THE SCIENTIFIC REVOLUTION
THE SCIENTIFIC METHOD
The Scientific Revolution was characterized by three great achievements. First, there was the shattering of the Aristotelian synthesis and a decisive break with its philosophical, speculative, and qualitative approach to doing science (even though science continued to be referred to as “natural philosophy”). Second, there was the realization of the importance of doing experiments and making precise measurements. This involved the growing use of artificial devices such as telescopes, pendulums, and vacuum pumps in investigating nature. (This second achievement also required a conceptual advance; in Aristotelian thought, machines had been regarded as causing things to move contrary to their “natural” motion.) And, finally, there was the mathematization of science. Of course, science had previously employed mathematics and even experiment, but it was in the 1600s that these tools came together to create a powerful new way of investigating the world, often called “the scientific method.”
The scientific method comprised (a) the collection of data by precise measurements and controlled, repeatable experiments, (b) the formulation of testable hypotheses to explain either regularities or anomalies in the data, and (c) the verification or falsification of those hypotheses by comparing their predictions with the results of new measurements or experiments. Unfortunately, the scientific method is sometimes spoken of as if it were an automatic or mechanical process. It is not a process, but an activity. Processes can be undertaken by machines. Activities require the imagination, insight, cleverness, initiative, creativity, and judgment of persons.
It is not sufficiently stressed in scientific education, where the main object is to master an enormous body of fact and theory, just how marvelously ingenious experiments and observations can be. Great experiments, like great theoretical ideas, are things of beauty. However, they tend to be more ephemeral. They are the scaffolding in the construction of the edifice of science and are too often forgotten after they have played their part.
Many also have an unrealistic understanding of scientific theorizing. Students are usually presented with a theory as a finished product, with all obscurities removed, essential ideas crystallized into precise concepts, fundamental principles identified, and logical structure clarified. This is indeed the best way for students to master the ideas. However, it can also lead them to lose sight of the messy, confusing, and arduous struggle by which the key insights were originally won.
A strong case can be made that a lack of appreciation of the actual methods of science has had harmful effects on philosophy. The two great contrary movements in European philosophy in the seventeenth and eighteenth centuries—rationalism and empiricism—were both to some extent inspired by the progress of science and considered themselves scientific in spirit. However, each exaggerated one pole of scientific thought at the expense of the other. The rationalists tended to think of all knowledge as advancing by a process of deductive reasoning from first principles, as in mathematics. They undervalued the empirical component of scientific progress. On the other hand, the empiricists underappreciated the role of hypothesis, abstraction, and theoretical construction in advancing scientific knowledge. Like the rationalists, they may also have been misled by a false analogy with mathematics. In mathematics, each conclusion must be firmly demonstrated before it can be used as the basis for further reasoning. Some have imagined that empirical science works in a similar way. They think that the existence of every entity posited by a theory and the truth of every theoretical proposition must be directly verified before the next step is taken. That is not how things work.
Take, for example, the theory of electromagnetism formulated by James Clerk Maxwell in the mid-nineteenth century. It posits the existence at every point in space and time of a three-component “electric field” and a three-component “magnetic field.” No one has ever directly verified the existence of such entities everywhere in space-time, nor would it be possible to do so. However, it is not necessary. That is not how the validity of Maxwell’s theory was established in the first place or why physicists still believe it to be true. Maxwell’s theory, like any sophisticated scientific theory, is a highly elaborate and abstract structure that presupposes the existence of many things, not all of which can be directly or separately observed. Rather, it is the theory as a whole that is verified on the basis of observations; and these observations, however numerous they may be, are necessarily few compared to the entities that the theory assumes to exist.
The empiricists seemed to think that humans built up a picture of reality by adding together a large number of sensory “impressions,” from which more complex ideas got generated by a process of “association.” However, one does not directly sense magnetic fields (unless one is a monarch butterfly, say). One infers their existence by their often very indirect effects, and even then only with the help of abstract theory. And yet these magnetic fields are as real and as physical as rocks and trees. (The same point is illustrated by the electromagnetic spectrum: we directly sense “visible light” with our eyes, but can only infer the reality of ultraviolet light and radio waves by indirect means. However, this difference is due simply to the characteristics of our sensory organs. Radio waves, ultraviolet light, and visible light are in themselves equally real, differing only in wavelength.) The crude notion of verification that one finds implicitly in the British empiricists of the eighteenth century also afflicted later versions of empiricism, such as the early-twentieth-century philosophical school called “logical positivism.” For the logical positivists, every meaningful statement had to be translatable into statements about sensory impressions.
