Section II, Chapter Five: The Culmination
At long last, humanity had a physical theory that could explain the motion of stars and rocks alike using a single mathematical and conceptual framework.
Note—this is a draft of Chapter Eight of a book I’m writing. The book is going to cover humanity’s deepest ideas from philosophy, physics, epistemology, and economics. Each chapter is meant to be short and digestible. Most of the chapters will explain just one or two ideas.
‘Chapter []’ indicates a future chapter that I’ve not yet written.
“The discovery of the laws of dynamics, or the laws of motion, was a dramatic moment in the history of science. Before Newton’s time, the motions of things like planets were a mystery, but after Newton there was complete understanding. Eve slight deviations from Kepler’s laws, due to the perturbations of the planets, were computable. The motions of pendulums, oscillators…could all be analyzed completely after Newton’s laws were enunciated.”
-Richard Feynman, The Feynman Lectures on Physics, Volume I
Principia Mathematica marked the culmination of the Scientific Revolution that began with Copernicus’ 1543 book, On the Revolutions of the Heavenly Spheres, in which he overturned the Ptolemaic model of the solar system with his heliocentric model (see Chapter Four). As we’ve seen, the fourteen centuries between Copernicus’ and Newton’s books witnessed not just improvements in our scientific understanding of the universe, but also refinements in how to reason more broadly (see Chapters Four and Five): new modes of criticism, rejection of arguments by authority, experimentation, rigorous mathematics, and abstracting away irrelevant details gradually fixed themselves in intellectuals’ toolkit as they investigated the nature of reality. Newton’s theory of classical mechanics not only built on the scientific work of his predecessors, but he took full advantage of the aforementioned tools of reason that his predecessors had developed since the middle of the sixteenth century.
In Principia Mathematica, Isaac Newton laid down his famous three laws of dynamical motion, as well as his law of universal gravitation. At long last, humanity had a physical theory that could explain the motion of stars and rocks alike using a single mathematical and conceptual framework—the physics of Copernicus’ solar system and Galileo’s inclined plane were one and the same. Moreover, and crucially, Newton’s theory of classical mechanics was testable, universal for all physical systems, and hard to vary (see below). That an idea with such robust characteristics was eagerly accepted by the broader culture meant that science as we now understand it was here to stay.
We’ll review classical mechanics and its implications in more detail in subsequent chapters, but here we will simply sketch out its primary laws in broad strokes.*
Newton’s First Law: “Each body perseveres in its state of stillness or uniform rectilinear motion unless it is forced to change that state by forces applied to it.”
In Chapter Six, we saw that Galileo came close to this law, but he failed to identify the agent that could change a body’s (read: physical system’s) uniform motion—force. That is, bodies move in straight lines at constant speeds unless acted on by an external force.
Newton’s Second Law: “The change of motion is proportional to the applied driving force, and occurs along the straight line with respect to which the force is itself exerted.”
The change in an object’s motion, its acceleration, is proportional to the force acting on it. In algebraic terms, Newton’s Second Law is written as F = m*a, where F is the external force, m is the mass of the object (also called the inertial mass–see Chapter []), and a is the acceleration caused by the force F.
Newton’s Third Law: “To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.”
When object A exerts a force F on object B, then object B necessarily exerts a force -F on object A. F and -F are equal in magnitude but opposite in direction (see Chapter []).
Newton’s Law of Universal Gravitation: F = G*m*M / r2, where F is the attractive force between two objects, G is a fundamental constant of Nature, m and M are the masses of each object, and r is the distance between the objects.
According to Newton, all massive (here, massive just means ‘having mass’) objects exert an attractive force on all other massive objects. For example, consider the gravitational force between two arbitrary stars. Its magnitude increases as the stars’ masses increase but decreases as the distance between the stars increases (see Chapter []).
What makes classical mechanics testable? Any of the above four laws can be (and has been) tested, but consider the Second Law as an example. If F = m*a for any object, then, generically, if we know two of the three variables in the equation, then Newton’s Second Law predicts what the third, unknown variable must be (force is measured in units called ‘Newtons’, mass in kilograms, and acceleration in meters per second squared). For instance, if we know that the force acting on object A is 10 Newtons and the resultant acceleration is 5 meters per second squared, then Newton’s Second Law predicts that the object has a mass of 2 kilograms.
While Newton’s Laws are ‘directly’ testable in this way, their consequences can also be checked against reality. For instance, one may use Newton’s laws to predict an object’s velocity and position at an arbitrary time t, provided one knows the object’s velocity and position at an earlier time t0, the forces acting on it from time t0 to time t, as well as the object’s mass (see Chapter []).
Another (important) example is predicting acceleration due to gravity. Consider a rock of mass m falling towards the surface of the Earth (whose mass we label M). Newton’s Law of Universal Gravitation tells us that Earth’s gravity pulls the rock downwards with a force whose magnitude we can calculate. And Newton’s Second Law tells us that this force is proportional to the acceleration that the rock undergoes (ignoring air resistance and other complicating factors) on its descent. We may combine the equations of these two Laws to deduce that a = G*M / r2 (we’re also making the reasonable approximation that r, the distance between the falling rock and the Earth’s center of mass, is a constant). Notice that the rock’s mass does not appear in the equation. So, Newton’s Laws not only allow us to quantitatively predict, say, the acceleration due to Earth’s gravity, but it also tells us that this value will be the same for all objects (provided m is much less than M, which is a fine assumption for most of the objects near the Earth’s surface).
In what sense is classical mechanics hard to vary? In The Beginning of Infinity, physicist David Deutsch writes, “[G]ood explanations…are hard to vary in the sense that changing the details would ruin the explanation.” An explanation is hard to vary if “all its details play a functional role.” If you replace even one conceptual or mathematical element of classical mechanics, the entire explanation loses its coherence. For instance, if you replace acceleration with velocity in Newton’s Second Law, then Newton’s First Law wouldn’t work, either, since then objects ought to slow to a halt in the absence of external forces. One can play with the elements of the theory in this way, permuting them as one wishes, only to find that any permutation would render other parts of the theory problematic (to say nothing of the disintegration of the theory’s predictive powers). Newton’s ‘version’ of the theory as he presented it is coherent, and delicately so—it is hard to vary whilst retaining its ability to explain (and accurately predict) the dynamics of massive objects.
Finally, classical mechanics is universal in the sense that it explains the dynamics of all massive objects (it turns out that this isn’t quite right, as classical mechanics is only a limiting case of yet deeper theories—see Chapters [] and []). As philosopher Dennis Hackethal writes in A Window on Infinity, “When [an explanation] solves all problems in a single domain–or at least can do so–it has universal reach within that domain. That is universality.” Prior to classical mechanics, physicists conjectured more fragmented explanations of the motion of the stars and planets on the one hand and that of terrestrial projectiles on the other. Newton’s explanation unified both realms, allowing us to solve any problem whose solution requires solely understanding the dynamics of massive objects.
*English translations of Newton’s original Latin were taken from The Fundamentals of Newtonian Mechanics, by Maurizio Spurio.