Induction and Experimental Method

Author’s note: The following is adapted from a chapter of my book in progress, “Induction in Physics and Philosophy.” The book is based on Leonard Peikoff’s lecture course of the same title.

The scientific revolution of the 17th century was made possible by the achievements of ancient Greece. The Greeks were the first to seek natural (as opposed to supernatural) explanations, offer comprehensive theories of the physical world, and develop both deductive logic and advanced mathematics. However, their progress in physical science was impeded by the widely held view that higher knowledge is passively received rather than actively acquired. For many Greek thinkers, perfection was found in the realm of “being,” an eternal and immutable realm of universal truths that can be grasped by the contemplative mind of the philosopher.

In contrast, the physical world of activity was often regarded as a realm of “becoming,” a ceaselessly changing realm that cannot be fully understood by anyone. The modern scientist views himself as an active investigator, but such an attitude was rare among the Greeks. This basic difference in mindset—contemplation versus investigation—is one of the great divides between the ancient and modern minds.

Modern science began with the full development of its own distinctive method of investigation: experiment. Experimentation is “the method of establishing causal relationships by means of controlling variables.” The experimenter does not merely observe nature; he manipulates it by holding some factor(s) constant while varying others and measuring the results. He knows that the tree of knowledge will not simply drop its fruit into his open mind; the fruit must be cultivated and picked, often with the help of instruments designed for the purpose.

Precisely what the Greeks were missing can be seen by examining their closest approach to modern experimental science, which was Claudius Ptolemy’s investigation of refraction. Ptolemy conducted a systematic study in which he measured the angular deflection of light at air/water, air/glass, and water/glass interfaces. This experiment, when eventually repeated in the 17th century, led Willebrord Snell to the sine law of refraction. But Ptolemy himself did not discover the law, even though he did the right experiment and possessed both the requisite mathematical knowledge and the means to collect sufficiently accurate data.

Ptolemy’s failure was caused primarily by his view of the relationship between experiment and theory. He did not regard experiment as the means of arriving at the correct theory; rather, the ideal theory is given in advance by intuition, and then experiment shows the deviations of the observed physical world from the ideal. This is precisely the Platonic approach he had taken in astronomy. The circle is the geometric figure possessing perfect symmetry, so Ptolemy and earlier Greek astronomers began with the intuition that celestial bodies orbit in circles at uniform speed. Observations then determined the deviations from the ideal, which Ptolemy modeled using mathematical contrivances unrelated to physical principles (deferents, epicycles, and equants). Similarly, in optics, he began with an a priori argument that the ratio of incident and refracted angles should be constant for a particular type of interface. When measurements indicated otherwise, he used an arithmetic progression to model the deviations from the ideal constant ratio.1

Plato had denigrated sense perception and the physical world, exhorting his followers to direct their attention inward to discover thereby the knowledge of the perfect ideas that have their source in a non-physical dimension. Unfortunately, Plato explained, these perfect ideas will correspond only approximately to the ceaselessly changing and imperfect physical world we observe.

Ptolemy’s science was superficially anti-Platonic in that he emphasized the role of careful observation. However, at a deeper level, his science was a logical application of Platonism; in astronomy and in optics, he started with the “perfect” model and then merely described without explanation the inherently unintelligible deviations from it. Thus Ptolemy regarded experiment not as a method of discovery but instead as the handmaiden of intuition; he used it to fill in details about a physical world that refuses to behave in perfect accordance with our predetermined ideas. This approach is a recipe for stagnation: The theory is imposed on rather than derived from sensory data; the math is detached from physical principles; and, without an understanding of causes, the scientist is left with no further questions to ask.

The birth of modern science required an opposite view: Experiment had to be regarded as the essential method of grasping causal connections. The unique power of this method is revealed by examining how it was used by the geniuses who created the scientific era.

Galileo’s Kinematics

Legend has it that modern physics began in a church.

On a Sunday in 1583, the 19-year-old Galileo let his attention wander from the sermon and focus instead on a hanging cathedral lamp that swung in the draft. As he watched, he noticed with surprise that swings through small arcs seemed to take the same time as swings through larger arcs. Using his pulse beat as a clock, he timed the oscillations and confirmed that the period seemed independent of arc size.

At this point, Galileo had made an interesting observation and a crude measurement. But he had proven nothing. In order to prove the amplitude independence—and then go further to discover a law relating the properties of a pendulum to its motion—he had to design and perform a series of experiments.

He began by constructing two similar pendulums, both of the same length and with the same lead bob weights. He pulled both pendulum bobs from the vertical by different angles (e.g., thirty degrees for one and fifteen degrees for the other), and released them at the same time. Galileo then observed that the two pendulums swung back and forth in nearly perfect unison despite their different amplitudes, and he noted that they continued to do so as he counted more than ten swings. With all other relevant factors held constant, doubling the amplitude had no discernible effect on the period.

The next steps of Galileo’s investigation were guided by his thoughts concerning the cause of this surprising result. He realized that the pendulum bob gains speed only because it is falling toward the Earth, that is, only because of the downward component of its motion—and this component obviously increases as the amplitude of the swing is increased. Thus when the bob traverses a greater distance it does so at a greater speed; the two factors compensate, keeping the period constant.

But is this compensation exact for all amplitudes? Galileo approached the question in two ways: experimentally and mathematically. First, he repeated the above experiment using an amplitude of nearly ninety degrees for one pendulum and only ten degrees for the other. In this extreme case, he measured the period of the large swings to be about 10 percent greater than the period of the small swings. Despite the discrepancy, this result did not dissuade him from the “law of amplitude independence.” He thought it likely that the relatively small increase in period is caused by the larger and faster swing encountering greater air resistance, and he was deliberately abstracting from the effects of such “impediments.”

It was impossible for Galileo to directly determine by experiment the effects of air resistance; a vacuum pump would have enabled him to remove the air, but it was another half century before the instrument was invented. So, instead, he turned to mathematics for a proof of the amplitude independence. Starting from the idea that the pendulum bob’s acceleration is proportional to the sine of the amplitude—because this factor isolates the efficacious, downward component of the motion—he attempted to prove that no matter where a bob is released along the circular arc, it reaches the bottom at the same time.

Here Galileo encountered another obstacle: In general, the mathematical concepts and methods of his day were inadequate to deal with a moving body that continuously changes direction. So he simplified the problem by replacing the circular arc with a chord (a line segment between the uppermost point of the swing and the point at the bottom of the arc). Now he had a problem that he could solve, and he successfully proved that frictionless movement down all such chords takes the same amount of time. The result was tantalizingly close to what he wanted, and yet irreconcilably different: Motion down the straight chord is not the same as motion down the circular arc. But Galileo could not make any further progress with the analysis. In the end, he decided to present the law of amplitude independence as valid, while privately expressing his dissatisfaction with the lack of a proof.2

After eliminating amplitude as a relevant factor, he turned to consider properties that might affect the rate at which the pendulum swings. He asked: Does the material or weight of the bob affect the period? The question was easily answered by experiment. Starting with his two identical pendulums, he replaced the lead bob of one with a cork bob. He then pulled the bobs aside by twenty degrees or more and released them at the same time. The amplitude of the cork pendulum diminished more rapidly, and he correctly attributed this difference to air resistance. Given the results of his earlier experiments, he could disregard the amplitude difference because it has a negligible effect on the period. The crucial observation was that the lead and cork bobs continued to swing back and forth in nearly perfect unison. Galileo confidently and logically generalized from this result, concluding that in all cases the material and weight of a pendulum bob have no effect on its period.

