the same time both intelligible and sensible in a different respect”. Thus, he considered that “mathematics, as it is vulgarly taken and called, is co-extended and made equal with physics itself”.Barrow was not alone in entertaining this view. For John Wallis, for example, who worked on mechanics (a mixed science par excellence), it was obvious that physics was intimately related to mathematics. In the course of a discussion with Oldenburg, Wallis noted that he was surprised to learn that “the Society [’s members] in their present disquisitions have rather an Eye to the physical causes of motion, & the principles thereof, than to the mathematical Rules of it”. He then commented that he considered his hypothesis on motion to be indeed of the Physical Laws of Motion, but Mathematically demonstrated. For I do not take the physical & Mathematical Hypothesis to contradict one another at all. But what is Physically performed is Mathematically measured. And there is no other way to determine the Physical Laws of Motion exactly, but by applying the mathematical measures & proportions to them.
For Wallis the physics of motion was mathematical and he could simply not understand what Oldenburg meant by separating them or even giving the impression of opposing them. And Huygens, whose Orologium oscillatorium published in 1673 gave new examples of the geometrization of natural philosophy, was still complaining to the Marquis de l’Hôpital in December 1692 that:
We find so few occasions to apply geometry to physics that I often find that surprising. For this, with mechanical inventions, is what merits most of our attention; otherwise, as Seneca said somewhere, we lose our intelligence in playing with futile calculations.Huygens thus makes explicit the relation between mathematics, mechanics and practical utility that is often present in the tradition of the mixed sciences. Finally, Newton himself, who followed Barrow’s lectures and succeeded him in the Lucasian Chair at Cambridge University, made clear the continuity between the mixed sciences and natural philosophy in his Optical Lectures of 1670–72 when he said about the use of mathematics that just as astronomy, geography, navigation, optics and mechanics are held to be mathematical sciences, though they deal with ... physical things, so although colours belong to physics, nevertheless scientific knowledge of them must be considered mathematical, in that they are treated through mathematical reasoning.
Whereas discourses concerning the mathematization of nature have been largely discussed, following Koyré’s lead, through the lens of Platonic philosophical influences, the above considerations suggest that a more fruitful approach would be to see this process as the extension to other fields of the tradition of the mixed science. As recent work has shown, this process had important repercussions on the transformation of the disciplinary boundaries between mathematics and natural philosophy in the seventeenth century. But here, I would like to concentrate on what could be called the long term unintended consequences of the use of mathematics in physics, which have received scant attention from historians of science. It is thus the effects rather than the causes (or reasons) of the mathematization of physics . My starting point will be the publication of Newton’s Principia which marks, conceptually, a radical departure from the then dominant tradition of a mechanical philosophy that explained phenomena, most often qualitatively, by contact forces. I will defend the thesis that by taking the mathematical route to natural philosophy, Newton initiated, or at least accelerated, a series of social, epistemological and even ontological consequences which, over the course of a century, redefined the legitimate practice of physics. As we will see, although these consequences were indirect and often only confusedly perceived by the actors involved, they led finally to the state of affairs we now generally take for granted: that physics is mathematical in its formulation. Far from being obvious, this idea was long debated in the eighteenth and even in the first half of the nineteenth century as more and more domains of physics lent themselves to mathematical formulations. By concentrating their attention on the ‘winners’, that is those who have accepted the mathematical conception of natural philosophy and physics, historians have not analysed the resistances to mathematization. In a recent book, for example, John Henry wrote that after the publication of the Principia, readers “took for granted the validity of mathematics for understanding the working of the world” and that “although his book met with some fierce criticism, not a murmur was raised against it in this regard”. As we will see, this was far from being the case but to recover these murmurs, one must look at actors who are now unknown precisely because they rejected the mathematization of physics and were thus excluded from the field (and its history) as it evolved in the eighteenth and nineteenth centuries. And it may be significant that only medicine and chemistry have been examined as cases of resistance to mathematization, as if there could be little such resistance in physics after Newton’s Principia made the power of mathematics ‘obvious’.
Sketching the elements of a larger research program, this paper will focus on the major effects of mathematization mentioned above: (1) social: the use of mathematics had the effect of excluding actors from legitimately participating in discourses on natural philosophy; and (2) epistemological: the use of mathematics in dynamics (as distinct from its use in kinematics) had the effect of transforming the very meaning of the term ‘explanation’ as it was used by philosophers in the seventeenth century. A third unintended consequence of the progress of mathematization, which we will only broach in the last section, was ontological: by its ever greater abstract treatment of phenomena, mathematization led to the vanishing of substances. Not only Cartesian vertices but also the luminiferous ether were dissolved in the acid of mathematics, and I have suggested elsewhere that the same process was at work in the transformation of the concepts of mass and light (photons and wave–particle duality).
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