A Brief Guide to the Great Equations

by Robert Crease

  This was a modest step beyond Galileo and other close contemporaries of Newton such as Leibniz and Descartes, but thanks to its significance was monumental. It changed the ontology of nature – the way we conceive the most basic units of explanation of the reality we see. Most of Newton’s contemporaries conceived of these units as the bodies themselves, which affected other bodies through various mechanisms that brought about different kinds of changes. Newton transformed that, asserting that explanations of motion had to be in terms of the forces that changed the motion of a mass. The three basic terms in the ontology of motion were now force, mass (what resisted force), and acceleration (change in motion). And each of these was quantitative, measurable.

  Isaac Newton (1642–1727)

  The Principia, the most revolutionary single publication in science, much like Euclid’s Elements, lays out its contents as if they were deductions from self-evident axioms. It contains three Books (the first two called ‘The Motion of Bodies’, the third ‘The System of the World’) preceded by a Preface, eight Definitions, and a set of ‘Axioms, or Laws of Motion.’ In the Definitions, we see Newton, sometimes clumsily, developing the components that would be included in F = ma – especially the idea of force – from out of the ideas of his predecessors. Definition One is of mass, or quantity of matter; Definition Two of quantity of motion. Of the following Definitions, of force of various kinds, some court ambiguity and even confusion: Definition Three, for instance, is supposedly about ‘inherent force’, but is actually about what we call inertia, describing it in terms of impulse. This definition is best viewed, one commentator says, as ‘a concession to pre-Galilean mechanics.’17 Definition Four is the key one, and defines ‘impressed force’ in more modern terms as ‘an action exerted on a body to change its state either of resting or of moving uniformly straight forward.’ Newton thereby generalized the notion of force developed by his precursors, extending it from instantaneous forces to continuous forces, which was a product of Newton’s intuition, science historian I. Bernard Cohen notes, because it was ‘a step which he never justified by rigorous logic or by experiment.’18 And as Newton writes later, ‘this concept is purely mathematical, for I am not now considering the physical causes and sites of forces.’19 The remaining four definitions are of what amount to other aspects of force. In a commentary, Newton warns that ‘although time, space, place, and motion are very familiar to everyone’, and have a popular meaning that arises from sense perception, he will give them a technical meaning, and proceeds to describe what he calls ‘absolute time’, whose flow is unchanging, and ‘absolute space’, which remains homogeneous and immovable.’ Newton then distinguishes between absolute and relative motion. ‘Absolute motion is the change of position of a body from one absolute place to another; relative motion is change of position from one relative place to another.’20 Thus the motion of a sailor on a ship is the sum of three motions: his motion relative to the ship, the ship’s motion relative to the earth, and the earth’s motion relative to absolute space.

  After the Definitions come the ‘Axioms, or the Laws of Motion.’ The first law of motion is of inertia: ‘Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.’ This has been described as ‘the great factor which in the seventeenth century helped to drive the spirits out of the world and opened the way to a universe that ran like a piece of clockwork.’21 Then comes Newton’s second law of motion: ‘A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.’ Newton thus did not set this down in the form of an equation. Modern notation would express this as F ∝ Δ(mv) – or, since change in velocity is acceleration, F = ma. The first person to set this down as an equation was Leonhard Euler, almost a century later. Newton’s third law of motion: ‘To any action there is always an opposite and equal reaction.’

  Why does Newton call these axioms or laws? The word ‘axioms’ suggests that what follows is true a priori (by definition or prior to experience), while ‘laws’ suggests that what follows is an empirical fact, something discovered in each actual experience. In fact, Newton’s axiom-laws are both: they are true by definition of all possible motions, and empirical truths of laboratory measurements done under select conditions. These axiom-laws thus sketch out the meaning of objective knowledge, or what does not change despite different conditions. An object such as a baseball has a certain weight, but its weight is different on the moon and on Mars. How can this be? It’s the same ball! What is objective, unchanging, about it is not its weight but its mass, which is independent of what gravity it happens to be in. Similarly, when you throw that baseball you can get it moving at a certain velocity, but you can’t get a cannonball moving nearly as much even if you throw it just as hard. Physics has not changed: the relationship among force, mass, and acceleration has remained absolutely the same in this and in all other circumstances. Newton’s second law of motion states just this ‘invariance’, as scientists now say. This is why Cohen says that ‘the Second Law of Motion in its manifold applications lies at the very heart of the Newtonian system of physical thought.’22

