Copernicus

The Copernican heliocentric system

The great breakthrough that ultimately led to modern astronomy and cosmology came with Copernicus’ heliocentric system for describing the heavenly motions. But practically nothing one can say about Copernicus, either about his system or about its origins, is simple.

For one thing, the Copernican Universe is not the modern one, but the ancient cosmology in many of its most important respects. It is still a finite world bounded by a sphere of fixed stars. For another, Copernicus was not started on his path to his great discoveries by seeking a system that could use the Earth's motion to provide a more unified account of the apparent motions of the Sun and planets. Rather, he was dismayed by the appearance of the device of the equant point in Ptolemy's system and sought a means of eliminating it from an account of the heavens. Copernicus was even more devoted than the Greeks to the view that all the heavenly motions must be described in terms of pure circular motions in which everything moved uniformly about the circle with respect to the actual central point of the circle. Of course, hierarchies of circles, for example epicycles whose centers moved uniformly on deferent circles, were permissible.

The growth of theories

A very naïve view of science might go something like this: Scientists encounter a range of observable phenomena for which they have no explanatory account. Hypotheses are generated from the imaginations of the scientists who seek to explain the phenomena in question. These hypotheses are tested against the experimental results. If they fail to successfully account for those results, the hypotheses are rejected as unsatisfactory. But if they succeed in predicting and explaining that which is observed, they are accepted into the corpus of scientific belief. Then scientific attention is turned to some new domain of, as yet, unexplained phenomena.

This simple-minded picture of science has been challenged for a variety of reasons. Some are skeptical regarding the possibility of characterizing theory-independent realms of observational data against which hypotheses are to be tested. Others have noted the way in which the testing of hypothesis by data is a subtle matter indeed. It has often been noted, for example, that even our best, most widely accepted fundamental theories often survive despite the existence of “anomalies,” observational results that are seemingly incompatible with the predictions of the theories.

The background to the Principia

In Newton's great work dynamics is presented as being derivable in a systematic way from a small number of fundamental first principles. In this Newton resembles Descartes. But, unlike Descartes, the first principles are not alleged to be derivable a priori from “clear and distinct ideas.” They are, rather, painstakingly inferred from the known lower-level generalizations, themselves inferred from observation and experiment, as the best basic principles from which the known phenomena can be derived. And, very much unlike Descartes, and very much in the tradition of Galileo and Huyghens, Newton is extraordinary in his ability to apply the methods of mathematics to the description of particular dynamical situations so that detailed and exact characterizations of the situation can be formulated, and precise predictions made.

Newton's Principia appeared only after many years of the careful exploration of dynamics and its applicability to a theory of the world by its author. Most of this earlier work remained unpublished. There seem to be several reasons for this, including many forays by Newton into other fields, such as the invention of the calculus, brilliant experiments on light, including the discovery of the dispersion of white light into colors and basic interference phenomena, and such matters as the invention of the reflecting telescope. Newton's sensitivity to what he took to be inappropriate criticism of early public work may have also contributed to a reluctance on his part to publish.

Some Greek knowledge and speculation

The motion of the heavenly bodies, observed as lights in the sky, provides us with a remarkable spectacle of a phenomenon describable in a small number of terms and exhibiting an easily noted regularity in space and time. This spectacle caught the attention of many cultures in the beginnings of their attempts to characterize the world as a place of some describable order. Most of the cultures were unable to get beyond the ability to discover numerical formulae that allowed one to predict recurrences in the domain of the heavens, sometimes with astonishing accuracy. In ancient Greece, however, astronomy took science further. In particular, Greek astronomy involved deep connections with Greek attempts to construct a general dynamical theory of motion and its causes. This close connection between dynamics and astronomy persisted throughout the history of classical dynamics, as we shall see. It is necessary for us, therefore, to say a little bit about some of the aspects of Greek astronomy that impinged upon Greek theories of motion.