Since the pendulum bobs in the above experiment swung with the same period, the same cause must have been operative. It appears that the only relevant property the two pendulums have in common is length, and therefore length must be the causal factor determining the period. A little reflection shows that this idea integrates with a wealth of common observations; for example, a child’s swing hanging from a high tree branch takes more time to go back and forth than a shorter one hanging from a lower branch, and a long vine sways more slowly in the breeze than a short one. But precisely how does the length affect the period? Galileo discovered the answer by conducting a series of experiments in which he varied the length of a pendulum across a range from about two to thirty feet. He measured the period by comparing it to that of another pendulum of constant length or by using a water clock (a device that measures time by marking the regulated flow of water through a small opening). His data established an exact mathematical relationship between length and period, which he immediately generalized to a law applying to all pendulums anywhere on Earth in any era: The length is proportional to the square of the period.

It is instructive to note the experiments that Galileo did not perform. He saw no need to vary every known property of the pendulum and look for a possible effect on the period. For example, he did not systematically vary the color, temperature, or smell of the pendulum bob; he did not investigate whether it made a difference if the pendulum arm is made of cotton twine or silk thread. Based on everyday observations, he had a vast prescientific context of knowledge that was sufficient to eliminate such factors as irrelevant. To call such knowledge “prescientific” is not to cast doubt on its objectivity; such lower-level generalizations are acquired by the implicit use of the same methods that the scientist uses deliberately and systematically, and they are equally valid. Given such a context, Galileo quickly concluded that the period of a pendulum depends only upon its length.

In reaching this conclusion, he did overlook one relevant factor: the pendulum’s location. He knew that a pendulum swings because the bob falls toward the Earth. He also knew that the moon does not fall to the Earth, which might have suggested to him that the Earth’s gravity must diminish with increasing distance. So why not carry a pendulum to the top of a high mountain and see if it swings more slowly?

In Newton’s era, momentous consequences would follow when scientists used pendulums to discover such gravitational variations. But this possibility never occurred to Galileo because he lacked Newton’s concept of “gravity.” Galileo still thought in terms of the simpler concept of “heaviness,” which merely referred to the property of earthly objects that causes them to press downward and fall to the Earth. He lacked the idea of an unseen interaction between the object and the Earth. Given this modern idea of force, it is reasonable to think of the interaction weakening with increasing distance between the bodies. If, however, one thinks only of a “natural tendency” of the heavy object, the factor of distance to the Earth’s center cannot be grasped as relevant. The more advanced idea of “gravity” was necessary before scientists could discover that a pendulum’s period varies slightly with location—and then go further to discover all that this implies. Here we see how the lack of a crucial concept can halt progress, and how the concept’s formation can pave the way to further knowledge.

The pendulum provided Galileo with an excellent introduction to the experimental study of motion; the measurements were relatively easy because the bob stayed in one area while slowly repeating its motion for a long time. However, as Galileo discovered, the analysis of the motion is complicated by the fact that the bob continuously changes direction. By contrast, the simplest case of a body exhibiting its natural tendency to move toward the Earth is that of the body falling straight down.

Early in his career, when he was mathematics professor at the University of Pisa, he began his investigation of free fall by addressing an old question that was still a key point of confusion: How does the weight of a body affect the rate at which it falls? Galileo demonstrated the answer with his characteristic dramatic flair. He climbed to the top of the famous Leaning Tower and, from a height of more than fifty meters, dropped two lead balls that differed greatly in size and weight. The crowd of students and professors assembled below saw both objects hit the ground at very nearly the same time. Contrary to the common assumption, the rate that a body falls is independent of its weight.

Galileo then asked the next logical question: Does the rate of fall depend upon the material of the body? He repeated the experiment using one ball of lead and another made of oak. Again, when dropped simultaneously from a great height, they both hit the ground at very nearly the same time. Thus Galileo arrived at a very broad generalization: All free bodies, regardless of differences in weight and material, fall to Earth at the same rate.

On the surface, it seems too easy. It appears as though Galileo arrived at this fundamental truth of physics—one that had eluded the greatest minds in ancient Greece—merely by doing a few experiments that any child could perform. But a closer look reveals that Galileo’s reasoning was not so simple; it depended on his pioneering use of one valid concept and his rejection of certain widely accepted invalid concepts.

First, notice that the objects he dropped were not selected randomly. If Galileo had thrown a bale of hay and a straw hat off the top of the Leaning Tower, the event would not have been a landmark in the history of physics. Yet these objects are made of similar material and have greatly different weights, just like the two lead balls he actually used. Or consider Galileo’s second experiment: Imagine that he attempted to drop the lead and oak balls through water instead of air, perhaps thinking that it would be easier to investigate a slower motion. Again, the result would not have led to any important discovery. On the contrary, such experiments would be easily misinterpreted as evidence that weight is always an essential factor in determining the rate of fall.

Galileo chose the conditions of his experiments with a crucial criterion in mind: He wanted to minimize the effects of friction. Friction is “the force that resists the relative motion of two bodies in contact.” It is sometimes said that Galileo ignored this force, because the laws he discovered describe frictionless motion. But this is the opposite of the truth. In fact, he thought more deeply about air resistance and other forms of friction than any of his predecessors. He carefully distinguished the cases in which friction plays a minor role from the many cases in which it plays an essential role. Without this distinction, it is impossible to arrive at any law of motion; with it, Galileo successfully discovered the law of free fall.

Enrico Fermi, another great Italian physicist, paid tribute to this achievement with the following comment:

[I]t was friction itself that for thousands of years had kept hidden the simplicity and validity of the laws of motion. In other words, friction is an essential element in all human experience; our intuition is dominated by friction; men can move around because of friction; because of friction they can grasp objects with their hands, they can weave fabrics, build cars, houses, etc. To see the essence of motion beyond the complications of friction indeed required a great insight.3

A contemporary physicist sees the effects of friction everywhere around him. That is because he has been raised on the truths discovered by Galileo and Newton. Prior to the 17th century, natural philosophers viewed the motions they observed from a different perspective, a perspective tainted by the errors contained in the old Greek physics. For example, when Leonardo da Vinci studied pendulums, he did not grasp that the amplitude gradually diminishes because of air resistance. Instead, he analyzed the arc into a “natural” downward portion and an unnatural, or “accidental,” upward portion. He then invoked the widely accepted dogma that “accidental motion will always be shorter than the natural” to explain the damping of the swings. It never occurred to him to abstract from this effect, since he regarded it as fundamental to the nature of motion.

Just as valid concepts such as “friction” can propel science forward, invalid concepts can stop it. Da Vinci’s mistaken analysis of pendulums rested on the Greek concepts of “natural” and “violent” motion, which were formidable barriers to progress in physics. At the root of this distinction was the false idea that motion requires a mover, that is, a force. Rocks falling, smoke rising, and the moon circling were regarded as cases of natural motion in which the body is moved by an internal force inherent in its nature. Rocks thrown upward, smoke blown horizontally and birds flying were regarded as cases of violent motion in which the body is moved by external pushes against other bodies. Natural motions were held to be the true province of the physicist, since they resulted from the nature of the body; violent motions were typically dismissed as less interesting.