  In the Principia, Newton has in effect expanded Galileo’s thought experiment of an infinite plane without resistances, and produced a complete and abstract world-stage. Only forces, motions, and masses appear on it. There are no human purposes, no final causes. But because of this, certain aspects of motion appear and show themselves on this stage more clearly than in the messy human world. Aristotle had not found it necessary or even possible to imagine such a stage; he saw motion thoroughly woven into the world and unable to be understood apart from it. (In his commentary on the third law of motion, Newton describes a horse pulling a stone tied to a rope, and sets about explaining the forces involved. Imagine how puzzled this would have made Aristotle! He would have asked: Who tied the horse to the stone? Why? What purpose is being actualized here?) Galileo only glimpsed this stage from afar. Newton’s stage is spare, but therein lies its beauty and effectiveness. Nature is not to be looked at, as Aristotle had, as a cosmic ecosystem, in which qualitatively different kinds of things act in qualitatively different kinds of ways in qualitatively different domains. It is more like a cosmic billiard table, in which all space is alike, all directions are comparable, all events are motions, and in all changes of motion the same basic kinds of things exert the same basic kinds of forces. In this world, movement involves change in space, not attainment, actualization, or intensification of being. It’s a world where all chandeliers, trapezes, and swings are pendulums, all sport and dance instances of F = ma, all balls elastic, and all planes go on forever. One can move about this stage anywhere in space and over time – in a car, train, plane, roller coaster, bicycle – and the laws remain the same, ‘invariant under translation’, we would say. If you want to understand what’s happening on this world-stage, according to Newton, here’s what to do: First, quantify positions, speeds, and masses. Then follow the forces.23

  This is how F = ma can be both a definition and an empirically discoverable fact. It is a definition insofar as it is part of the warp and woof of the abstract world-stage. It is a fact insofar as, when connected with our world via the right concepts, assumptions, and measurement techniques, it states a quantitative relationship between values found in a laboratory situation. Theories become the vehicles by which to go back and forth between the world-stage and our own, between its ideal values and the real values of our world. To nonscientists, the abstract world might seem strange, something arbitrary and imposed upon nature, a world of fiction – an effective fiction, perhaps, but an invention nonetheless. To scientists, who have been trained to connect this abstract world with our own through concepts, procedures, and measurement practices, the temptation is just the opposite. They can move so confidently back and forth that they can forget how abstract this world-stage is – as if it were not an
invented part of the world they live in.

  At this final stop, where F = ma appears, therefore, we have traveled quite a distance from the beginning of the human understanding of motion. We see the heavens and the earth as the same place. We do not see an inherent distinction between natural and violent motion, or between natural and unnatural places, or between different kinds of forces. All things follow the same laws, and if we think that something behaves strangely, we assume that we do not yet see how the laws we know apply to it. Motion is not an action but a state. There is one kind of motion, and circular motions are to be explained as the result of combinations of components. Aristotle may be right that motions in the heavens seem different from motions down here, but that’s because the resistances in the heavens are different. Nature is a huge space-time determination of forces, accelerations, and motions whose blueprint we can seek through theories. This allows us to view nature as full of quantitative laws to be found through experimentation.

  ‘[Newton] is our Columbus’, Voltaire wrote in 1732, ‘he led us to a new world.’24 But it is a strange world. It is not found in our own like a concealed continent. Nor is it revealed by instruments, the way tiny worlds are seen in microscopes, or distant and gigantic ones in telescopes. Newton’s strange new world was found in our world – but it is not our world, either, nor one we could live in. We humans, even the scientists among us, inhabit what philosophers call the ‘lived world’, amid designs, desires, and purposes: we live in an Aristotelian world. The world Newton discovered is an abstract one that appears by changing what we look at and how we look at it. It’s a fishbowl-like world, seen from the outside, closed off from our own, whose events disclose to us much about our world. The equation F = ma is the ‘soul’ of that world, as Wilczek wrote, serving to define its structure, and part and parcel of every event that takes place in it.

  This is why F = ma is not as straightforward as it appears. When we learn it we are learning more than we think. We are inheriting the entire journey that led up to it.

  Interlude

  THE BOOK OF NATURE

  Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.

  – Galileo, The Assayer

  In 1623, Galileo crafted a famous image that is still often cited by scientists. Nature, he wrote, is a book written in ‘the language of mathematics’, and if we cannot understand that language, we are doomed to wander about as if ‘in a dark labyrinth.’

  Like other metaphors, this one is both true and untrue; it is insightful but it may be misleading if taken literally. It captures our sense that nature’s truths are somehow imposed on us – that they are already imprinted in the world – and underlines the key role played by mathematics in expressing those truths. But Galileo devised the image for a specific purpose. Taken out of its historical context and placed in ours, the image can be dangerously deceptive.

  The idea of a book of nature did not begin with Galileo. For centuries it had been an accepted part of religious doctrine that the world contained two fundamental books. Nature, the first book, is full of signs that reveal a deeper meaning when interpreted according to scripture, the second book, which supplies the ultimate meaning or syntax of nature’s signs. Understanding involved reading the books together, going back and forth between what one finds in the world and what one reads in scripture. As Peter Harrison has pointed out in his book The Bible, Protestantism, and the Rise of Natural Science, reading the Bible was once considered part and parcel of studying nature, and it is therefore wrong to equate serious Bible reading with literalism and antiscientific behaviour as we often do today.