The relationship between theory and observation in science is complex and dynamic. Theories are built on experiment, and experiments depend for their interpretation upon theory. This “theory-dependence of experiment” is much talked about in recent times and has provided an opening for some “postmodern” thinkers to claim that the scientific method involves a “vicious circle” that somehow vitiates the notion of scientific objectivity. One can see through such sophistry by a simple analogy: maps were made by explorers; and explorers had to make use of existing maps. This “circularity” obviously did not prevent better and better maps from being made, nor does the dynamic interaction of theory and experiment prevent better and better theories of the physical world from being made. Indeed, that is precisely how they must be made; and the recognition of this fact was the fundamental achievement of the Scientific Revolution of the seventeenth century.
Before we turn to the history of that revolution, it is worth saying a bit more about how scientific theories are verified. In mathematics a theorem may be proven rigorously, in which case it can be affirmed with certainty; or it can be disproved, in which case it can be denied with certainty. However, in natural science, as in life, one accepts a theory with a degree of confidence that is, as it were, a continuous variable. In some cases, the confidence may be so great that we can speak of virtual certainty—indeed, of a “scientific fact.” For instance, no scientist seriously doubts anymore that matter is made of atoms or that the earth rotates on its axis. In other cases, the confidence of scientists in a theory may be fairly strong, but not so strong that they would say it has been “confirmed.” In yet other cases, there exists simply what lawyers would call a “rebuttable presumption” in favor of a theory. It all depends on the quantity and type of evidence.
What counts as evidence for a theory? It is not only a matter of quantitative predictions confirmed by experiment. For one thing, a particular experimental number, or even many such numbers, might be accounted for in several ways. For example, one of the major successes of Einstein’s theory of gravity (the so-called general theory of relativity) was that it predicted accurately the “precession of the perihelion of Mercury” (the slow shift over time of the point of closest approach of Mercury to the sun). However, one could have accounted for the perihelion shift in several other ways. One way was to posit a certain amount of solar oblateness (i.e., a flattening at the sun’s poles). Another was to posit a certain density of matter filling the space between Mercury and the sun. Nowadays, enough is known about the sun and its environment that these alternative explanations are no longer viable. But it is possible to find other alternatives; one could simply add, for instance, a new term to Newton’s inverse-square law of gravity to account for the perihelion shift. So the successful prediction of that shift, though vitally important, was not the only reason that physicists began to believe in Einstein’s theory.
Many considerations influence scientists’ judgments about the plausibility or likelihood of a theory. These include the theory’s simplicity and economy, its provision of a more unified and coherent picture of nature, its explanatory power, its mathematical beauty, its grounding in deep principles, its prediction of new phenomena, and its ability to resolve theoretical puzzles or contradictions. Einstein’s theory of gravity, for instance, resolved the problem that Newton’s theory of gravity was not consistent with the principles of the special theory of relativity. It also explained why “inertial mass” is equal to “gravitational mass.” It predicted the phenomenon of the bending of rays of light by gravity. It flowed from the deep “equivalence principle.” It was based on the beautiful idea that gravitation is the result of the curvature of space-time. In other words, Einstein’s theory was supported by many converging lines of evidence and grounds of credibility.