Like square pegs in round holes, the facts resisted attempts to fit them into these invalid categories. Consider the simple case of a man throwing a rock. Why does the rock continue to move after leaving the thrower’s hand? As it flies through the air, where is the mover? Since a violent motion requires an external force, the proponents of the Greek theory were compelled to give an unconvincing answer: The thrower allegedly passes his moving power to the air, and then air currents push on the rock to cause its continued movement. According to this view, the primary role of air is not to resist such violent motions but to cause them.

During the Middle Ages, some thinkers began to reject the implausible claim that air pushes a projectile along its path. In discussing the case of a long jumper, 14th-century philosopher Jean Buridan wrote: “The person so running and jumping does not feel the air moving him, but [rather] feels the air in front strongly resisting him.”4 But nobody was yet ready to give up the idea that motion requires a force. So they “internalized” the force; they claimed, for example, that the thrower of a rock transfers his motive power directly into the rock, giving it a property called “impetus.” Despite their acceptance of a false premise, these philosophers did achieve a partial break with the errors of the past. They abandoned the distinction between natural and violent motion; in effect, their view transformed violent motions into natural ones by claiming that such motions are caused by the internal impetus of the body.

The medieval proponents of “impetus” supplied an answer to the Greek quandary about what makes a projectile go—but then they faced the question: Why does the projectile ever slow down or stop? What happens to its impetus? One response was to claim that impetus naturally dissipates over time. But this answer was inadequate; among other problems, it gave no clue as to why the rate of dissipation depends on the medium through which the body travels. So it was suggested by Buridan that a body loses impetus only when it works to overcome resistance. Buridan’s idea contained an important element of truth insofar as it identified the role of friction in opposing motion.

But it was Galileo who took the crucial step by combining this appreciation of friction with the experimental method. He did not merely acknowledge the existence of friction; he actively sought to control and minimize it. This is what enabled him to abstract from the effects of air resistance and thereby discover that all free bodies fall with the same motion.

The next reasonable question was: What is the nature of this motion? In particular, Galileo wondered how the speed of a falling body increases with time and with distance. Of course, he had no way to directly measure the speed. However, he realized that there was a closely related measurement that was difficult but not impossible: He could measure how the distance fallen varies with the elapsed time.

Using his water clock, Galileo timed a fall from about six feet and then another fall from twice that height. He found that doubling the height increased the time of fall by less than 50 percent. This result suggested that the distance fallen might be proportional to the time squared, just as the length of a pendulum is proportional to its period squared.

Galileo then realized that he could use the pendulum to check this idea. He devised a pendulum of easily adjustable length in which the bob impacted a fixed board placed at the bottom of its swing. He then drew the bob aside and released it while simultaneously dropping another weight to the floor. For a particular length of the pendulum, he adjusted the height of the falling weight so that its thud against the floor was simultaneous with the thud of the pendulum bob against the board. He found, for example, that a weight will fall nearly five feet in the time that a four-foot pendulum swings to the vertical. By repeating the procedure for several different values, he proved that the ratio of height fallen to pendulum length is always the same. Since the pendulum length varies as the square of the elapsed time, the distance traversed in free fall must also be proportional to the time squared. In this way, using his prior knowledge of pendulums and the experimental method, Galileo arrived at a generalization of impressive scope: For all free bodies on Earth, the height fallen is equal to the square of the elapsed time multiplied by a specific constant (the value of which depends on the particular units).

He realized the implications of his time squared law. Since the height fallen is equal to the average speed of fall multiplied by the elapsed time, the height can be proportional to time squared only if the speed is directly proportional to the time. Thus Galileo had found the answer to his original question about the increase of speed during fall: The speed increases in direct proportion to the elapsed time, that is, it increases by equal increments in equal time intervals. In familiar English units, we say that the speed increases by thirty-two feet per second during each second of fall.

In addition to the concept of “friction,” this discovery depended on Galileo’s prior development of two key concepts of motion. Throughout the above reasoning, he was using concepts of “speed” and “acceleration” that differed profoundly from those in common use at the time. Stillman Drake, a leading Galileo scholar, points out that “the Italian word ‘velocita’ . . . just meant swiftness, a vague qualitative concept. . . .”5 The alternative was the Latin word “velocitas,” then used by natural philosophers to mean “intensity of motion.” Galileo recognized that such qualitative ideas are dead ends in physics; the science of kinematics requires quantitative concepts of motion that are defined mathematically and can be identified by measurements.

Galileo faced two obstacles that prevented him from developing fully adequate concepts of motion. First, the Greek theory of proportions restricted him to ratios of “commensurable quantities,” for example, ratios of distances, or of times, or of speeds. An overly narrow concept of “number” had led the Greeks to reject ratios of “unlike quantities” such as distance over time or speed over time. Second, the crucial ideas of instantaneous speed and instantaneous acceleration are impossible without the mathematical concept of “limit,” which had not yet been developed. As a result, he could offer mathematically rigorous definitions only for motion at constant speed or for motion at constant acceleration.

Despite these restrictions, Galileo’s new concepts of motion were a crucial advance over those of his predecessors, and they were adequate for his purposes, since free fall occurs with a simple uniform acceleration. However, his direct experimental evidence for the law of free fall was open to one criticism: It was difficult to obtain repeatable and accurate measurements of such short time intervals. One could more easily investigate the acceleration of fall if there were a way of slowing it without changing its nature. This was the motive behind Galileo’s inclined plane experiments.

In the case of a ball rolling down an inclined plane, the movement is caused by the downward component of the heavy ball’s constrained motion. Galileo reasoned that since the cause of the motion is the same as in free fall and the ball’s direction is constant, the acceleration down the plane should be of the same nature as in free fall, but merely attenuated by the ratio of the height fallen to the total distance traversed. Thus rolling balls down a plane inclined at a small angle from the horizontal provided a way to study the acceleration of a falling body in a form that was greatly reduced in magnitude and therefore easier to measure.

Galileo rolled a bronze ball down a smoothly polished groove carved in a straight wooden plank that was about eight feet long. Using an angle of inclination of about two degrees, it took more than four seconds for the ball to roll down the plank. He then had an ingenious idea. He tied eight very thin strings around his plank. When the ball rolled over a string, it made a slight but audible bumping sound. As he repeatedly rolled the ball down the plank, he adjusted the location of the strings until the sounds occurred at regular intervals of a little more than a half second. Galileo was quite knowledgeable about music and he knew that the regularity of such beats can be judged very accurately (most people can detect a deviation of one sixty-fourth of a second). The positions of the strings that produced the regular beats were a record of the distance traveled by the ball as a function of time. His results proved that the distance is proportional to the square of the time, and therefore motion down an inclined plane is uniformly accelerated. He later confirmed this law with additional experiments in which he used longer inclined planes and made time measurements with his water clock.

Galileo also grasped some crucial implications of his idea that the ball’s acceleration down the plane is proportional to the sine of the inclination angle. First, he mathematically deduced that the final speed of the ball at the bottom of the plane depends only on its initial height, not on the length of the plane or its degree of inclination. The height of the plane, he showed, is proportional to the square of the ball’s final speed. Second, the acceleration of the ball must approach zero as the inclination angle approaches zero, implying that free horizontal motion must occur at constant speed.

