HoP 467 Written in Mathematics: Descartes’ Physics
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Hi, I'm Peter Adamson, and you're listening to the History of Philosophy podcast, brought to you with the support of the philosophy department at King's College London and the LMU in Munich, online at historyofphilosophy.net. Today's episode, Written in Mathematics, Descartes Physics.

Philosophy is written in this all-encompassing book that is constantly open before our eyes, that is, the universe. But it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures. Without these, it is humanly impossible to understand a word of it, and one wanders around pointlessly in a dark labyrinth.

These lines were written by Galileo in his treatise The Assayer, Il Sagittore, which appeared in 1623. As it happens, in that very same year, Descartes went to Italy as part of his youthful wanderings around Europe. Those travels had already brought him to the Netherlands a few years earlier, where he worked with Beekman, who, like Galileo, cherished the idea that the study of nature could best be pursued through mathematics.

Baikman would also have encouraged Descartes to embrace corpuscularianism, the idea that the physical objects around us are made of far smaller bodies, too tiny to see. As we've seen, Descartes was later reluctant to share credit for these ideas with Baikman, so I suppose he would be annoyed with me for pointing out that the two men were carrying on developments that had already been underway for decades.

Apart from Galileo himself, there were the humanists who laid foundations for his work, like Giovanni Battista Benedetti, Federico Comandino, Leon Battista Alberti, and Reggio Montanus. As I discussed way back in episode 361, these scholars devoted philological studies and commentaries to ancient mathematicians like Archimedes and Ptolemy.

They also explored both pure mathematics and applications to such diverse topics as perspective in painting, projectile motion, and timekeeping. As for corpuscularianism, that wasn't new either. For this, you can go back to episode 389, in which we learned how Daniel Sennert, writing in 1619, just when Descartes was collaborating with Baikman, drew on ancient atomism to explain chemical reactions.

Sennett was in turn pushing forward the corpuscularian approaches of Nicholas Taurellos and David Gourlaes, who said that even the human body is just an aggregation of smaller parts. And yet, I don't want to dismiss Descartes' claim to originality in physics, as when he wrote to his undisciplined ally Regius, I am the first to have considered extension as the principal attribute of body.

Actually, this way of putting it understates the boldness of what Descartes was suggesting because of the special technical sense he's here giving to the word attribute. He means that the very nature or essence of body is nothing other than extension. This would explain why, as Galileo said, the book of nature is written in mathematics. Out there in the world is nothing but spatial quantities.

The difference between the real world and mathematical constructions is simply that bodies are concrete, not abstract or conjured up by our minds, and that they're moving around. It's an idea that naturally appealed to the young Descartes, whose first great intellectual breakthroughs were in the area of pure mathematics. A major outcome of his work together with Baikman was a step towards what they envisioned as a universal mathematics.

Descartes realized that he could show an equivalence between algebraic and geometrical phenomena, like when you express a curve, not by drawing it, but as a formula with variables. In a sign that Descartes' project was indirectly rooted in the mathematical scholarship of the humanists, he used this insight primarily to solve problems from ancient geometry. These were also the sorts of challenges that Mersenne and his friends discussed in their letters.

It's even been argued that Descartes was not interested in the study of curves as such, but only in problem solving. After him, though, other mathematicians would show how the method could be generalized. In any case, you can see why it would appeal to Descartes to suppose that, as Cartesian physics specialist Dan Garber has put it, the bodies of physics are the objects of geometrical demonstration made real. Or, we might quote Descartes himself,

I recognize no matter in corporeal things apart from that which the geometers call quantity and take as the object of their demonstrations, that to which every kind of division, shape, and motion is applicable. My consideration of such matter involves absolutely nothing apart from these divisions, shapes, and motions. In a way, this gives Cartesian matter more explanatory power than matter as understood by the Aristotelians.

For them, the most fundamental basis of physical things is prime matter, a pure potentiality for acquiring all the features that belong to bodies, including quantitative features like size, as well as place and motion, along with sensible qualities like color. Admittedly, this had always been controversial, given the difficulty of grasping what prime matter is if it has no determinate properties of any kind.