Another good example of how a theory comes to enjoy acceptance is the Dirac equation, invented by P. A. M. Dirac in 1928 to describe electrons in a way consistent with both special relativity and quantum theory. Dirac was led to the equation primarily by considerations of mathematical beauty. But the equation also resolved a puzzle—namely, that the magnetic moment of the electron was twice as big as previous theory had held it ought to be. The Dirac equation predicted a new phenomenon: the existence of anti-particles. And it shed light on a property of electrons called spin. Eventually it was used to make many very precise experimental predictions that were later confirmed. This example illustrates the fact that, while many factors can lead theorists to entertain a certain hypothesis or build their confidence in it, ultimately it is usually a significant number of precise and correct quantitative predictions that “clinches it.” (That, at least, is the case in sciences where controlled and repeatable experiments are possible. However, it is unreasonable to demand the same kinds of confirmation in all fields. For instance, there is a great deal of converging evidence that the evolution of species has taken place, but one cannot predict how particular lineages will evolve. Similarly, one may learn what causes earthquakes without being able to forecast them accurately.)
FROM COPERNICUS TO NEWTON
In a certain sense, one could almost say that Sir Isaac Newton (1643–1727) was the Scientific Revolution. There is much truth in Alexander Pope’s famous couplet:
Nature and Nature’s laws lay hid in night.
God said, “Let Newton be,” and all was light.
Newton was a towering peak. There was no one to rival him in physics until the twentieth century. One may think of everything that went before Newton as having set the stage for his great breakthroughs, and everything that came after him—until the twentieth century—as having exploited those breakthroughs.
Three lines of development led to the achievements of Newton: in astronomy, the discovery of Kepler’s laws of planetary motion; in physics, Galileo’s discovery of the law of falling bodies; and in mathematics, the development of analytic geometry and the use of coordinates by Descartes (1596–1650).
In astronomy, the line that led to Newton began with Copernicus (1473–1543), who sparked the Scientific Revolution with his heliocentric theory of planetary motion. It proceeded through the extremely precise observational work of the great Danish astronomer Tycho Brahe (1546–1601). And it culminated in the discovery by Johannes Kepler (1571–1630) of his three great laws of planetary motion (which would have been impossible without Brahe’s data). Galileo (1564–1642) was not important in this particular line of development—indeed, he firmly rejected Kepler’s crucial idea of elliptical planetary orbits. Rather, Galileo’s great contribution to astronomy was the use of telescopes, by means of which he made a series of dramatic discoveries—such as the phases of Venus, the moons of Jupiter, and sunspots—that helped undermine Aristotelianism and the Ptolemaic system. In astronomical theory, however, it was Copernicus and Kepler, not Galileo, who made the key advances. On the other hand, in physics Galileo made the important breakthrough, when he discovered the law of falling bodies by applying to terrestrial phenomena the powerful combination of experimentation (carried out with inclined planes and pendulums) and mathematics.
COPERNICUS, NICOLAUS (1473–1543) was born in Torun, Poland, and studied astronomy at the University of Cracow. His uncle, the bishop of Ermland, obtained for him a position as canon of the Cathedral of Frauenburg, an administrative office. Copernicus studied civil and canon law at the University of Bologna and medicine at the University of Padua before obtaining a doctorate in canon law from the University of Ferrara in 1503. He thereupon returned to Ermland, where he acted as advisor to his uncle the bishop and took up his duties as canon. He acquired a wide reputation as an astronomer and was visited in 1539 by Georg Joachim Rheticus, professor of mathematics at the University of Wittenberg, who persuaded Copernicus to publish his ideas on heliocentric astronomy. Copernicus finished his epoch-making work De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) shortly before his death, the published copy being presented to him on his deathbed. This is the book that sparked the Scientific Revolution.
In Kepler’s planetary laws and Galileo’s law of falling bodies we have examples of precise mathematical laws that apply to specific systems or to a narrow range of phenomena. Today we would call these “empirical relationships” or “phenomenological laws.” The genius of Newton enabled him to see behind these relationships the operation of laws of much greater generality and depth—namely, the law of universal gravitation and the three universal laws of motion.
Like all great advances in science, the articulation of these laws led to profound unifications. The first unification was of terrestrial and celestial phenomena. Until Newton, the general and deeply ingrained belief was that the heavens and the earth were wholly disparate realms governed by fundamentally different principles and even composed of different kinds of matter. The “crystalline” heavens appeared eternal, untouched by the kinds of change (“generation and corruption”) that characterized the “sublunary” world. It was therefore seen as fitting that the “natural motion” of heavenly bodies should be in perfect circles, for such motion is without beginning or end (as we have seen, even Galileo could not completely free himself from these ancient ideas). What Newton showed, however, is that the very same forces govern both the celestial and the terrestrial realms. The orbits of the planets around the sun, the swinging of pendulums, and the falling of dropped weights all obey the same equations of gravity and mechanics. Indeed, Newton showed that ocean tides could be explained by the gravitational forces exerted by the moon and sun. Interestingly, this had been suggested much earlier by Kepler, but had been ridiculed as superstitious by Galileo.