Galileo designed an experiment that made use of the first implication in order to test the second. His inclined plane was mounted on a table about three feet high. At the bottom of the incline he devised a curved deflector so that the ball made a smooth transition to roll briefly along the horizontal tabletop before flying off and hitting the ground some distance away. He chose an initial height of the plane and then measured where the ball landed. Armed with his knowledge of the relationship between height and speed and with his hypothesis of constant horizontal speed, Galileo could then calculate where the ball would land for any other height of the inclined plane. He made his calculations, performed the experiment, and found that his predictions agreed with his measurements.

The inclined plane provided Galileo with a bridge between vertical and horizontal motion, and it cast light upon the nature of both. He used it to study acceleration of a falling body and to provide a projectile with known and easily variable horizontal speed. The results of his experiments led inexorably to the crowning generalization of his kinematics: Free vertical motion occurs with constant acceleration, whereas free horizontal motion occurs with constant speed.

The experiment described above led to yet another crucial discovery. Galileo had not merely measured the distance from the table to the ball’s impact point; he had also observed and carefully drawn the trajectory of the ball through the air. He had an intimate knowledge of Greek geometry, and therefore the shape of the trajectory struck him as familiar: It looked like a semi-parabola. This observation started a chain of thought that led to the realization of why the trajectory is necessarily a parabola.

He had proven that a free body falls through a vertical distance that is proportional to the square of the elapsed time. He had also proven that the horizontal distance traversed by a free body is directly proportional to the elapsed time. Furthermore, his experiment showed that the horizontal motion is unaffected when the body is simultaneously falling. Thus the vertical and horizontal motions occur independently, each rigorously following its own law while remaining oblivious to the other. By combining the two separate laws, Galileo derived the conclusion that the change in the height of a projectile is proportional to the square of the change in horizontal position—and he knew that this relationship describes the curve of a parabola. Without any of the hand-wringing and arbitrary doubting of a skeptic, he concluded that all free projectiles follow parabolic trajectories.

While discussing concept-formation, Ayn Rand explained that “perceptual awareness is the arithmetic, but conceptual awareness is the algebra of cognition.” She ended the discussion with a challenge to the skeptics: Those who deny the validity of concepts must first prove the invalidity of algebra. Here we are dealing with inductive generalizations, but a similar challenge applies: Those who deny the validity of induction by declaring that it is impossible to find “all” in the “some,” must first prove the invalidity of kinematics by showing that it is impossible to find a causal connection between the referents of the concepts “projectile” and “parabola.” Such skeptical arguments are futile, since Galileo found just that connection.

Galileo thought of a simple way to demonstrate the above principle. When a ball is rolled across a smooth inclined tabletop, it moves with constant speed across the table and with constant acceleration down the table—and hence it traces out a parabolic path. He immediately put the principle to practical use by solving several long-standing military problems; for example, he showed how a cannon’s range depends upon its firing angle, and how to calculate the firing angle for a target at some specified height above the ground.

It was a feat of genius to grasp that projectile motion could be analyzed into independent horizontal and vertical components. Stillman Drake points out that Galileo’s predecessors had thought quite differently:

Medieval impetus theory, like Aristotelian physics, supposed that when two different tendencies to motion were present in the same body, only the stronger would determine its actual motion. When the stronger tendency was violently imparted, as in a ball thrown horizontally, it was assumed that conflict between this and the natural tendency to fall weakened the horizontal motion until the constant vertical tendency became stronger and brought the ball to earth.6

Even today, this false view remains influential. Consider an untutored man who is told that a bullet will be fired horizontally from the muzzle of a gun while a second bullet is dropped simultaneously from the same height. When asked to guess which bullet will hit the ground first, the man will invariably choose the dropped bullet. Galileo was the first to grasp that the horizontal motion of the fired bullet is irrelevant and therefore both bullets hit the ground at the same time (assuming, of course, that the Earth’s surface can be approximated as flat).

Galileo’s analysis led to a new synthesis. According to the old medieval view, two causes are necessary to explain the rise and descent of a projectile. Once the projectile is in the air and moving freely, its rise is caused by the upward impetus that has been imparted to it. After this impetus dissipates, a second cause becomes operative: the body’s natural tendency to fall toward the Earth. In contrast, Galileo recognized that the same cause and the same effect are operative throughout the trajectory: The projectile is always accelerating toward the Earth at the same rate while simply conserving its horizontal speed.

Experiments enabled Galileo to reach and validate his mathematical description of motion, and thereby achieve a perspective that was more abstract and more integrated than that of his predecessors. His abstraction from the effects of friction, his analysis of motion into horizontal and vertical components, his mathematically precise definitions of uniform speed and uniform acceleration, his application of the Greek knowledge of parabolas—these were among the key conceptual steps that raised him to a height from which he could see the same principles at work in many superficially different motions. From this new perspective, he saw a swinging pendulum, a falling apple, a ball rolling down a hillside, and a cannonball rising in the air as variations on the same theme: the constant acceleration of heavy bodies toward the Earth. One historian of science puts it this way:

Galileo introduced a classificatory system where very different looking things . . . were regarded as all belonging to the same category, and hence were analyzable in a coherent and comparable manner. They were seen as instances of the same thing, in much the same way as a moving compass needle, patterned iron filings, and induced current in a moving conductor—observationally all very different—are seen as indicators of one thing, a magnetic field.7

The various motions Galileo studied were not related merely in hindsight, as a result of his laws; as we have seen, they were connected during the discovery process, and the connections were essential to discovering the laws. At each step along the way, Galileo made use of the full context of knowledge available to him. The pendulum played a crucial role in the study of free fall, and then both pendulum and free-fall investigations led to the study of inclined plane motion, which in turn led to the understanding of projectile motion. Galileo’s kinematics was developed and validated not as a conglomeration of separate pieces, but as a unified whole.

Integration is the process of uniting a complexity of elements into a whole. Cognitive integration is the very essence of human thought, from concept-formation (an integration of a limitless number of concretes into a whole designated by a word), to induction (an integration of a limitless number of causal sequences into a generalization), to deduction (the integration of premises into a conclusion). An item of knowledge is acquired and validated by means of grasping its relation to the whole of one’s knowledge. A thinker always seeks to relate, grasp hidden similarities, discover connections, unify. A conceptual consciousness is an integrating mechanism, and its product—knowledge—is an interconnected system, not a junk heap of isolated propositions.

Galileo integrated his knowledge not only within the subject of physics but also between physics and the related science of astronomy. Copernican astronomy claimed that the Earth spins rapidly about its own axis and hurtles around the sun at astonishing speed. According to the old views regarding motion, this claim was simply preposterous. If the Earth is moving, people asked, what would happen when a rock was tossed straight up in the air? The Earth would move underneath it, and, contrary to experience, the rock would come down miles away. Furthermore, what is causing the alleged motion of the Earth? Nothing is pushing it, and the materials on Earth exhibit no natural inclination to revolve in circles. The only motion natural to heavy bodies is to fall straight down. If the Earth were not already in its natural place, it would simply fall toward that place. And since the Earth is very heavy, it would fall very quickly; any lighter bodies that were not fastened down—including ourselves—would be left behind! Galileo’s theory of motion nullified these objections, and thus it served as the foundation for his defense of the new astronomy. This crucial integration had the inverse benefit as well: Galileo could point to the abundant observational evidence in favor of the heliocentric theory as further support for his kinematics.

