From place to space

A stone that keeps flying

Throw a stone. It leaves your hand, climbs, slows, curves over and drops. Lesson 1 left us with Aristotle’s picture of place: each thing belongs somewhere, and its natural motion is a return there. This lesson follows what happened when thinkers in the 1500s and 1600s stopped asking where a thing belongs and asked only where it is, and when. The place to start is the thrown stone, a long-standing puzzle for Aristotle’s physics.

Here was the trouble. In Aristotle’s physics a falling stone needs no explanation, since earth goes down to its natural place. A stone thrown sideways is forced motion, and forced motion needs a mover in contact (Course 3 tells the rule). Once the stone has left the hand, what pushes it? Aristotle kept the push in the air: the air that has been pushed goes on pushing the stone, or, in his own subtler version in a later book of the Physics, the hand gives the air a power to act as a mover, passed on and growing weaker. Philoponus, a commentator in Alexandria whose work is dated to 517 CE, said instead that the thrower impresses a force on the stone itself, which wears itself out. In the 1300s Jean Buridan called this an impetus, which lasts until something resists it.

None of these answers said what shape the path has. In 1537 the mathematician Niccolò Tartaglia, in his New Science, described a cannonball leaving the gun in a straight line, then bending along a round arc, then falling straight down. He also found that a gun tilted at 45 degrees shoots farthest.

Galileo Galilei (1564–1642) gave a different answer. Treat the flight as two motions that do not disturb each other. Sideways, the stone keeps going at a steady speed. Downward, it falls, speeding up as every falling body does. Together they draw a parabola. By about 1609 he had defined the parabolic path, and it was published in 1638, in the Fourth Day of his Two New Sciences, not in the better-known Dialogue of 1632. The plate shows his simplest case, a body leaving a horizontal surface.

Everything hangs on the fall. Galileo found that a body falling from rest covers distances in proportion to the squares of the times: in twice the time, four times as far; in three times, nine times as far. Equal steps of time then give falls of 1, 4, 9 and 16 units, as in the plate.

How could he know? A falling stone is too quick to time with the clocks of the 1600s, so he slowed the fall down. In the Third Day of the same book he describes a beam about 12 cubits long (a cubit is about half a meter), with a parchment-lined groove a little more than a finger wide. Down it he rolled a hard, smooth bronze ball. For a clock he used water running through a thin pipe, and weighed what ran out during each roll: time turned into weight. He rolled the ball the whole length, then a quarter, a half, two thirds and three quarters of it, “a full hundred times,” and reports that the timings never differed by as much as a tenth of a pulse beat. The distances were in proportion to the squares of the times. Course 4 walks through the ramps; here we need only the result.

Two cautions. The medieval “mean speed theorem,” worked out at Merton College, Oxford, and by Nicole Oresme in the 1300s, already described how far a body goes when its speed grows steadily; Galileo applied it to falling bodies and tested it. And he was not the first to doubt that heavier bodies fall faster: Philoponus attacked the rule in 517 CE, and Simon Stevin said in 1586 that he had shown different weights fall alike.

Motion shrinks to change of place

Look at what Galileo did to the word motion. For Aristotle, kinēsis meant change of three kinds: in quality (a leaf turns red), in size (a child grows) and in place (a stone falls). Change of place was one kind among three. He defined motion as the actualizing of a potential, as building is what bricks do on the way to becoming a house.

Galileo’s mechanics keeps only the third kind. A motion is where a body is at each moment: a position that depends on time, and nothing more. This course argues that this made motion value-free. In the older picture, a motion belonged to a thing’s nature and its fulfilment; in the new one it is a change in numbers, and the numbers do not say whether the thing is flourishing or falling apart. That is a thesis, and it needs hedges. Aristotle did not think all motion improved things: he allowed forced motion against a thing’s nature, and change for the worse, such as illness and decay. And the clean split between facts and values came later; the usual landmark is David Hume’s remark of 1739–40 that you cannot derive an “ought” from an “is.” The thesis is about where the long road led, not about what Galileo intended.