But Descartes could still rightly take himself to be overthrowing a long-standing consensus, when he wrote in his treatise The World, that matter is not featureless, but a real, perfectly solid body, which uniformly fills the entire length, breadth, and depth of this huge space, in the midst of which we have brought our mind to rest. Thus, each of its parts always occupies a part of that space which it fits so exactly that it could neither fill a larger one nor squeeze into a smaller one.

nor could it while remaining there allow another body to find a place there. In another way though, Cartesian bodies have much less explanatory power than Aristotelian ones. If body is nothing but extension, we have to account for all properties of bodies in terms of an underlying physical geometry. There are no forms or qualitative accidents, as in Aristotelianism,

So if we're trying to explain why a campfire is good for roasting marshmallows, we can't just say that fire has a substantial form that makes it hot. For Descartes, there's no such thing as the form of fire or the quality of heat. Discussing this very example of burning wood, albeit without the marshmallows, he says, "...I am afraid of mistakenly supposing there is anything more in the wood than what I see must necessarily be in it, and so I am content to limit my conception to the motion of its parts."

The fire's capacity to toast the marshmallow, or burn it to a crisp if you don't pay attention, is nothing more than the agitation of the tiny extended parts in the flaming wood and their causal interaction with the tiny extended parts of the marshmallow. Descartes wants to find similar explanations for such familiar phenomena as falling bodies.

He dismisses the Aristotelian claim that the nature of a stone gives it a propensity to move downward, toward the center of the earth. This would be like saying that there's a little soul in the stone, as if the stone knows where it is headed and would really like to get there as fast as possible. As Descartes' contemporaries noted, his new theory is somewhat reminiscent of ancient atomism, which likewise made no use of forms or qualities, but explained physical phenomena in terms of the interaction of invisibly small particles.

Yet again, though, Descartes was affronted by the notion that his ideas had been anticipated by anyone else. Someone who thinks that there's no difference from the ancient atomist Democritus must just be unfamiliar with Descartes' works. And here he actually has a good point. For one thing, Democritus and other ancient atomists, like the Epicureans, assumed that their atoms have weight, a natural tendency to move down.

This is not true of Descartes' corpuscles. Instead, as we'll see, they just move in whatever direction they have already been moving, unless something interferes to make them change course. More fundamentally, Cartesian corpuscles are not atoms. Since body is just extension and every extension is divisible, Descartes emphatically rejects the possibility of an indivisible body.

On this score, he agrees, for once, with the Aristotelians, who also held that bodies are, in principle, infinitely divisible. Whether that means that they're divisible in practice is of course another matter. Descartes also agrees with the Aristotelians in rejecting the void. Democritus and Epicurus thought that atoms move around in empty space. But for Descartes, this is just a basic confusion. Extension is the same thing as body, so how could there be an extension that is unoccupied by any body?

He thus writes to Mersenne, "It is no less impossible that there be a space that is empty than that there be a mountain without a valley." The Aristotelians shouldn't get too comfortable though, because Descartes is eager to accuse them of basic confusion also. We just saw him offering what we might call a reductionist account of the heat in fire, which turns out to be nothing more than the motion of his parts. He adopts the same tactic when it comes to colors, sounds, tastes, and smells.

An Aristotelian would say that our experiences of these things are due to sensible qualities. The untoasted marshmallow looks white because there is a form of white in it, which sounds commonsensical enough. Things are colored white if white color is in them. For Descartes, though, that is precisely the problem. As Garber remarks, the scholastic world, as Descartes understood it, is simply a metaphysical elaboration of the world of commonsense.

What the Aristotelians are doing is to take the way things look, sound, smell, taste, and feel to us, and project those experiences into the world as really existing physical qualities. In fact, Descartes insists, these experiences are just the result of the way that extended bodies causally interact with our sense organs.

Of course, it doesn't seem like that. When you look at the marshmallow, you don't see a swarm of geometrical shapes. You see white color. And color is not quantitative or geometrical at all. That's why Aristotle and his followers distinguished in their theory of categories between qualities, like color, and quantities, like length. But this proves nothing, according to Descartes. Consider how words signify without needing to resemble what they signify, or how tears can represent sadness, even though sadness isn't wet.

Just so, sensible experiences are the result of a causal interaction, but do not wear the underlying cause on their sleeve, so to speak. This is why we can have experiences and be mistaken about their causes. Descartes gives the example of a soldier who thinks he's been wounded, but then discovers that it's just a belt buckle pressing into his flesh. Similarly, in the meditations he points out that a tugging sensation in the stomach need not signify that it's time to eat.