Now let us turn from Galileo’s triumphs to mention the problems that he confronted but could not solve. Today, the errors of great scientists are often cited as reasons for doubting the validity of scientific method. If even the best practitioners of this method make mistakes, it is argued, how can we trust any scientific results? In order to see that such doubts are baseless, it is worth examining Galileo’s errors. We will see that they provide no foothold for skepticism; on the contrary, they illustrate that a rational process is self-correcting.

Let us start with the analysis of pendulum motion. Later scientists proved that the slightly greater period of a pendulum swinging in a larger arc is not caused by air resistance, as Galileo had supposed. Even when the air is removed, a pendulum bob swinging along a circular arc requires more than 10 percent more time for very large swings than for small swings. Galileo did not have the experimental or mathematical means to identify the cause in this case. Ideally, he should have openly acknowledged the small amplitude dependence he had discovered, and then merely suggested air resistance as a possible cause.

But the mistake was of little consequence. The conditions for a truly isochronal pendulum were discovered one generation after Galileo published his theory of motion. In 1659, Christian Huygens proved that the period of a pendulum is independent of amplitude when the bob moves along the arc of a cycloid rather than a circle (a cycloid is the curve traced by a point on the rim of a rolling wheel). In order to solve the problem that had stumped Galileo, Huygens made use of two new developments in mathematics: the recently discovered properties of the cycloid and the technique of “infinitesimals.” By starting with Galileo’s law of inclined plane motion and then treating the curved path of the pendulum bob as a series of infinitesimally small inclined planes, he demonstrated that the bob always descends in the same time only when the curve is a cycloid. Thus the very knowledge that Huygens inherited from Galileo—when combined with the new mathematics—enabled him to correct Galileo’s mistake.

The same point is illustrated by an oversight in Galileo’s analysis of inclined plane motion. His bronze ball moved down the inclined plane by rolling—rather than sliding—because of the friction between the ball and the wood surface. He never suspected that the acceleration of a rolling ball is about 28 percent less than that of a sliding ball. His theorem relating the acceleration on an inclined plane to the acceleration of free fall is true only for frictionless sliding, yet he implied that it is true for the rolling balls used in his experiments. This was an understandable error on a subtle point. The mechanics of rolling balls is complex; Galileo lacked the dynamical and mathematical concepts that are required to grasp the subject. Eventually, in the 18th century, it was the powerful combination of Newton’s dynamics and Euler’s mathematics that rendered the behavior of rotating bodies fully intelligible. Again, scientists stood on Galileo’s shoulders in order to reach a height from which his error was seen and corrected.

The fundamental error in Galileo’s physics is found in his treatment of horizontal motion. His evidence that free horizontal motion occurs with constant speed came primarily from laboratory experiments and secondarily from field observations of short range projectiles. Thus the evidence was limited to a domain of short distances over which the Earth can be approximated as flat. However, Galileo speculated about how his principle would apply to motions over very large distances. He argued that in such cases “horizontal motion” can only mean motion at constant altitude, from which he deduced that free horizontal motion is ultimately circular motion around the spherical Earth. This was his concession to the Greek idea of “natural circular motion,” and he supported it with an a priori argument handed down from Plato (an argument based on nothing more than the alleged perfection of circles).8

Galileo was left vulnerable to committing this error because he lacked the concept of “gravity.” Since he never formed the idea of an attractive interaction that diminishes with distance, he could not abstract from the Earth’s gravity. With this abstraction he might have arrived at Newton’s first law of motion, which states: In the absence of forces, a body remains at rest or moves with constant speed in a straight line. However, Newton’s ability to abstract from gravity depended on his grasp that it is a variable force that can diminish to insignificance at sufficiently large distances; such an abstraction makes no sense on Galileo’s view that free, heavy bodies simply fall at a constant rate of acceleration—as an omnipresent effect. So, in the absence of a crucial prerequisite concept, Galileo’s mind could only remain at rest or move in a wrong direction on this issue; he could not arrive at the principle that later became Newton’s first law.

Earlier we saw Leonardo da Vinci make a similar error in his analysis of pendulum motion. Lacking any clear concept of friction and dismissing the air as an omnipresent background, he never identified or abstracted from the effects of air resistance. As a result, he was unable to explain the damping of the pendulum, and he filled the void in his understanding by appeal to an arbitrary dogma. In a parallel way, Galileo did not possess the concept of gravity and could not abstract from its effects, and he filled the void in his understanding by appeal to a baseless Platonic argument.

Galileo erred on other issues that cannot be understood without the idea of gravity. An obvious example is his attempt to explain the ocean tides without reference to the gravitational force of the moon and the sun. Less obvious examples are his failure to accept Kepler’s law that planets move in elliptical orbits, and his suggestion that comets might be atmospheric phenomena rather than celestial bodies. Both of these latter errors were caused by Galileo’s concession to “natural circular motion,” which I have argued was a consequence of his inability to abstract from the effects of gravity. On one topic after another, Galileo was stopped by the same barrier. We can see why Newton’s concept of gravity was so central to the development of modern physics.

Of course, it was Galileo who paved the way for his successors—not merely by presenting the knowledge he discovered but also by providing insight into the proper method of discovery. The latter was the most valuable part of his legacy. The old adage applies here: “Give a man a fish and he eats for a day; teach him to fish and he eats for a lifetime.”

Regrettably, Galileo’s published works do not give an entirely accurate portrayal of his discovery process. It is often the case that a scientist presents his theory in a form that obscures the steps by which he arrived at it. Galileo occasionally created a misleading impression of his method by presenting deductive arguments from “plausible first principles” or from thought experiments, while giving less emphasis to inductive arguments from actual experiments. In regard to method, his practice was better than his presentation. Thus he did not take full advantage of the opportunity to teach his successors how to acquire scientific knowledge.

As a result, the role of experimentation was not adequately grasped by the generation of scientists who followed Galileo. This left the door open for René Descartes, who led a Platonist revival. Descartes explicitly rejected the method of inducing causes by observing their effects, and he criticized Galileo for using such an approach. “He seems to me very faulty in . . . that he has not examined things in order,” Descartes wrote, “and that without having considered the first causes of nature he has only sought the reasons of some particular effects, and thus he has built without foundation.”9 In contrast, Descartes explained that his goal was “to deduce an account of effects from their causes . . .”10 How do we know the fundamental causes? With Plato, Descartes claimed that he had direct access to them by means of “clear and distinct” innate ideas.

Thus even Galileo’s spectacular achievement was not enough to institutionalize the experimental method and discredit Platonism among scientists. That task was left to the man who completed the scientific revolution: Isaac Newton.

Newton’s Optics

When Newton began his battle to establish a proper inductive method in physics, he was working in the field of optics, not kinematics or astronomy. In his early years, well before the Principia brought him fame, he conducted a study of light and colors that has been described as “the preeminent experimental investigation of the seventeenth century.”11

We live in a colorful world. Typically, the colors we see are produced by reflection of ordinary (white) light from bodies, and the specific color reflected depends on the nature of the body. By the second half of the 17th century, it was also known that colors can be produced by refraction. While an undergraduate at Trinity College, Newton learned about the sine law of refraction (discovered by Snell in 1621), about colors that result from white light passing through wedges of glass (prisms), about the idea that rainbows are somehow caused by light refracting within raindrops, and about the fact that refracting telescopes produce blurred images with colored edges.