The second step is smaller. Once motion is position over time, a body can be treated as a point, and its position as three numbers, x, y and z: how far along, how far across, how high. Its size and surroundings drop out. Be clear about what this is: Newtonian bookkeeping. In general relativity (Lesson 3) coordinates are only labels with no physical meaning, so “every body is a point with coordinates” describes how Newton’s physics keeps its books, not how modern physics sees the world. What the point leaves out is what this course calls place, and Lesson 4 returns to it.

Matter is extension

The philosopher René Descartes (1596–1650) pushed in the same direction as far as it would go. In his Principles of Philosophy (1644) he declared that the nature of body consists in extension: in being long, broad and deep, and nothing more. Not weight, not hardness, not color: hardness, he argued, is only a sensation. Lesson 1 named extension, the size of things, as one of three experiences of space; Descartes made it the whole of matter. That was new in the 1600s, but not unprecedented: Aristotle reports that Plato, in the Timaeus, said matter and space are the same.

Several things follow, and he accepted all of them. First, there is no void. A vacuum in the strict sense, extension with nothing in it, is a contradiction. If God took everything out of a jar, he argued, its walls would come together, since a space with nothing in it would have no size. Second, the world is a plenum, full to the brim, so bodies can only push one another around, and the planets are swept round the sun in great whirlpools, vortices, of fine matter. Third, he stated the law of inertia: a thing stays in its state until something acts on it, and motion of itself runs in straight lines. Newton’s first law owes him a debt, though Newton’s Principia later refutes the vortices in its Book II. Fourth, physics is geometry: Descartes wrote that the principles of geometry and pure mathematics were the only ones he needed, since the only features of matter he admitted were divisions, shapes and motions. In practice his physics contains almost no equations.

It failed, and the reasons are instructive. He had no separate idea of mass: quantity of matter was volume, and “quantity of motion” was size times speed. His rules for collisions covered only perfectly hard bodies alone in the universe, and the fourth says that a smaller body striking a larger one at rest can never move it, however fast it goes. That is false.

He also changed how we draw. His La Géométrie (1637), published as an appendix to the Discourse on the Method, joined algebra to geometry: a curve can be written as an equation, and an equation drawn as a curve. The grid of two perpendicular axes that we call Cartesian is not in the book. Descartes and his contemporary Pierre de Fermat each used a single axis, measuring lengths off it, often at a slant; the perpendicular pair came later. Fermat had the method by 1636, but it was printed only in 1679. (A popular story says Descartes thought of coordinates while watching a fly on his ceiling. It is a legend with no known source.)

One more claim needs care. Some writers say general relativity is Descartes’ idea at a higher level. There is a real echo: in Einstein’s field equations of 1915, the geometry of space and time is tied to matter and energy and no longer sits apart from them. But an echo is not an identity. The same equations have solutions with no matter at all that still have a shape, such as the empty space around a star or black hole (the Schwarzschild solution, found in 1915). Descartes’ world had matter everywhere; Einstein’s equations allow spacetime where there is none. Lesson 3 returns to this.

How the void came back

If matter is extension and nothing is empty, what became of the atomists’ void? Aristotle had argued against it (Lesson 1), and in medieval Europe his physics was taught in the universities. The void came back by a surprising door. On 7 March 1277, Étienne Tempier, Bishop of Paris, after a request from Pope John XXI, condemned 219 propositions. One said that God could not move the heavens in a straight line, because a vacuum would be left behind. Aristotle had argued that a heaven moving in a straight line would leave a void, which is impossible. The bishop answered that God’s power is not bound by Aristotle’s physics: if God wished, he could make a void. The historian Edward Grant found that after 1277 a void beyond the cosmos was discussed, either as space created by God or as an aspect of God’s presence everywhere.

Space as something of God was not a new idea even then. Nicole Oresme, in his Livre du ciel et du monde, commissioned by King Charles V of France and completed about 1377, held that beyond the heavens lies an infinite, imaginary space, which he identified with God’s immensity. The Jewish philosopher Hasdai Crescas (about 1340–1410) defended the void and called God the place of the world.

