The World of Descartes

Michael S. Mahoney
Princeton University

Plenary address to the 7th Annual Conference of the Association for Core Texts and Courses, University of Notre Dame, 5-8 April 2001; published in The Wider World of Core Texts and Courses (ACTC, 2004), 73-83

When Scott called to invite me to speak at this conference, I had to think a bit about how to respond. I notice that I'm talking about science and core texts at a conference on "The Wider World of Core Texts and Courses", but I hope I don't have to argue for science as part of the narrow world as well, whatever that might be and however narrowly it might be construed. Science forms the core of modern society (and will of the postmodern as well). It is deeply ingrained in the way Western culture has thought about the world, society, and the self, at least since the seventeenth century. In some aspects it is fundamental to Western culture itself. As someone originally trained as a medievalist who has spent a lot of time teaching the history of science from antiquity through the seventeenth century, I have difficulty separating science from the rest of society. Indeed, "science" may not be the right word for what historians of early science study, since they extend their aegis over logic, the quadrivium, and natural philosophy, i.e. the bulk of the traditional curriculum. They tend to be less imperialistic as they march forward from 1700 and the natural sciences gradually form their own institutions and courses of study. But I still take it as axiomatic that a culture's understanding of and interaction with nature is an integral and inseparable element of the culture as a whole and hence that the texts that deal with nature are among the culture's core texts. That is why perhaps, never having taught a core course proprie dictu, I have covered a good number of the core texts.

I haven't taught a core course, but I have always tried to teach from primary sources. That desire has led to an interesting split in my career between subjects for which a set of core texts lies readily to hand and subjects for which the texts are less "core" or for which the core is not textual. On the one hand I have taught courses on "The Scientific World View of Antiquity and the Middle Ages" and "The Origins of Modern Science, 1500-1700", not to mention simply "The History of Europe from 400 to 1700". For those courses, it is not hard to find core texts and indeed common core texts. I had no trouble finding a place for the Scientific Revolution in the history of early modern Europe. It's hard to see how one can understand the one without the other. So in my teaching ancient and medieval science have emerged from a reading of Plato's Timaeus, Aristotle's Physics and History of Animals, Ptolemy's Almagest, and excerpts from St. Thomas Aquinas and William of Ockham. My students have traced the Scientific Revolution through Copernicus' On the Revolutions, Galileo's Two World Systems and Two New Sciences, Descartes' World (to which I shall return presently), Harvey's On the Motion of the Heart and Blood in Animals, Newton's Principia, and several other less familiar but no less important texts. These were the books that made the history under study. To read them is to engage directly with the thoughts of the people who set us on our modern way of viewing the world.

But an interesting thing happened one year toward the end of the course on the Scientific Revolution. As I came into class, a student was sitting there, looking quite perplexed. I asked him what was bothering him. He said, "I can't make any sense at all of Newton's Principia. I don't understand a thing." I said, "How did you do with Galileo's Two New Sciences?" "Fine," he replied, "no trouble." "Well, there you have it," I said, " that's what the Scientific Revolution was about. Only fifty years separate those two works." Then I hastened to assure him that we would try to enhance his understanding of the Principia in lecture and discussion.

Now in encountering the limits of his knowledge in the seventeenth century, the student learned a valuable historical lesson, one that can only be learned by wrestling with the past directly. But it should be clear that in reaching it at that point in history, he had also effectively reached the end of his reading of the core texts in the history of physics for the three centuries that separated the science of his world from that of Newton's. If one aims one's courses in the history of science since 1700 at a general undergraduate body, rather than at science majors, one successively meets the points in the various sciences at which the technical demands of the original sources, the core texts, exceed the training of the students. One reaches for popularizations and thus begins to treat science vicariously.

But something else happens, I think. It becomes increasingly hard to identify core texts, texts that play the role that the works of Aristotle, Copernicus, Galileo, and Newton play for the earlier period. There are, of course, notable exceptions: Lyell's Principles of Geology and Darwin's Origin of the Species keep the science of the nineteenth century open to students with general backgrounds, precisely because those works and others like them were addressed to broad audiences. But as the sciences grow in technical sophistication, and especially as they draw ever more heavily on mathematics, the mode of scientific communication shifts. The journal article and the research monograph replace the comprehensive treatise. One can point to seminal publications, indeed one can follow their impact through the citation indices, but one cannot make them do the work of core texts.