If we were angels, we would experience bodies in purely quantitative terms, which reminds me of the scene at the end of The Matrix when Neo begins to see the virtual reality around him as code instead of as sensible objects. But of course, we're not angels, and all but one of us are not Keanu Reeves. We're normal humans. So apart from our direct awareness of extension like the size and shape of the marshmallow, all our sensory experiences simply arise from the way moving bodies impinge on our own bodies,

Which, by the way, gives us another parallel to Galileo, who says much the same thing in that treatise, The Essayer. For Galileo, the primary and real attributes of bodies are shape, size, location, and motion, and the words we use to describe sensory qualities are just empty names. It must be said that all this is not so much a complete physical theory as a research program, and a mighty ambitious one at that. It's one thing to assert that all natural properties can be reduced to extension and motion, another actually to do it.

A frequent complaint about Descartes is that he didn't get very far with his project. In vain would you page through his published writings for a detailed account of how taste and smell arise from the agitation of particles, or for a correlation between the speed of corpuscular agitation in a fire and its degree of hotness. His theory of the cosmos, which we will get to shortly, is likewise a number-free description of how the universe is structured.

The historian of science, Alexander Coiré, went so far as to say that Descartes' physics is, in fact, as little mathematical as that of Aristotle. In defense of Descartes, though, his unpublished writings, especially his letters, do make more of an attempt to quantify phenomena like falling bodies and sound. If he was reluctant to pursue the full implications of his mathematical physics, especially in the public domain, it was perhaps because he knew that he lacked the necessary experimental data.

Think of something as simple as a falling rock. How was Descartes to put numbers on the air resistance encountered by the rock, so as to calculate the speed it would have if it were falling in a void? On the occasions when he does try to provide a quantitative analysis, he's largely forced to make things up as he goes along. A good example would be color, for which he attempts a sketchy explanation, hazarding the guess that our experience of each color derives from the ratio of straight to circular motions in the corpuscles streaming off the visible object.

But Cartesian physics looks better when we go up to a higher level of abstraction and consider its fundamental principles. In several works, including the one that is indeed called Principles, Descartes identifies three general rules that govern motion. He calls them natural laws, a phrase we still use for such generalities of physics.

We probably don't reflect much on the idea that they are "laws," but for Descartes, the terminology is significant, because these regularities are indeed laid down by a lawmaker, namely, God. It is God who has decreed that, as stated in the first law, bodies continue to move as they have been doing unless they encounter resistance from another body. This law is another idea Descartes shared with Baikman, and another departure from scholastic philosophy.

Again, the mechanics of projectile motion were a matter of controversy in the Aristotelian tradition, but one popular idea was that if you throw a rock, you impart a power for continued motion into the rock, which we can call impetus. It's as if the act of throwing has filled the rock with a bit of virtual fuel for moving through the air. As the rock flies along, the impetus is used of and it duly falls back to earth.

For Descartes, by contrast, the rock would just continue on its way forever if it were passing through an infinite void. Of course, there is no void, so that could never happen. Instead, its motion is lessened by the resistance offered by the air. This brings us to the second law, which is that, when bodies get in each other's way, there is a kind of pushing contest between them, with the more powerful motion winning out.

Here, the power of the motion is determined by size and speed. One motion will be twice that of another, if it is the same speed but in a body double the size, or in a body of the same size but double the speed. As evidence, Descartes points to the whistling sound made by projectiles, like our rock hurtling through the air. This is the sound of the air exerting resistance on the rock and slowing it down, but still losing the pushing contest and being shoved aside.

The third law, finally, is that motion tends to be rectilinear, but can be deflected into circular motion by resistance. Descartes gives the nice example of a stone whirling around in a sling. The stone will fly off in a straight line as soon as it is released, but until then, it is constrained to go in a circle. Notice that all of these laws only concern spatial or "local" motion, which for Descartes is the only kind of motion there is.

If this seems quite commonsensical, that just goes to show how influential Descartes has been. Because for Aristotle and his followers, motion, in Greek, kinesis, could refer to any kind of change. For example, Aristotle would call a change in heat or color a motion, meaning by this a change in qualitative features. But then, for Descartes, there are no real qualitative features anyway.