Newton’s early interest in the subject is evident from his detailed study of Robert Boyle’s book Experiments Touching Colors (1664). Two scholars offer the following description of Newton’s notes on the book:

[The notes] comprise data concerning the ways in which the colors of objects are changed under a wide variety of circumstances. They record the effects of heat, the characteristics of various sublimates, acids, and precipitates, the ways objects change in various lights and positions, the effects of dyes, of solutions, and salts, and the changes wrought on colors by various combinations of these “instruments.” Although Newton’s aim is to increase his basis of information, the entries are more than a haphazard miscellany. Each bit of information pertains in some way to either the difference in the appearance of a body’s color when it is looked on, in contrast to when it is looked through, or the ways in which a body’s color can be changed.12

During these undergraduate years, Newton kept a notebook titled Certain Philosophic Questions in which he recorded his questions and first groping thoughts about a wide range of topics in physical science, including the topic of light and colors. For example, he asked why materials differ in transparency, why refraction is slightly less in hot water than in cold water, why coals are black and ashes are white. He asked whether light moves with a finite speed, whether rays of light might “move a body as wind does a sail,” and whether refraction at glass surfaces is the same when the surrounding air is removed. The questions show an extraordinarily active mind that had absorbed most of the available knowledge on the subject.

In his notes on colors, Newton referred to some of the proposed explanations that he encountered in the literature. He wrote: “Colors arise either from shadows intermixed with light, or by stronger or weaker reflections. Or, parts of the body mixed with and carried away by light.”13 He then quickly ruled out the first possibility, simply by citing many cases in which black and white are mixed without producing any color, and by pointing out that the edges of shadows are not colored. At this early stage, he had no theory of his own—and he was realizing that nobody else had one either (despite a few boastful claims by others to the contrary).

It wasn’t long before Newton purchased a prism and began his own investigation. He started as one might expect a curious child to start: by looking at various objects through the prism. His first important observations were of colors that appear along the boundaries between light and dark objects. For example, when a thin strip of white paper is placed against a dark background and viewed through a prism, one edge of the paper will appear blue and the other edge red.

From observing rainbows and toying with prisms, it was widely known that blue and red were on opposite sides of the color spectrum. Some scientists proposed that when a beam of white light enters water or glass at an oblique angle, one edge of the beam is affected differently than the other, causing the light on one side to turn blue while the light on the other side turns red. However, another possibility occurred to Newton: It struck him that rainbows and “boundary colors” might result if blue light is refracted slightly more than red light. In other words, he thought to ask: Are blue and red light seen on opposite sides not because they originate from different locations within the beam but instead because they are bent at different angles by the water or glass?

The question could be answered only by experimentation. Newton took a thread and colored half of its length blue and the other half red. When he laid the thread in a straight line against a dark background and viewed it through a prism, the blue and red halves looked discontinuous, with one appearing above the other. The prism shifted the image of the blue half of the thread more than it shifted the red half. From this one experiment, Newton reached a universal truth: Upon refraction, blue light is bent more than red light.

Since the various colors of light emerge from the prism at slightly different angles, the color spectrum will spread out as the light moves farther away. This thought led Newton to another experiment. With the windows of his room shaded, he allowed sunlight to enter through one small hole. He placed a prism near the opening so the light would pass through it and be displayed on the far wall about twenty-two feet away. He observed that the narrow, circular beam of white light was bent by the prism and transformed into a full, elongated spectrum of colors in the order red, orange, yellow, green, blue, and violet. The spectrum spread out in the same direction that the glass bent the light, and it was five times as long as it was wide.14

After observing this striking change in a beam of incident white light, it was natural to ask what effect a prism has on an incident beam of colored light. To find the answer, Newton performed a series of experiments in which he generated a spectrum with one prism and then passed the individual colors one at a time through a second prism. In contrast to white light, he found that only the direction of the colored light was altered. His measurements confirmed that blue light was bent more than red light, as he expected. More important, however, was the crucial new fact that the color is always unchanged and the beams are not spread out by the second prism.

Apparently, the individual colors are redirected but otherwise unaffected by prisms. But if prisms do not affect colors, then how can they create them from a beam of white light? Perhaps, Newton thought, the colors are not created by the prism; perhaps they are in the white light and merely separated by their variable angle of refraction. In other words, he was struck by a radical idea: Maybe a mixture of all the colors is experienced by us as white light.

He knew that in some cases a mixture of two colors is seen as a different color. In his early notes, he had recorded his observation that a yellow candle flame viewed through blue glass appears green. He also knew that white light viewed through a combination of red glass and blue glass appears purple. Such evidence had already convinced him that a mixture of colors can appear quite different from any of its components. However, it was a bold step to suggest that whiteness—which had long served as the very symbol of “purity”—was in fact a mixture of all the colors in the spectrum. It was bold, but nevertheless grounded in observational evidence.

Newton was never satisfied with merely suggesting a possibility—he settled for nothing less than experimental proof. If white light is composed of all the colors, then it should be possible to bring the separated colors back together and form white light once again. He realized that this can be done by using a combination of prisms or a focusing lens, and he showed that when the entire color spectrum is made to converge, it appears white. Furthermore, when the spectrum is allowed to continue through the focal point and diverge again on the other side, the colors reappear in the reverse order. The conclusion was now inescapable: The individual colors are the “pure” and simple components, whereas white light is a mixture of them.

He next applied this new insight to understand why objects around us appear with their characteristic colors. The basic implication of his theory was clear: When an object is illuminated by white light and yet it appears some particular color, the reason must be that the object reflects that color strongly while absorbing or transmitting much of the light in the rest of the spectrum. Based on his prior discoveries, Newton did not expect colors to be created or changed by reflection; they should merely be separated to the extent that one color is reflected more than the others.

In order to test this experimentally, he took a piece of paper and painted the left half blue and the right half red. In a shaded room, he illuminated the paper with only the blue light from a prism. As he expected, the entire paper appeared blue, but the color was intense on the left half and faint on the right half. When the paper was illuminated with red light from the prism, he again saw the expected result: The entire paper was red, but now the left half was faint and the right half intense. Colors are not changed upon reflection; blue paint reflects blue light strongly and red light weakly, whereas red paint reflects red light strongly and blue light weakly. These observations were a simple yet powerful confirmation of his theory.

He offered other convincing demonstrations that the colors composing white light are separated by unequal amounts of reflection or transmission at surfaces. For example, he darkened his room and then allowed a beam of white light to illuminate a very thin gold foil. He found that the reflected light on one side was the usual brownish-yellow color of gold, whereas the transmitted light on the other side was a greenish-blue color.

Newton’s theory of colors integrated and explained an enormous range of observations. For instance, he was able to explain all the essential properties of rainbows; for example, the greater brightness of the sky within the rainbow, the angular position and width of the primary and secondary rainbows, the reversed order of colors in the two rainbows, and so on. The theory also enabled him to understand why simple refracting telescopes produce blurred images with colored edges. Since the colors in white light are refracted at slightly different angles, they do not converge to form a sharp image. He solved the problem by inventing a new type of telescope that focused the light by means of reflection from mirrors rather than by refraction through glass. Thus he wasted no time putting his theory to practical use, and the superior performance of his reflecting telescope provided further confirmation of his theory.