Let me give an example from my own experience. About twenty years ago I shifted focus from the world before 1700 to the world of the last century. Drawn for various reasons to the history of technology, I found the computer more interesting than I had when I first encountered it as a programmer in 1959-60. My current research deals with the formation of theoretical computer science as a mathematical discipline and with the so far frustrated efforts to establish software engineering as an engineering discipline. These concerns have in turn led me back around to topics in the history of modern mathematics and science, indeed to current issues in the relation of mathematics to science when the science is done using a computer. My teaching has reflected that shift of focus. The history of technology and the history of computing have both led me away from core texts, the former for a reason to which I want to return at the end of my talk, the latter for the reason I was just discussing. For computing, one can cite seminal articles and papers, one can describe the core ideas, but it is hard to find a core text. There is a sense, I suppose, in which one might want to call John von Neumann the Newton of computing. But von Neumann wrote no Principia. Alan Turing's "On Computable Numbers, with an Application to the Entscheidungsproblem" must count as a seminal text of twentieth-century science, but very few students have the training or talent to read it. The same may be said of Norbert Wiener's Cybernetics and Claude Shannon's Mathematical Theory of Communication, the twin classics of 1948 that form the roots of today's information revolution. If, as is frequently claimed, the computer is changing the way we work, think, and indeed live, then the question of how to incorporate its intellectual foundations into the core curriculum is worth pondering -- but not now.

All that said, in a strange way, this work on more recent topics has led me back on several occasions to the core texts of the seventeenth century. That should not surprise me, for among the reasons for my turn to the history of computing was my sense in the early 1980s that mathematics and the mathematical sciences were entering a period of fundamental, perhaps even revolutionary, change. We seemed to be coming to the end of the Age of Analysis, as analytical mathematics, the continuous mathematics of the calculus, was giving way to combinatorial and computational methods based on the discrete mathematics of modern abstract algebra. (I was on the Princeton School Board at the time, and this had implications for the teaching of mathematics.) Having spent my career to that point writing about the origins of algebraic analysis and analytic mechanics in the seventeenth century and about the revolution in scientific thought they embodied, it seemed to a good idea to look closely at what was happening around me. I might learn something about scientific revolutions from watching one happen, or even from taking some small part in it.

The motto of my secondary school is Finis origine pendet, "The end depends upon the beginning." And sure enough, the end that we seem to be witnessing --in complex adaptive systems, in artificial life, in cellular automata-- has repeatedly brought me back to the beginnings, deepening my understanding of both. And for the beginnings, there are core texts. Some of them are available in modern translations and editions, others I have translated and made available to students, most recently by way of the web. Some will be familiar to you, e.g. Newton's Principia or Huygens' Pendulum Clock, others perhaps not. Among the latter, one in particular to which I frequently return is Descartes' The World, or a Treatise on Light, my translation of which was published in 1979 and is now available through my website. Let me say a few things about it.

For me, this has been a core text for two reasons in particular. First, understanding it and its genesis shaped my course on the Scientific Revolution over the years that I taught it. Second, and perhaps more importantly, I have repeatedly been drawn back to it as I have wandered into other areas, in particular recent work on computer modeling and its interaction with scientific thinking. Conceived at a particular time and place in history, Descartes' World has come to embody a way of thinking about the physical universe to which scientists since have returned again and again. In a strange way that perhaps on reflection is not so strange, the model of the world that Descartes describes there is better suited to the modern resources of the computer than to the analytic modes of mathematical analysis which he did so much to establish. It is a self-generating world that begins with chaos and shapes itself in an almost emergent way into a world that looks like the one we inhabit. In the spirit of the computational modelers at the Santa Fe Institute, it is a world that is synthesized and then studied for the behavior it exhibits. Let's look at it.

In November 1633 Rene Descartes wrote to Marin Mersenne to explain what had happened to a treatise that he had been working on for the previous four years and had repeatedly promised to send. "In fact," he wrote,

I had proposed to send you my World as a New Year's gift; less than two weeks ago I was still resolved to send you at least a part of it if the whole could not be transcribed in that time. But I shall say to you that, having inquired over the last few days in Leiden and Amsterdam whether there was a copy of Galileo's System of the World (because I seem to have heard that it had been published in Italy last year), I was told that it was true that it had been published but that all copies of it had been burned at Rome at the same time and he condemned to some punishment. This has so astonished me that I am almost resolved to burn all my papers, or at least not to let anyone see them. For I cannot imagine that he, who is an Italian and even well loved by the Pope, as I understand, could have been made a criminal for anything other than having wanted to establish beyond doubt the motion of the earth, which I know well to have been once censured by some cardinals. But I thought I had heard it said that since then one had not stopped teaching it publicly, even in Rome. I confess that, if it is false, all the foundations of my philosophy are also. For they clearly demonstrate it. It is so linked to all part of my treatise that I cannot detach it without rendering the rest completely defective.