By the time of the Greek classical era, many important facts were well known and widely agreed to. Whereas early Greek speculation about the shape of the Earth thought of it as flat, perhaps a disk of land surrounded by a circumventing ocean, it was soon an accepted fact that the Earth had the form of a sphere. Observations on how the elevations of stars changed as one moved north or south, how ships disappeared a little at a time over the horizon and how the length of a day varied with latitude could be explained only by invoking such a shape for the Earth. By analogy, models of the Moon and Sun as illuminated disks were soon replaced by accounts of these heavenly bodies as also spherical. (It is remarkable how spherical the Moon looks, in fact, when seen during a total eclipse.)

From special problems and ad-hoc methods to general theory

Our outline of the development of mechanics has proceeded as if we could tell a story with a single line of development. But our account up to now has been, as we shall see, somewhat misleading. Even prior to the great Newtonian synthesis, other approaches to the solution of the problems of dynamics were simultaneously being explored. Most of these approaches remained fragmentary and partial until the eighteenth century. For that reason we have neglected them, reserving discussion of them until the more extended discussion of how those programs became solidified, generalized and systematized in the later history of dynamics.

At this point, however, it becomes impossible to deal with matters in a strictly chronological manner. In the years following the publication of Newton's Principia dynamics followed a number of distinct, although deeply related, patterns of development. It will be essential to deal with each of these in turn, forcing us to go over the same temporal period from several perspectives. This chapter is preliminary to those that follow. In it I will try to lay out something of the problem situation facing the great developers of mechanics and outline the basic structure of the multiple approaches suggested to deal with that array of problems. We may then proceed to explore the several approaches in detail one at a time.

From least time to least action

The third developmental stream of dynamics that came to fruition in the eighteenth century is quite unlike the first two streams in significant ways. It did not arise out of repeated attempts at solving particular difficult special cases of dynamical problems. Nor, when it was discovered, was its primary importance its ability to provide new methods for solving such problems. Its importance for the discipline was, rather, more of a “theoretical” kind, providing new and deep insights into the fundamental structure of the theory. Whereas the other two streams of development carried with them philosophical issues already present in the standard controversies over the mode of explanation in dynamics familiar to Newton and his critics, this third developmental stream opened up entirely new controversial issues concerning what could count as a legitimate explanation in science. It was also curious that these developments in dynamics had their origins not in contemplation of mechanical issues, but, rather, in explanatory accounts offered in the theory of light, in optics, dating back to ancient Greece.

The Greek mathematicians had become aware of “minimization” problems quite early. For example, it was well known that the circle was the shortest curve bounding an area of specified size. Indeed, this is one of the “perfections” of the circle taken by Aristotle to account for the cosmic orbits. The use of a minimization principle by the ancient Greeks that eventually fed into the important role such principles play in physical explanations, though, was the proof by Hero that the law of reflection, namely the angle of incidence of the light on the surface of the mirror equaling the angle of reflection, could be derived by postulating that the distance taken by a light beam to go from a point to a point on the mirror and then to a third point of reception was least when the mirror point was such that the distance traveled by the light along the path was less than that which would be traveled for any other point on the mirror taken as intermediary. The proof is from plane geometry and is very simple.

Newton's masterful achievement was constructed under the influence of much previous philosophical discussion and controversy that went beyond the limits of scientific debate narrowly construed. Much that Newton says in the Principia also ranges beyond the confines of experimental, or even theoretical, science and passes into the realm of what we usually think of as philosophy. Newton's work gave rise, possibly more than any other work of science past or future, excepting just possibly the work of Darwin and Einstein, to vigorous philosophical as well as scientific discussion. Let us look at some of the philosophical issues behind, within and ensuing from Newton's work.

It is convenient to group the discussions into three broad categories. First, there is the “metaphysical” debate over the nature of space, time and motion. Next there is the debate over what can be properly construed as a scientific explanation of some phenomenon. Lastly, there is the controversy over what the appropriate rules are by which scientific hypotheses are to be credited with having reasonable warrant for our belief. We will discuss these three broad topics in turn.