BRAHE, TYCHO (1546–1601) was born of a Danish noble family. While studying at the University of Copenhagen, his interest in astronomy was piqued by a predicted eclipse that took place in 1560. Studying existing astronomical charts, he found them all to be in disagreement with one another. At age seventeen he decided that “what is needed is a long-term project with the aim of mapping the heavens conducted from a single location over a period of several years,” an effort to which (along with alchemy) he devoted his life.
In 1572, Tycho observed the appearance of a “new star” (a supernova). He was able to show that it lay far beyond the atmosphere, contradicting the Aristotelian principle that the celestial realm was unchanging. This impressed the king, who built him an observatory that Tycho named “Uraniborg” (castle of the heavens). Later, Tycho had another, subterranean observatory built nearby named “Stjeleborg” (castle of the stars).
Tycho’s were the most accurate astronomical observations ever made (or that can be made) with the naked eye. He rejected the heliocentric Copernican theory because he understood that the motion of the earth around the sun would lead to small shifts in the apparent positions of the stars in the sky (“stellar parallax”), and he was unable to observe this. (The stars are so distant that the parallax effect was not seen until 1838.) Thus, he proposed his own geocentric model. He was aided in his last years by Kepler, who succeeded him as “Imperial Mathematicus.” The vast wealth of precise data Kepler inherited from Tycho allowed him to discover that the orbit of Mars was an ellipse, not a circle, and to formulate his three great laws of planetary motion.
Tycho was an exotic figure. While a student, he lost part of his nose in a duel and wore a prosthesis made of gold and silver for the rest of his life. At his ancestral castle in Knudstrup, where he entertained on a grand scale, he kept a court jester named Jepp, a dwarf to whom Tycho attributed clairvoyance. Tycho also kept a tame moose, who, after imbibing too much beer one night at dinner, tumbled down a flight of stairs to an ignoble death.
MATHEMATICS IN A NEW ROLE
The Scientific Revolution, as well as the modern science to which it gave birth, was characterized not only by a happy marriage of mathematics and experiment but also a different way of looking at mathematics and its application to the physical world. A common view at the time of Copernicus and Galileo was that mathematics is useful for describing the quantitative aspects of things, but not particularly relevant to understanding those things or their causes. For example, the techniques of geometry could be used to predict accurately where heavenly bodies would appear in the sky at particular times, much as a modern train schedule is useful for predicting when trains will arrive at various stations. But just as a train schedule does not tell you what makes the trains go or why, the view of many Aristotelians was that mathematics gave no real insight into phenomena and their underlying physical causes: that was the job of “natural philosophy.” (It is significant that scientists were called “philosophers” in Galileo’s time, but astronomers were called “mathematicians.”)
That is one reason that the heliocentric system of Copernicus did not create much of a stir before the time of Galileo. It was generally seen as merely an alternative method of computation which, while having certain advantages, involved no claim that the earth was really in motion. The motion of the earth in the Copernican system was widely understood to be merely “hypothetical,” with no more reality than the constructions geometers made to prove their theorems. As long as any computational scheme correctly predicted where things would appear in the sky—“saved the appearances,” as they put it—it was regarded as no better and no worse than any other scheme, except in terms of convenience.