Wanting neither to draw down the Church's disapproval nor to "cripple" his treatise, Descartes had decided to suppress it, assured that Mersenne "would not send a bailiff to collect his debt". Descartes would lie low for a while.

That treatise did not appear for another thirty years, not until 1664 and even then in truncated form. As published, it broke off in the middle of Chapter 15. From descriptions of the text, for example in Part V of the Discourse on the Method, we know it originally included what was separately published as his Treatise on Man, which seems to explain why that text begins with Chapter 18. Those descriptions also suggest that the missing chapters 16 and 17 contained something close to what appeared in 1637 as two of the Essays to which the Method served as introduction, namely the Optics and the Meteorology. More importantly, perhaps, the core of the treatise, a description of a mechanistic world of matter in motion, governed by mathematical laws of motion and impact, appeared in expanded form in the Principles of Philosophy in 1644, albeit phrased in a way to mask the motion of the earth.

So when The World finally saw the public light, it was of historical interest only, albeit of sufficient interest to warrrant a second printing thirteen years later. Why then treat it as a core text? Well, precisely for historical interest. In it we have Descartes' original vision of a mechanistic universe, composed in the years immediately followed the discovery that provoked it and suggested a new path to the true nature of the physical world. Scientists since then have embraced that vision and followed that path, even as they have modified it in detail. Where some, like Newton, have disagreed with it, the disagreements have hinged on momentous issues, such as whether the world is self-organizing or requires an external guiding hand or designer.

I said I had most recently been drawn back by the new style of synthetic modeling: create a world, say, of artificial life and then see how its behavior matches that of the world around us. That is how Descartes sets up his World, inviting the reader to listen to a story - and thus inviting his critics to condemn his physics as no more than a belle romance (Huygens). "For a short time, then," he wrote at the beginning of Chapter Six containing the "Description of a New World", "allow your thought to wander beyond this world to view another, wholly new one, which I shall cause to unfold before it in imaginary spaces." It was meant as a sketch. After laying out in Chapter Seven the "laws of nature" governing this world, Descartes assured the reader that they accounted for all that occurred in it. "Nonetheless," he concluded,

in consequence of this, I do not promise you to set out here exact demonstrations of all the things I will say. It will be enough for me to open to you the path by which you will be able to find them yourselves, whenever you take the trouble to look for them. Most minds lose interest when one makes things too easy for them. And to compose here a setting that pleases you, I must employ shadow as well as bright colors. Thus I will be content to pursue the description I have begun, as if having no other design than to tell you a fable.

A fable, yet time and again Descartes will remind the reader of how closely this imagined world of particular matter interacting according to the laws of motion resembles in its behavior the world we see around us. Beginning with a chaotic soup of particles, it will develop a sun and stars, planets and comets, which will all move in the same patterns as ours. Its sun and stars will generate light, which will propagate through the heavens, reflect from planets, and refract through lenses according the same laws as ours. It will strike our eyes as does the light in our world and produce the same sensations. Containing no more than undifferentiated matter in motion, it will look and feel the same as the world that Aristotle and his followers have filled with forms and qualities, spirits and intelligences, of potential and actualization.

Why the charade, the dissembling? There's little doubt that Descartes believed he was describing our world as it is, not as it could be. Let's look further.

The World is a core text in another sense. It reveals the scientific core of Descartes' philosophy. It carries the subtitle, Treatise on Light, which points to its origin in Descartes' solution in 1625 of the longstanding problem of the law of refraction. From that law followed a science of lenses that reinforced a theory of vision in which rays of light from an object are refracted by the lens of the eye on a focal point on the retina. What results is a one-to-one mapping of points of the field of vision onto an image. That is all that vision is: a pattern of light on the retina, or rather two patterns of light on the two retinas. All else, all that we "see" is a construction of the mind, as it translates those patterns into objects with depth, color, texture, shape, position with respect to other objects. "In proposing to treat here of light," he begins,

the first thing I want to make clear to you is that there can be a difference between our sensation[1] of light (i.e. the idea that is formed in our imagination through the intermediary of our eyes) and what is in the objects that produces that sensation in us (i.e. what is in the flame or in the sun that is called by the name of "light"). For, even though everyone is commonly persuaded that the ideas that are the objects of our thought are wholly like the objects from which they proceed, nevertheless I can see no reasoning that assures us that this is the case. On the contrary, I note many experiences that should cause us to doubt it.