His predecessors had assumed that colors were the result of some modification of “pure” white light. Then, without any supporting evidence, they speculated about the specific nature of the modification. Descartes had claimed that light was a movement of certain small particles, and that colors are caused by rotation of the particles: The light particles that rotate most rapidly are allegedly seen as red, whereas those that rotate most slowly are seen as blue. The prominent English scientist Robert Hooke offered a different theory. He supposed that white light was a symmetrical wave pulse, and he claimed that colors result when the pulse becomes distorted. According to his theory, light is red when the leading part of the wave pulse is greater in amplitude than the trailing part, and it is blue when the reverse is true. He assumed that the other colors were a mixture of red and blue.

Newton saw these “theories” for what they were: fictions based only on the fertile imaginations of their creators. He rejected their speculative approach and refuted their basic assumption. He proved that colors do not result from any modification of white light; rather, they are the elementary components of white light.

The most radical aspect of Newton’s theory did not consist of what he said, but of what he refrained from saying. He presented his results and conclusions without committing himself to any definite view regarding the fundamental nature of light and colors. He reasoned as far as the available evidence could take him—and no farther. Many scientists reacted to Newton’s original paper with surprise and confusion because they were accustomed to the Cartesian method of deducing conclusions from imagined “first causes.” Here was a paper about colors in which the author simply ignored the controversy about whether colors were rotating particles or distorted wave pulses or something else. Were some pages missing, they wondered?

Rather than omitting anything, Newton had supplied what was omitted from the method of Descartes: objectivity. Newton induced his conclusions from the observed results of experiments; he did not deduce them from “intuitions.” He was careful to identify the epistemological status of his ideas, and to distinguish clearly between those he regarded as proven and those based on evidence that was inconclusive. He knew only too well the painstaking effort that is required to discover basic truths about nature, and he had no patience for those who attempted to shortcut the process with empty speculation.

Newton once said that he “framed no hypotheses,” a statement that became both famous and widely misunderstood. Explaining his terminology, he wrote: “The word ‘hypothesis’ is here used by me to signify only such a proposition as is not a phenomenon nor deduced from any phenomena, but assumed or supposed—without any experimental proof.”15 Unfortunately, this did not make his meaning entirely clear. He did not intend to reject out of hand all hypotheses that lacked full experimental proof; in actual fact, he used the term “hypothesis” to refer to an arbitrary assertion, that is: a claim unsupported by any observational evidence.

Newton understood that to accept an arbitrary idea—even as a mere possibility that merits consideration—undercuts all of one’s knowledge. It is impossible to establish any truth if one regards as valid the procedure of manufacturing contrary “possibilities” out of thin air. As he explained in a letter to a colleague: “If anyone may offer conjectures about the truth of things from the mere possibility of hypotheses, I do not see by what stipulation anything certain can be determined in any science; since one or another set of hypotheses may always be devised which will appear to supply new difficulties. Hence I judged that one should abstain from contemplating hypotheses, as from improper argumentation.”16

Here, while defending his theory of colors, he introduced a crucial new principle of inductive logic. It is the proper response to nearly all the claims made by Descartes and his ilk, the only response they warrant: outright dismissal. Newton recognized that the attempt to refute an arbitrary assertion is a fundamental error. To grasp the nature of the world, one’s thinking must begin with information received from the world: sensory data. An arbitrary idea is detached from such data; to consider it is to leave the realm of reality and enter a fantasy world. No knowledge can be gained by taking such an excursion. One cannot even achieve the misguided goal of disproving an arbitrary idea, because such claims can always be shielded by further arbitrary assertions. Upon entering the fantasy world, one is caught in a proliferating web of baseless conjectures, and there is only one way out: to dismiss all such claims as non-cognitive and unworthy of attention.

The senses provide our only direct contact with reality. Without such contact, there may be brain action but there is no thought. The mental gyrations of Cartesian physics are like the spinning wheels of an elevated car—despite the motion, there is no chance of taking the road anywhere. As to those scientists who agree with Plato and Descartes and therefore reject the road because it is dirty and noisy and degrading to their elevated tires, they forfeit their means of ever going anywhere.

Newton’s theory of colors received a hostile reaction from such scientists. At first, Newton patiently reiterated how to conduct the experiments and what conclusions could be inferred with certainty from the results. Finally, he laid down an epistemological law in his effort to preempt all discussions that were not based on the observed facts. He declared that any valid criticism of his theory must fall into one of two categories: Either it argues that his observations are insufficient to support his conclusions, or it cites further observations that contradict his conclusions. As he put it:

The theory which I propounded was evinced to me, not by inferring it is thus because not otherwise, that is, not by deducing it only from a confutation of contrary suppositions, but by deriving it from experiments concluding positively and directly. . . . And therefore I could wish all objections were suspended from hypotheses or from any [grounds] other than these two: of showing the insufficiency of experiments to prove . . . any part of my theory, . . . or of producing other experiments which directly contradict me, if any such may seem to occur.17

Galileo had fought the Church in order to expel religious faith from the realm of science; Newton fought his fellow scientists in an effort to expel the arbitrary as such, including arbitrary secular claims.18 The appeal to faith is the demand that ideas be accepted on the basis of emotion rather than evidence, and it is therefore a species of the arbitrary. It makes little difference whether or not the idea is in the Bible—for example, whether one is asked to accept that Joshua lengthened the day by commanding the sun to stand still, or whether one is asked to accept that particles of white light become colored when they rotate. No evidence exists for either claim, and to consider either is to reject the mind’s only means of knowledge: reasoning from observed facts.

Although Galileo pioneered the experimental method, Newton was the one who established its fundamental role in modern physics. As two historians of science note: “Experiment became a principle as well as a method with Newton, who came to see the experimental foundation of his philosophy as the feature that set it apart from other natural philosophies and made it superior to them.”19 His experimental work in optics serves as a model of how physical science ought to be done.

The Methods of Difference and Agreement

Our first generalizations are based on causal connections that are directly perceived.20 Thus the first step toward the modern physicist’s understanding of gravitation is the generalization: “Heavy things fall.” A child grasps the idea of “heaviness” by holding objects and feeling the downward pressure they exert against his hand. He notices that some things press down more than others; he implicitly omits the measurements (i.e., the specific weights), and calls the attribute “heaviness.” The concept “fall” is also based directly on perceptual data: It refers to the downward motion of things that occurs spontaneously, without pushing. When a child feels the heaviness of a thing and then lets it go and sees it fall, he immediately grasps: “Its heaviness (that which made it press down on his hand) is what made it fall.”

No deliberately applied method is required to grasp such first-level generalizations. The measurement-omission process is subconscious and automatic, and the causal connection is given in the perceptual data. The need for a method arises when we attempt to establish relationships that involve higher-level concepts.

Recall Galileo’s study of pendulum motion. In contrast to the connection between the heaviness of a pendulum bob and its descent, we do not directly perceive the causal connection between the pendulum’s length and its period. “Length” is a first-level concept (i.e., it is directly perceivable), but “period” is not. The application of the concept “period” in this context presupposes knowledge that a pendulum engages in repetitive motion that can be related quantitatively to other motions by means of a unit of time. Here we need prior conceptual integrations—and an explicit method.

In this case, Galileo discovered the causal relationship when he built and compared two pendulums that differed in only one relevant factor—the length of their arms—and then measured the resulting difference in their periods. By isolating and varying the length, he created a situation in which the difference in length could be identified as the only possible cause of the difference in period. This is the same method he had used earlier to eliminate other possible causal factors. For example, he observed that two pendulums have the same period when they differ only in the weight of their bobs or in the amplitude of their swings. In these experiments a single difference was introduced, but with negative results—the difference made no difference in the period, and therefore the varied factor is causally irrelevant.