In Newton’s day an influential voice was Henry More (1614–1687) of Cambridge, who had corresponded with Descartes. In his Enchiridion metaphysicum (1671), More argued against Descartes that whatever exists is extended, spirits included, and that space is infinite, immovable and incorporeal. He gave it about twenty divine names, among them One, Eternal and Omnipresent. He was not the first to link space and God, but he made the case in Newton’s own university.

Then came Newton. In the Principia (1687) he wrote that absolute space, the unmoving box that is there whether or not anything is in it, is “without relation to anything external” and “remains always similar and immovable” (Motte’s translation, 1729). But an empty, motionless box raises the question of what it is, and his answers came from God. In the General Scholium, added in 1713 to the second edition, he wrote of God: “By existing always and every where, he constitutes duration and space.” In the same passage God is not duration and space; he endures and is present. In an unpublished manuscript, De Gravitatione, from the 1670s or 1680s, space is “an emanative effect” of the first existing being. And in the Opticks (Query 28 in the 1717 edition) God sees things “as it were in his Sensory,” meaning in infinite space. A sensorium is an organ of sensation.

Leibniz pounced. In a letter of November 1715, passed to the theologian Samuel Clarke by Caroline, Princess of Wales, he wrote that Newton makes space “an Organ, which God makes use of to perceive Things by.” If God needs an organ, he said, things do not depend wholly on him. The same letter mocked a God who must “wind up his Watch.” The jibe had a target: Newton thought the solar system needed divine care, and in 1694 David Gregory recorded him saying that “a continual miracle is needed” to keep the sun and fixed stars from rushing together. Clarke replied that Newton’s words are a similitude, as it were, and that God perceives without any organ. The exchange ended with Leibniz’s death in 1716. Whether Newton helped Clarke write his replies is debated.

This course argues that the void needed God before it could be respectable. Aristotle’s physics said nothing can be empty, so empty space entered physics only after a bishop said God could make a void, and it held together once space was tied to God. That is a thesis, and it is contested. Pierre Gassendi, a Catholic priest who revived Epicurean atomism with the void as a principle, held that space and time are there before any body and whether or not any body is in them, and his space did not need God to hold it up. If so, God was one route to empty space, not the only one.

What can be detected? Newton believed absolute space and time cannot be seen directly. His Corollary V says that the motions of bodies inside a space are the same whether that space is at rest or moves uniformly in a straight line, the idea behind Galileo’s ship cabin (Course 20). No experiment inside can find absolute rest, or absolute steady speed. But he thought absolute acceleration, spinning included, shows itself by its effects. How? The bucket, in Lesson 3.

Kant locks the box

By the late 1700s absolute space worked beautifully in physics, and nobody could say what it was. The German philosopher Immanuel Kant (1724–1804), whom you met in Course 19, gave a new kind of answer in his Critique of Pure Reason (1781, second edition 1787). Space is neither a self-standing thing, as Newton’s was, nor only a set of relations among things, as Leibniz held. It is the form of outer intuition: the way we must take in anything outside us. We do not learn about space from experience; we need it in order to have experience of outer things. Kant mocked the picture of space and time as eternal, infinite, self-subsisting “non-entities.” If Newton built the box that holds the world, Kant locked it: the box now came with the knower.

The story of his hands shows how far he moved. In 1768 Kant noted that a left hand and a right hand are alike in every inner measurement, yet one cannot be swapped for the other, and argued that this supports a Newtonian absolute space, against Leibniz. By 1770 he thought space was “subjective and ideal,” and in the Prolegomena of 1783 he used the very same hands to argue that space is a form of intuition. The same example served opposite conclusions.

Why did he go that way? Kant took mathematics and the science of nature as given, and asked how they are possible; Part Two of the Prolegomena is titled “How is pure natural science possible?” For geometry his answer was that its truths come from intuition, known beforehand and with certainty, so Euclid’s geometry describes the form of our intuition of space. Kant’s space is not Newton’s dressed in philosophy: it keeps a Euclidean, Newtonian structure but makes it a form of human knowing. This course argues that after Kant, “where and when?” became the first question to put to anything that claims to be known. That is the course’s reading; Kant’s own claim is narrower, that space and time are conditions of how things appear to us. Lesson 3 shows what happened when the geometry in the box turned out not to be the only one.