Here is the mind being separated from the body, the world of ideas from the world that our ideas are about. Once the separation is made, the latter world need consist of no more than is necessary to stimulate our senses and thus provoke those ideas. And all that is necessary is something to push on our nerves -matter-- and a changing pattern of the pushes -motion.

That is all there is in Descartes's world, the world outside our mind: matter --space-filling matter, for Descartes, like Aristotle, cannot conceive of space as other than the place occupied by a body-- in motion. All the other entities, qualities, and relations that constitute our experience of the world reduce in reality to configurations of matter in motion. It is a radically reduced ontology born of scepticism, and the first five chapters of The World are aimed at persuading the reader that it is at least plausible. Chapter One softens resistance by appealing to the then common notion that words are related (as Mersenne put it) "neither by rhyme nor by reason" to the things they name. Nor is there any reason why our ideas should resemble their objects. Chapter Two challenges the idea that heat and cold are real qualities or forms. Here's a passage in which a lot is going on.

When flame burns wood or some other similar material, we can see with our eyes that it moves the small parts of the wood and separates them from one another, thus transforming the subtler parts into fire, air, and smoke, and leaving the grosser parts as ashes. Hence, someone else may, if he wishes, imagine the form of "fire," the quality of "heat," and the action that "burns" it to be completely different things in this wood. For my part, afraid of misleading myself if I suppose anything more than what I see must of necessity be there, I am content to conceive there the motion of its parts. For, posit "fire" in the wood, posit "heat" in the wood, and make the wood "burn" as much as you please. If you do not suppose in addition that some of its parts are moved or detached from their neighbors, I cannot imagine that it would undergo any alteration or change. By contrast, remove the "fire," remove the "heat," prevent the wood from "burning:" provided only that you grant me that there is some power that violently moves the subtler of its parts and separates them from the grosser, I find that that alone will be able to cause in the wood all the same changes that one experiences when it burns.

Chapter Three reduces hardness and liquidity to the relative motion of the parts of a body to one another: if they are mutually at rest, it takes great force to get them moving, and the body as a whole seems hard. If they are moving about, the body seems more liquid; the faster and more random the relative motions, the more fluid the body.

None of these arguments is conclusive, just plausible. But for Descartes, it's enough. If you're already disposed to distrust the senses, to refuse to assume anything until you have a clear intuition that it must be so, that it could not be otherwise, then cruising around the "real" qualities of Aristotle's world and showing again and again that they can be explained in terms of matter in motion begins to acquire persuasive force.

Aristotle is not the only target. Switching to the language of 'elements', Descartes again shaves reality down to the bare necessities. There is only one kind of matter, but it clumps together in three basic forms: there are big chunks, what we usually think of as matter; there are small, round pieces that swirl around the big chunks, or rather that form vortices in which the big chunks float; and there are pieces that are as small as need be to fill the spaces among the fluid particles and that in a pure unmixed conglomerate move exceedingly fast. The last are the source of light, and they make up stars; the second particles are the bearers of light from the stars to our eyes. They bear it as a fluid bears a wave; they transmit it instantaneously as a pressure.

Well, clearly, if matter is undifferentiated except by the form of its clumping, then motion is going to have to do a lot of work in Descartes' world. And indeed it does. In Chapter Seven he lays out the laws of motion, which proceed from the immutability of God and are reinforced by the principle of sufficient reason. At the beginning God put a certain quantity of motion into the world. (I have the irreverent picture of a cube of ice given a sharp crack to break it into pieces and then a strong swirl to get it all going around.) Each particle will proceed at its present speed straight in the direction in which it is headed (the world is conserved from moment to moment, and all one can know about a body's motion in a moment is how big it is, how fast it's going, and where it's headed). But there is no void in which such motion can get anywhere. Every body is pushing up against every other, and the world as a whole is turning in circles. So how do bodies fall in straight line? What is weight? Descartes has explanations for this based on the notion of centrifugal force and on the manner in which faster moving bodies seeming to push their way outward from the center of their vortex by conserving their rectilinear path will push past slower ones, forcing them by extrusion toward the center. That is gravity, or weight.