  During the Renaissance, however, scholars came to appreciate more keenly that the truths of nature were not always easy to discern. Rather, such truths were often cleverly encoded in nature and so required a special training to unlock. Meanwhile, the Protestant Reformation brought about changes in the understanding of texts, emphasizing the truths in them that were exact and self-contained rather than symbolic or allegorical.

  Galileo, building on these scientific and religious changes, then appropriated the ‘two books’ image for his own purposes, transforming its meaning.

  For in 1623, Galileo was in a jam. His troubles had begun 10 years earlier, when a student of his had discussed his work at the Pisan court, and a participant noted the apparent conflict between scripture and Galileo’s scientific claims, especially regarding the motion of the earth. Meanwhile, the authorities were threatening to put De Revolutionibus, by his intellectual ally Copernicus, on the official index of forbidden books for similar reasons. Worried for himself and for other scientists, Galileo wrote a letter to the Grand Duchess Christina about the connection between science and scripture. In that letter, he appealed to the traditional image that God reveals himself to humanity in two books, nature and scripture. He suggested that both books express eternal truths and are compatible because they have the same Author – God is saying the same thing in two different ways.

  Yet Galileo’s novel crafting of the image would prove explosive. Galileo had insisted that the book of nature was not written in ordinary words; its characters were fundamentally different from the words of the scriptures, of Aristotle, and of any textual author. ‘It is necessary for the Bible’, Galileo said, though he might as well have said it of the books of Aristotle, of the Church Fathers, or of any author, ‘in order to be accommodated to the understanding of every man, to speak many things which appear to differ from the absolute truth so far as the bare meaning of the words is concerned. But Nature, on the other hand, is inexorable and immutable; she never transgresses the laws imposed upon her, or cares a whit whether her abstruse reasons and methods of operation are understandable to men.’1

  Galileo’s arguments seem to have convinced Christina, but not the authorities. In 1616, De Revolutionibus was put on the Index, followed by Kepler’s textbook on Copernican astronomy, Epitome, in 1619, and Galileo himself came under attack. Partly in response he wrote The Assayer, containing the famous passage that ‘the grand book of the universe...cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed...the language of mathematics.’ Those versed in mathematics and physics, in other words, can know aspects of God’s handiwork that others cannot.

  Galileo chose his image carefully, and its roots were deep in Western metaphysics and theology. First, it used the traditional idea that God revealed his power, glory, and truth in the world. Second, it relied on the equally traditional notion that the Bible cannot go against clear demonstrations of logic or the senses. Finally, it appealed to the time-honored metaphor of nature as a book. Galileo was on solid theological ground.

  In fact, however, Galileo – perhaps without his being fully aware of it – had stood the old image on its head. The image of the book of nature now implied something almost opposite what it had before – that the signs of nature had their own self-contained meaning. To understand nature one did not need to rely on the Bible as an allegorical aid; studying nature was an independent activity best carried out by a separate, professional class of scholars. If anything, the book of nature now became the primary text – the blueprint written in technical language – and scripture the user’s manual, written in popular language.

  Galileo thereby used the image to defend not only himself but also all scientists, suggesting that they were as authoritative as the clergy. ‘The book of nature and those natural philosophers who interpreted it...assumed part of the role previously played by the sacraments and the ordained priesthood’, writes Harrison.

  But the image of the book of
nature can haunt us today. One reason is that it implies the existence of an ultimate coherent truth – a complete text or ‘final theory.’ While many scientists may believe this, it is ultimately only a belief, and it is far likelier that we will endlessly find more in nature as our concepts and technology continue to evolve. Furthermore, the image suggests that the ‘text’ of the book of nature has a divine origin. The idea that the world was the oeuvre of a superhuman author was the precursor of the idea that it was the engineering project of an intelligent designer. This implication has led some contemporary sociologists of science to succumb to the temptation of characterizing scientists as behaving, and seeking to behave, in a priestlike manner.

  The most important lesson to be found in Galileo’s image is the need to keep developing and revising the metaphors with which we speak about science.

  3

  ‘The High Point of the Scientific Revolution’:

  NEWTON’S LAW OF UNIVERSAL GRAVITATION

  DESCRIPTION: Gravity exists in all bodies universally, and its strength between two bodies depends on their masses and inversely as the square of the distance between their centres.

  DISCOVERER: Isaac Newton

  DATE: 1684–87

  The high point of the Scientific Revolution was Isaac Newton’s discovery of the law of universal gravitation. All objects attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of their separation. By subsuming under a single mathematical law the chief physical phenomena of the observable universe Newton demonstrated that terrestrial physics and celestial physics are one and the same.