Our concern in this book is with dynamics, namely the science of motion, its description and its causes. But mechanics traditionally had two branches, dynamics and statics, the latter being the science of the unmoving, that is of how forces can jointly result in unchanging, equilibrium, states of systems. Although we shall deal with statics only in a brisk and cursory manner, we must pay some attention to its historical development, since principles developed in statics played a fundamental role in the foundations of some important approaches to general dynamical theories. Before moving on to the development of dynamics beyond Newton's great synthesis, then, we shall have to spend at least a little time surveying some aspects of the development of statics prior to the eighteenth century.

Two basic areas of investigation constituted the initial exploration of statics in ancient Greece. The major set of problems that gave rise to statics consisted in attempts at describing the general laws governing the equilibrium conditions for simple machines. In particular, the lever and the inclined plane were the characteristic problems tackled. A second branch of statics began with considerations of the static behavior of objects immersed in fluids, consideration of which constituted the first efforts at understanding the statics of fluids, hydrostatics. It is the former problem area, though, that is of most interest to us.

Throughout this book so far our attention has been firmly directed to issues arising out of consideration of the fundamental postulates of dynamical theory. What should those posits be? How are the alternative choices for foundational axioms related to one another? What are the basic concepts utilized by those posits and how are they to be interpreted? What is the epistemic status of the basic postulates? And, finally, what kind of a world, metaphysically speaking, do the posits demand?

In this chapter, however, we will look not at the fundamental posits but, instead, at how the consequences of these posits are developed in particular applications of dynamical theory. We have generally been avoiding issues of application. But some of these realms of application are quite fascinating from an historical point of view, for it is often an extremely difficult task for the scientific community to figure out how the elegant basic posits of the theory are actually to be used in describing the complex systems we find in nature. Great efforts in classical dynamics were devoted to such issues as the appropriate frameworks for applying the theory to the motion of rigid bodies, to fluids of ever more complex nature, going from the non-viscous to the viscous and from the incompressible to the compressible, and to more complex continuous media. Difficult special cases, such as those of shock waves, where familiar continuity assumptions cannot be maintained, also led to the development of subtle and sophisticated application programs for the dynamical theory.

The variety of explanatory modes in dynamics

We have seen that dynamics is a theory with a multiplicity of explanatory structures that can be used to formulate its lawlike conclusions and to provide explanations of the behavior of systems within its purview. We have also seen that the threads of some of these structures can be traced back to the earliest days of the development of the theory.

One pattern of explanation in dynamics we might call the “Newtonian.” Here one must first posit an appropriate structure that admits a preferred metric of time and the existence of the preferred inertial reference frames to which all motion is to be referred. Inertial motion with constant speed and direction is taken as the “natural,” “unforced,” state of a body.

Inertial mass and force are introduced. The former is an intrinsic property of a piece of matter representing its resistance to having its state of motion changed, and the latter is the (vector) measure of the influences that can change the state of motion of a system. The fundamental law, of course, is the proportionality of the linear momentum change of the system to the force applied to it. This initial Newtonian framework must be supplemented, as was first seen by Euler, by a corresponding notion of moment of inertia as intrinsic resistance to change of state of rotation and of moments of forces (or torques) as the measure of the influences generating changes of angular momentum.

The theory this book is exploring, Newtonian dynamics, is, of course, a false theory of the world. Several scientific revolutions have shown us that it must be replaced as our fundamental theory of motion and its causes.

The special theory of relativity rejects even the basic kinematics of the Newtonian account. Even at the level of the very description of motion, at the level of our construal of the spatial and temporal structure of the world, the Newtonian account is rejected. At the level of dynamics, the level at which the causes of changes of motion are explored, the theory, once again, replaces the Newtonian account with a novel theoretical structure. The general theory of relativity proposes even further, highly dramatic, changes in our very notions of space and time. Its most direct contact with the Newtonian picture of the world is to replace Newton's famous account of the origin, nature and effects of gravitational force with a new account of the gravitational interaction of the material of the world.