Although the laws are stated quantitatively, Descartes uses them qualitatively. Although he has reduced the world to a mathematical ontology, he cannot yet make his system work with mathematical precision. He has no measure of centrifugal force by way of which to explain, say, the law of falling bodies. He cannot derive from his laws of motion a gradient of speeds across the vortex that will have the planets moving in accordance with Kepler's laws. But he believes that it can be done. He has done it in one important realm. Having discovered the law of refraction in 1625, he has worked out a derivation of the law from his laws of motion. It requires a change of model from an instantaneously propagated pressure to a body moving through space over time, but he can show how that transformation works. Indeed that is where he seems to be heading in Chapters Thirteen through Fifteen, as he explains how light is engendered by the stars, transmitted through the second element, refracted by moving from one vortex to another, and reflected on its encounter with solid bodies. To see how it works out in quantitative detail, we must turn to the Optics. (Not now. There isn't time; you'll have to take my word for it.)

There is an agenda here: a laying out of problems to be solved, a ranking of their priority, a specification of what constitutes a solution. The agenda is to move the mathematical analysis of mechanical experience --of pendulums, of colliding bodies, of bodies whirled in a sling, of bodies accelerating in free fall-- down to an underlying microworld of elementary particles and to build up from there back up to the experienced world. Descartes was wrong in the details, as his successors did not hesitate to point out, sometimes brutally. But that doesn't matter historically. In the hands of successors such as Christiaan Huygens and Isaac Newton, who followed Descartes' lead in mechanics even as they extended it and corrected its details, that mathematics became the basis an increasingly powerful method for modeling the physical world. It provided a means of taking apart the world machine, the clockwork universe, of resolving it into its constituents and reassembling them into new configurations, translatable into experiments. Newton made the clockwork universe tick. But at the same time that his title page chided Descartes --Newton was setting out the Mathematical Principles of Natural Philosophy-- he was pursuing Descartes' agenda.

In short Descartes' World was a persuasive vision. We know that from the way people at the time responded to it. It thus serves as a core text, not only in conveying the birth of a mechanistic universe but also in enabling the student to discover by through close reading and exegesis what Descartes thought would make his vision persuasive to his readers. Like any core text, Descartes' World is as much about its intended audience as it is about its author.

I said earlier that I've been drawn back to The World recently in the context of what some are beginning to refer to as "postmodern" science. There are many ironies to the text. It was the inspiration for an analytic approach to a mechanistic world, written by one of the creators of modern analytical mathematics. But Newton could make it work only by rejecting some of its fundamental assumptions, in particular the principle that mechanical causes can be propagated only by direct contact and that therefore space had to be filled with some sort of tenuous fluid, an ether. Newton addressed the problem directly in the Principia and said the fluid simply wouldn't work. The alternative to a gravity that extended instantly over empty space was a fluid that did not behave as any known fluid. The argument has continued down to the present.

What makes the problem difficult is that fluids are hard to analyze mathematically, indeed they are analytically intractable owing to non-linear interactions. One of the motivating factors for the development of the computer was to calculate the solutions of hydrodynamic equations for which no analytical solution was known. By the end of World War II it formed a bottleneck for the development of physics. As computers have grown in speed and capacity, numerical approximations have given way to simulations in which the behavior of the system is generated by defining the interactions of neighboring particles and then letting the computer generate the result. A lot of problems have been solved that way, but at the loss of analytical insight. As one looks back on Descartes' World, one sees a prime candidate for such modeling. Descartes's World may have been a complex world in ways we are only now beginning to understand. It's worth thinking about, and that, after all, is the function of core texts.

That said about texts, let me take a moment in conclusion to make good on a promise that I made earlier. I said that teaching the history of technology had led me away from core texts, because the primary sources are not texts. Here is what I mean. Among the people who shaped American society as profoundly as any politician, philosopher, novelist, or social theorist were Thomas Edison and Henry Ford. Indeed, much of the work of the thinkers was aimed at understanding what these men of action had created. But how do we do that? Are there core texts for technology? I tried looking when I first started my course, but I gave up when I decided that I was looking in the wrong place. Henry Ford persuaded me of that when I read his admonition in his 1924 book, My Life and Work.

There is an immense amount to be learned simply by tinkering with things. It is not possible to learn from books how everything is made -- and a real mechanic ought to know how nearly everything is made. Machines are to a mechanic what books are to a writer. He gets ideas from them, and if he has any brains he will apply those ideas.

I could say much more on the subject, but my time is up. So let me leave you with a question. What, I wonder, can machines be to the humanistic scholar?