Now, as a matter of fact, some of Galileo’s telescopic discoveries (in particular, the phases of Venus) showed that the Ptolemaic system was no longer able even to “save the appearances,” whereas the Copernican system could. That was not enough, however, to prove the earth really moved, for there was on the market an alternative, proposed by Tycho Brahe, to the Copernican and Ptolemaic systems. The system of Tycho saved all the appearances just as well as that of Copernicus, but without having to suppose that the earth moves. (It was therefore embraced by the Jesuit astronomers of the time). In fact, Tycho’s system was simply the Copernican system as viewed from the earth (or as we might say now, it was the Copernican system as it appears in the “frame of reference” in which the earth is at rest). From a purely “mathematical” point of view, there was no way to decide between the Copernican and Tychonic systems, and there still isn’t, if one understands the role of mathematics as most did in Galileo’s time—that is, if one divorces mathematics from physical causes.
KEPLER, JOHANNES (1571–1630) was born in Weil der Stadt, Germany. After entering the University of Tübingen to study for the Lutheran ministry, he learned of the ideas of Copernicus and became enchanted with astronomy. He taught mathematics in Graz, but was driven out by the advancing Catholic Counter-Reformation. Finding work with Tycho in Prague, he was eventually forced to leave there for the same reason. (Kepler suffered vexations from his fellow Lutherans, too. They excommunicated him for his views on the Eucharist, and, on another occasion, prosecuted his mother for witchcraft.) Using Tycho’s data, Kepler discovered his three great laws of planetary motion. He was enabled to make these discoveries not only by his persistence and mathematical skill, but also by his sound physical intuition, which told him that the sun, as the largest body in the planetary system, somehow exercised a controlling influence on the other bodies. His astronomical thought was deeply influenced by his Pythagorean mysticism as well as Christian theology, which led him to see analogies between the Trinity and the interrelations of the heavenly bodies. It was Kepler who opened the door to the new science, though Copernicus and Tycho had given him the keys. At the end of his book Harmonices mundi (The Harmonies of the World), in which he announced his third law, Kepler exulted “I thank thee, Lord God our Creator, that thou allowest me to see the beauty in thy work of creation.”
All that changed with Newton, for his laws of motion and his law of gravity provided the critical link between the mathematical description of motion and the physical causes of motion. Specifically, it related acceleration to force.
To appreciate this point, it may be helpful to consider a simple example. Suppose that a two-hundred-pound man is swinging a little ball around himself in a circle at the end of a long, elastic string. Mathematically, one can just as well consider the ball to be at rest and the man to be in circular motion around the ball. Why is the first description physically more sensible (as, indeed, it obviously is)? The reason is that, in the first description, we understand the forces that are at work, i.e., the “dynamics.” Circular motion involves acceleration, and given the speed of the ball and the radius of its path we can calculate its acceleration. We can also calculate the force exerted on the ball by the string, because we understand strings. (Specifically, we can measure how much the string is stretched, and there is an empirical law, called Hooke’s law, that relates the amount a string stretches to the force it exerts.) And what we would find in the sensible frame in which the ball is going round the man is that the force of the string just matches the acceleration of the ball times its mass, satisfying Newton’s famous law, F=ma.
However, in the second description—where the ball is considered to be at rest, with the man going round the ball—the forces cannot be explained in a physically sensible way. Because the man’s mass is so large, the string’s force is not enough to constrain him to move in the circle. Therefore, to satisfy Newton’s law in the frame of reference where the ball is at rest, large additional forces acting on the man must be assumed. Where do they come from? Nowhere. There is no intelligible physical origin for them; they must be introduced ad hoc. In modern terminology, they are “fictitious forces,” and the need to postulate them is a symptom that one is not describing the situation in a physically sensible (or “inertial”) frame of reference.
Note, then, that given the correct “dynamical laws,” one may begin to understand physical reality and physical causes through a mathematical analysis. It is only by measuring the stretching of the string, its “elastic coefficient,” the mass and speed of the ball, and the radius of its path, and then applying the dynamical laws of Newton to these quantities by means of the requisite mathematical calculations, that one arrives at a proper understanding of what is physically happening and why. (In the same way, the knowledge of Newton’s law of gravity and Newton’s laws of motion allows one to see that it really is the earth that is in motion about the much more massive sun.)
This is a mathematization of science far more profound than was understood before Newton, except by a few who had inklings of it, such as Kepler, Galileo, and perhaps Copernicus. When Galileo said that “the great Book of Nature is written in the language of mathematics,” he heralded a radically new approach to the physical sciences.