Descartes, René (1596–1650)

DOI: 10.4324/9780415249126-DA026-1
Version: v1,  Published online: 1998
Retrieved January 23, 2019, from

11. Physics and mathematics

To his contemporaries, Descartes was as well-known for his system of physics as he was for the metaphysical views that are now more studied. Indeed, as he indicates in the Preface to the French edition of the Principles, metaphysics constitutes the roots of the tree of philosophy, but the trunk is physics.

Descartes’ physics was developed in two main places. The earliest is in the treatise Le Monde (The World) which he suppressed when Galileo was condemned for Copernicanism in 1633, though summarized in Part V of the Discourse. Later, in the early 1640s, he presented much of the material in a more carefully worked-out form, in Parts II, III and IV of the Principles. Like the physical thought of many of his contemporaries, his physics can be divided into two parts – a general part, which includes accounts of matter and the general laws of nature, and a specific part, which includes an account of particular phenomena.

The central doctrine at the foundations of Descartes’ physics is the claim that the essence of body is extension (discussed in §8 above). This doctrine excludes substantial forms and any sort of sensory qualities from body. For the schoolmen there are four primary qualities (wet and dry, hot and cold) which characterize the four elements. For Descartes, these qualities are sensations in the mind, and only in the mind; bodies are in their nature simply the objects of geometry made real. Descartes also rejected atoms and the void, the two central doctrines of the atomists, an ancient school of philosophy whose revival by Gassendi and others constituted a major rival among contemporary mechanists. Because there can be no extension without an extended substance, namely body, there can be no space without body, Descartes argued. His world was a plenum, and all motion must ultimately be circular, since one body must give way to another in order for there to be a place for it to enter (Principles II: §§2–19, 33). Against atoms, he argued that extension is by its nature indefinitely divisible: no piece of matter in its nature indivisible (Principles II: §20). Yet he agreed that, since bodies are simply matter, the apparent diversity between them must be explicable in terms of the size, shape and motion of the small parts that make them (Principles II: §§23, 64) (see Leibniz, G.W. §4).

Accordingly, motion and its laws played a special role in Descartes’ physics. The essentials of this account can be found in The World, but it is set out most clearly in the Principles (see Motion 2§). There (Principles II: §25), motion is defined as the translation of a body from one neighbourhood of surrounding bodies into another. Descartes is careful to distinguish motion itself from its cause(s). While, as we have seen, motion is sometimes caused by the volition of a mind, the general cause of motion in the inanimate world is God, who creates bodies and their motion, and sustains them from moment to moment. From the constancy of the way in which God sustains motion, Descartes argues, the same quantity of motion is always preserved in the world, a quantity that is measured by the size of a body multiplied by its speed (Principles II: §36). To this general conservation law he adds three more particular laws of nature, also based on the constancy by which God conserves his creation. According to the first law, everything retains its own state, in so far as it can. As a consequence, what is in motion remains in motion until interfered with by an external cause, a principle directly opposed to the Aristotelian view that things in motion tend to come to rest (Principles II: §§37–8). According to the second law, bodies tend to move in rectilinear paths, with the result that bodies in circular motion tend to move away at the tangent (Principles II: §39). The first and second laws together arguably constitute the first published statement of what Newton, later called the law of inertia. Descartes’ third law governs the collision between bodies, specifying when one body imposes its motion on another, and when two bodies rebound from one another without exchanging motion. The abstract law is followed by seven specific rules covering special cases (Principles II: §§40–52). Though the law of collision turns out to be radically inadequate, it casts considerable light on Descartes’ conception of the physical world. One of the determinants of the outcome of a collision is what Descartes calls the ‘force’ of a body, both its force for continuing in motion, and its resistance to change in its motion (Principles II: §43). The role of such forces in Descartes’ mechanist world has generated much discussion, since they would seem to be completely inconsistent with Descartes’ view that the essence of body is extension alone.

These general accounts of matter and motion form the basis of Descartes’ physical theories of particular phenomena. The Principles goes on to explain how the earth turns around the sun in an enormous fluid vortex and how the light that comes from the sun is nothing but the centrifugal force of the fluid in the vortex, with ingenious explanations of many other particular phenomena in terms of the size, shape and motion of their parts. Other works contain further mechanistic explanations, for example of the law governing the refraction of light (Dioptrics II) and the way colours arise in the rainbow (Meteors VIII).

Descartes’ hope was that he could begin with an assumption about how God created the world, and then deduce, on the basis of the laws of motion, how the world would have to have come out (Discourse V, VI; Principles III: §46). But this procedure caused some problems. It is not easy to specify just how God might have created the world – whether the particles that he first created were of the same size, for example, or of every possible size. Furthermore, any hypothesis of this sort would seem to be inconsistent with the account of creation in Genesis (Principles III: §§43–7). These difficulties aside, it seemed obvious to Descartes how to proceed. For example, from his denial of the vacuum it would seem to follow that bodies in motion would sort themselves out into circular swirls of matter, the vortices which were to explain the circulation of the planets around a central sun. Similarly, Descartes used the tendency to centrifugal motion generated by the circular motion of the vortex to explain light, which, he claimed, was the pressure of the subtle matter in the vortex. But the very complexity of the world militates against the full certainty that Descartes originally sought, particularly when dealing with the explanation of particular phenomena, such as the magnet. Indeed, by the end of the Principles, it can seem that he has given up the goal of certainty and come to accept the kind of probability that he initially rejected (Principles IV:§204–6).

Central to Descartes’ physics is his rejection of final causes: ‘When dealing with natural things we will, then, never derive any explanations from the purposes which God or nature may have had in view when creating them. For we should not be so arrogant as to suppose that we can share in God’s plans’ (Principles I: §28). The emphasis on efficient causes was to prove very controversial later in the century.

One especially curious feature of Descartes’ physics, however, is the lack of any substantive role for mathematics. Descartes was one of the great mathematicians of his age. While it is, perhaps, anachronistic to see modern analytic geometry and so- called Cartesian coordinates in his Geometry (1637), there is no question but that it is a work of real depth and influence. In it he shows how one could use algebra to solve geometric problems and geometry to solve algebraic problems by showing how algebraic operations could be interpreted purely in terms of the manipulation and construction of line segments. In traditional mathematics, if a quantity was represented as a given line, then the square of that quantity was represented as a square constructed with that line as a side, and the cube of the quantity represented as a cube constructed with that line as an edge, effectively limiting the geometric representation of algebraic operations to a very few. By demonstrating how the square, cube (and so on) of a given quantity could all be represented as other lines, Descartes opened the way to a more complete unification of algebra and geometry. Also important to his mathematical work was the notion of analysis. Descartes saw himself as reviving the work of the ancient mathematician, Pappus of Alexandria, and setting out a methodology for the solution of problems, a methodology radically different from the Euclidean style of doing geometry in terms of definitions, axioms, postulates and propositions, which he regarded as a method of presentation rather than a method of discovery. According to the procedure of analysis, as Descartes understood it, one begins by labelling unknowns in a geometric problem with letters, setting out a series of equations that involve these letters, and then solving for the unknowns to the extent that this is possible.

Unlike his contemporary, Galileo, or his successors, Leibniz and Newton, Descartes never quite figured out how to apply his mathematical insights to the physical world. Indeed, it is a curious feature of his tree of knowledge that, despite the central place occupied by mathematics in his own accomplishments, it seems to have no place there.

Citing this article:
Garber, Daniel. Physics and mathematics. Descartes, René (1596–1650), 1998, doi:10.4324/9780415249126-DA026-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
Copyright © 1998-2019 Routledge.

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