Instead, modifications of heat or color are going to be explained as a change in spatial motion. For instance, the rate or direction of the motions of the tiny particles in fire or on a colored surface. Descartes said that the scholastic definition of motion as "actuality of the potential" was just "magic words." We all know intuitively what motion is, and throwing around this sort of technical vocabulary is worse than useless.

As the scholastics might say, it simply actualizes our potential for being confused. Despite such bluster, Descartes' own account of motion is not so straightforward. If you learned in math class about Cartesian space or Cartesian coordinates, you might assume that for Descartes, physical space is an unmoving frame of reference through which bodies can move, like a real-life version of a three-dimensional graph in geometry. Motion would then be defined as a change in the coordinates occupied by a given body.

Actually, though, that conception would be more applicable in the physics of Isaac Newton, which is still some years away. For Descartes, motion is instead defined as relative to nearby bodies. If a giraffe is swimming in the Gulf of Mexico, it counts as moving because all its parts are traveling together while the surrounding water is being left behind. If, by contrast, the giraffe is standing on the bed of the Mississippi River, we would say that it is at rest because it is not moving relative to the riverbed.

Rather, the rushing water is in motion because its parts are changing place in comparison to the floor of the river, the banks, and the giraffe. Descartes' theory has a couple of surprising results. First, things only count as separate bodies insofar as they move independently of one another. When Descartes is still enjoying his daily lie-in at noon, he and his bed are not moving relative to one another, so there's no reason to think of them as two distinct bodies.

It's only at the fact that he eventually gets up and starts moving without the bed going along with him that we distinguish him as a separate physical object. Likewise, in the words of another expert on Cartesian physics, Denis Deschenes, "Every part of a body is itself a potential body. All some part needs to do is begin to move separately, and it will be just as much a body as any other, whether this part is a grain of sand or the whole earth."

Things like giraffes, marshmallows, and beds have no privileged status in this physics the way they do in Aristotelian physics because they are not unified by forms. When we think of them as independent bodies, this is just because of their tendency to move around as a whole within their surroundings. This feature of Descartes' theory attracted criticism, and from no less a figure than Leibniz. He complained that the Cartesian universe is effectively just one big thing with lots of internally shifting parts.

No one part forms an independent substance in its own right, but only earns the honor of being a discrete body because of the way it is presently moving. Which is, I think, exactly right, but may not be so much a telling criticism of Descartes as a perceptive summary of what he was trying to say. Having dispensed with the substantial forms of the scholastics, he has no stake in the idea of robustly individual substances. For him, the material world is an inert jumble of interchangeable parts.

It's not a universe populated by giraffes and marshmallows, but by quantifiable extensions that interact with one another by means of the laws of nature. Now let's move on ourselves, to a second surprising consequence. In light of his theory of motion, Descartes is able to say that the Earth is "at rest." Though he is a committed Copernican, he thinks the Earth is being carried along by heavenly matter in a spiral around the Sun, which he calls a vortex.

This is just a larger scale version of the same phenomenon we see in water, when whirlpools form and carry around bits of straw with them. Relative to its surrounding body, the Earth is not moving any more than a bit of straw would be moving relative to the water when it is floating in a fast-moving current. Whirlpools and the vortex of the solar system are illustrations of that third natural law. Left to their own devices, bits of straw and planets would, just like everything else, keep going straight when set in motion.

But just like everything else, they are not left to their own devices, but are always getting pushed around, in this case into a circular path that results from constantly applied resistance. Now, scholars have often assumed that with his insistence that the earth is, in his very special sense, at rest, Descartes was sneakily trying to stay on the right side of the authorities. He was, however, insincerely agreeing with the church that the earth is unmoved, so as to avoid the fate that befell Galileo.

It's even been suspected that Descartes devised his whole relativistic account of motion specifically for this purpose. But this is hard to believe, given the other details of his cosmology. He states that our sun is no different from the visible fixed stars. They too lie at the centers of their own vortices and are presumably being orbited by other planets. Perhaps these planets even have people living on them. Furthermore, stars eventually die after radiating their substance out in the form of light.