Many threads were finally woven together in the great Newtonian synthesis from which all further developments in classical dynamics followed. In this chapter we will outline a few of the major contributions to dynamics that followed the Copernican revolution in astronomy, but preceded the final accomplishment of a full theory that we can recognize as classical dynamics. Three names dominate this early work on dynamics, those of Galileo, Descartes and Huyghens, but, as we shall see, important contributions were made by less well-known figures as well.

Galileo

Almost everyone would give Galileo credit for initiating the great revolution in dynamical theory that ultimately led to Newtonian classical dynamics. But, as is the case in the work of many originators of a new science, there is no simple way of characterizing Galileo's contribution. His work, while being innovative in a revolutionary way, retains much that in retrospect seems quite conservative in its nature, borrowing much in the way of concepts, views and arguments from his predecessors. His exposition is sometimes quite informal, presented in the form of a charming dialogue in some cases. This sometimes makes it rather difficult to say exactly what Galileo believed, since, in some crucial cases, he seems to hold a number of distinct opinions simultaneously. In some crucial cases we would now be inclined to think that Galileo got things very wrong indeed. But even in his errors he provides deep enlightenment.

The realization that a branch of knowledge could be presented in a form in which the entire contents of the field of investigation could be expressed by positing a small number of basic truths and by claiming that all the other truths of the discipline followed from these basic posits by pure deductive reasoning alone predated any serious development of dynamics or of other branches of physical science. The axiomatization of geometry has its origin at such an early date, in fact, that we have no good record of when or how the very idea of presenting geometry as a deductive formal discipline arose.

This early discovery of a branch of mathematics as a formal science had many consequences for the history of science and the history of philosophy of science. The entire history of the rationalist approach to knowledge in philosophy is founded on the early discovery that geometry could be structured as a set of consequences logically deducible from apparently “self-evident” first principles. Closer to our concerns, it is clear that Newton's Principia is itself structured to resemble as closely as possible the standard presentation of geometry. But there is, of course, no pretence on Newton's part that his first principles could themselves be established without reference to empirical experiment. (It was left to Kant to fall into the trap of trying to establish Newtonian dynamics as a fully a-priori science!)

The nineteenth-century reconstructions of dynamics we have just surveyed were certainly not motivated by philosophical reflections on the traditional foundations of the theory. Nor were they formulated in response to any felt disquietude about the traditional theory. Hamilton's equations arise out of an understanding, developing at the time, that a single second-degree differential equation could be replaced by a pair of coupled first-degree equations. Hamilton–Jacobi theory has a less purely mathematical motivation, in that it followed from a deep understanding of the degree to which the formalisms of geometrical optics and of dynamics bore interesting parallels to one another. Here the primary inspiration comes from the understanding that the Principle of Least Time in the former theory and the Principle of Least Action in the latter were sufficiently similar that other formal similarities, such as advancing wave fronts and trajectories as rays orthogonal to these, might be found as well. The bracket formulation of dynamics is, once again, a purely formal manipulation of the theory.

Each of these reconstructions, as we have seen, has manifold consequences. They do play some role in extending the ability of the theory to be applied to difficult special cases. But they also provide deeper understandings of the hidden internal structures of the theory, as in, for example, the realization of the fascinating algebraic structure among the generalized variables revealed by the bracket notation or the “wave-front” structure in configuration space revealed by Hamilton–Jacobi theory. And they also provide just the resources that will later be needed in going beyond classical dynamics to newer theories, as in the application of Hamilton–Jacobi theory to the foundations of the wave-theory version of quantum mechanics and the application of the bracket notations in the formalizing of the matrix version of quantum theory.