Their vortices then collapse in on themselves, leaving the stars to be captured as planets of other solar systems. As yet another Descartes scholar has dryly commented, it scarcely needs remarking that this is hardly the theory of someone who is seeking to placate the Roman Inquisition. But if Descartes was hardly following the lead of the Church, neither was he trying to eliminate God from his vision of nature. God may have his purposes in creating and overseeing the world, but they surely lie beyond our understanding.

Thus Descartes dismisses all talk of final causes or purposes in the study of nature. His God is like a divine architect or engineer who has created the universe as a vast machine. This attracted criticism from more conventionally pious thinkers like Pascal, who wrote: "I cannot forgive Descartes. In all his philosophy he would have been quite willing to dispense with God." But from another point of view, one might wonder whether Descartes dispenses with everything other than God.

We've already seen that God lays down the fundamental laws of physics. The reason that these laws never change is that God never changes, and the regularity of motion is a testament to God's eternal invariability. As DeChene remarks, "It is a nice trick to make the continual changing of things an argument for divine immutability." In addition, God is the one who set bodies in motion in the first place, bestowing upon them the rectilinear motions that then become curves as bodies get in each other's way.

Thus Descartes calls God the general cause of motion, even if collisions and mutual resistance become a further cause of individual motions. The total amount of motion in the universe, as measured in size and speed, remains constant, since bodies simply trade motion with one another when they get into those pushing contests. This too is a sign of God's immutability. An important but difficult question arises here.

There's no doubt that Descartes credits God with creating this whole system and literally setting it in motion. But is God also directly creating every motion, in every body, at every moment of time? That's a view we're going to find in other thinkers who were inspired by Descartes, the so-called occasionalists. They will make God not just a general cause, but the only cause for everything that happens. Scholars disagree about whether this view can already be found in Descartes.

He certainly thinks that God at least sustains bodies in being, but that is a pretty standard thing to say, one any scholastic would agree with. The question is whether God directly creates motion and rest in each bit of extension, thereby ensuring that the laws of nature are followed. Descartes may have changed his mind about this across his career. In earlier treatises like The World and Discourse on the Method, he seems to be saying that God conserves things in existence that allows them to interact causally with one another.

Later, in his Principles, though, he says that the laws of nature just are the causes for motion, and since natural laws don't really sound like the sort of thing that could move a body, it's plausible to assume that Descartes really ascribes the motions to the author of those laws, the legislating engineer who is God. We also find Descartes writing to Elizabeth that God would not be perfect if anything ever happened that did not derive from him, a remark that sounds clearly occasionalistic.

Nonetheless, I tend to think that Descartes wants to make God responsible only for the general system, without making him the cause of every event that takes place within the system. God sets up and enforces the rules, but the bodies are still allowed to play by those rules. And in any case, even if God is authoring all physical motions, it is also possible for us humans with our rational souls to introduce new effects within nature. That would be a significant qualification to any occasionalist tendency in Descartes.

Still, you can certainly see how thinkers like Malebranche would have taken inspiration from him in developing their occasionalism. Speaking of seeing things with certainty, there's a final feature of Descartes' physics that is worth noting. This is his extraordinary level of confidence that the picture he's developed is broadly correct, even if some of the details, or in fact almost all of the details, still need to be worked out.

As I mentioned last time, he says that the principles on which his physical theory is founded are simply obvious, and that his special achievement is to have realized that these obvious truths could be used as principles of physics. Among the points that he considers evident are: that two bodies cannot occupy the same place or interpenetrate, that body is distinct from God and from the mind, and even that body is the same thing as extension.

which is a bit disconcerting because these claims are in fact highly controversial, or have controversial implications. The Stoics for example would have said that God is a body pervading the body of the universe, while the Epicureans would have denied that extension is the same as body, since there can also be void space. So how can Descartes simply declare his preferred assumptions to be perfectly obvious and proceed to build his whole philosophy upon them?

As we're going to see, contemporaries like Gossandie objected to Descartes assuming that the principles of atomism are obviously false, but they could certainly not accuse him of being insufficiently concerned with the status of principles, or of the ideas that seemed to him, at least, clear and distinct. In fact, this question is central to his philosophy in general, and to the meditations in particular. Don't let anything get in the way of your joining me for that, next time on The History of Philosophy Without Any Gaps.

Thank you.