Following the terminology of John Stuart Mill, this method of identifying causal factors is widely called the Method of Difference. The investigator introduces one new factor (A) and as a result he sees the effect (B), which was absent prior to the introduction of the new factor. The method rests on the fact that the isolated factor is the only relevant difference. All other factors are eliminated as the cause because they are present even when the effect is not. When using this method, a scientist identifies the difference among all the similarities—the difference A that stands out as making the difference B. Assuming he has not overlooked a relevant factor or condition, he can conclude: A caused B—and then, upon identifying the causal connection conceptually, he arrives at the generalization: Cases of A lead to cases of B.

Most experiments employ the Method of Difference (with positive or negative outcomes). All of Galileo’s free-fall experiments used this method: He isolated and varied the weight and then the material in order to prove that these properties did not affect the rate of fall, and then he isolated and varied the height in order to establish its relationship to the total time of the fall. He used the same approach in his investigation of free horizontal motion: He introduced differences in a single variable—the initial horizontal speed—and then he measured the corresponding differences in distance traversed during a constant time interval.

Newton too used this method throughout his experimental investigation of colors. He began by introducing a difference in the color of a thread and then he observed through a prism the resulting difference in the thread’s image location. His later experiments directly revealed the causal relationship between an isolated change of color and the subsequent change in angle of refraction. Similarly, in his experimental investigation of reflection, Newton held constant the color of incident light while introducing a change in the color of the reflecting body, and as a result he observed a change in the intensity of reflected light.

Like explicit knowledge of the laws of logic or the law of causality, the explicit statement of the Method of Difference is not known to most people. But just as people know causality implicitly, and (much of the time) think and act on its basis, so they know the Method of Difference implicitly, because it is a corollary of cause and effect. When a child observes that a thing is changed in a single respect (while surrounding conditions are unaltered) and then he sees a change in the thing’s action, he concludes that the first change caused the second. This is how a toddler discovers that a lamp switch causes the light to turn on and off (in this case, the toddler will repeat the action several times, thus eliminating the possibility of a mere coincidence). We all perform such simple “experiments” and use such reasoning throughout our lives. At an advanced stage of knowledge, however, an enormous effort from an ingenious scientist may be required in order to create the crucial experiment that reveals a causal connection. But whether it is a child or a scientist who uses the Method of Difference, the point is that he observes an isolated difference against a background of similarities and then sees its effect.

The other fundamental method used to identify causal relationships is called the Method of Agreement. Rather than seeking to discover the different factor that stands out as leading to the difference in effect, here we seek to discover the similar factor in two or more cases that stands out (against a background of differences) as leading to the similarity in effect.

When using the Method of Agreement, we observe that two or more cases of a certain effect (B) agree in only one relevant antecedent factor (A). The factor A can then be identified as the cause of B; all other factors are eliminated because the effect occurs even when they are absent. And then, assuming that we have formed the relevant concepts properly, we can arrive at a universal truth: Cases of A lead to cases of B.

For example, recall that Galileo compared two pendulums that differed in all the potentially relevant factors except length, and yet the period remained the same. By the Method of Agreement, he concluded that length is the causal factor determining the period. Or, consider an experimental investigation seeking to discover the cause of the final speed of a ball rolling down an inclined plane. The weight, size, and material of the ball, as well as the length and angle of the plane, are varied while a single factor is held constant: the initial height of the ball. The final speed of the ball is always found to be the same, and therefore the height through which the ball descends is the causal factor determining its speed.

Galileo’s study of free fall also illustrates the Method of Agreement. Considered separately, each of his experiments used the Method of Difference; however, when the series of experiments are considered together we see that he varied many factors (e.g., the weight, size, density, and horizontal speed of the body) while holding a single factor constant: The heavy body was always free to fall unimpeded toward the Earth. The observed result was always the same, leading to the generalization that all free bodies fall to Earth with the same constant acceleration.

The same procedure was used by Newton to support the broad generalization that integrated his observations in optics. Many factors differed in his various observations involving white light interacting with prisms, lenses, raindrops, and reflective surfaces. From case to case, the total angular deflection of the light beam, its intensity, and the distance through which it traveled all changed; furthermore, sometimes the light was refracted through glass or water, whereas other times it was reflected and traveled through only air. Nevertheless, a similarity unites all these very different instances: One initial factor remained constant—the light was white; and one aspect of the outcome remained constant—the white light was decomposed into colors. By the Method of Agreement, this vast range of data led to the conclusion that white light is a mixture of elementary colors that can be separated by means of refraction or reflection.

The methods of Difference and Agreement often work hand in hand. Typically, one observes a difference making a difference, thus isolating some X factor as causal; and then one observes that the X factor alone is present in two or more of the observed cases of the effect. Of course, it can happen in the reverse order, when one first identifies a causal similarity against a background of differences, and later observes that removing the cause eliminates the effect. In either case, the two methods are used to complement and confirm each other. However, this conjunction of methods is not always necessary. Either method alone, properly performed, is conclusive.

In the experimental work of Galileo and Newton, one striking feature is the speed with which they arrived at generalizations. Galileo did not conduct a laborious study of a hundred different pendulums or projectiles before reaching his conclusions; Newton did not find it necessary to experiment with dozens of different prisms, lenses, or light sources. It is obvious that the validity of induction has nothing to do with the number of instances one observes. We can now see what induction does depend upon: the grasp of similarities and differences in a causal context. When using the Method of Agreement, this may be possible on the basis of just two cases; when using Difference, only one instance of the effect may be needed. Always, what counts is the grasp of a uniquely effective similarity or difference. Here the process of generalization parallels the process of concept-formation; one does not need to see a hundred tables in order to form the concept “table”—one can grasp the necessary pattern of similarities and differences merely by seeing two tables in contrast to a chair.

In concept-formation, the grasp of differences and similarities in relation to each other is the starting point and base of every concept; it is essential at every level of the hierarchy. Generalization is a form of conceptualization: It is measurement-omission applied to causal connections. Just as a concept, through measurement-omission, integrates an unlimited number of particular existents of a certain kind into a single word; so a certain union of concepts integrates through measurement-omission an unlimited number of particular causal sequences of a certain kind into a single proposition that subsumes them all: a generalization. Therefore the process of generalizing also rests on the grasp of differences and similarities; above the first level, it proceeds by the Methods of Difference and Agreement. The very type of relationship that makes possible concept-formation is what makes possible the grasp of causal relations and thus of generalizations.

The validity of the Methods of Difference and Agreement should be regarded as beyond dispute. The correct application of the methods may be difficult in a complex case, but the methods themselves follow as immediate implications of the law of causality. Yet their validity has been widely attacked and rejected by contemporary philosophers of science. The most common criticisms derive from a failure to understand the two essential components in the inductive proof of any high-level generalization: the role of perceptual evidence and the role of the conceptual framework.

First, it is crucial to grasp Newton’s point that some evidence—grounded in observation—is required before one is entitled to suggest a factor as a possible cause. In the absence of such evidence, the assertion of a possibility must be dismissed without contemplation. Otherwise, skepticism is unavoidable.

Today, intellectuals manufacture arbitrary possibilities as a counterfeiter manufactures illegitimate money. They are actually worse than counterfeiters, who at least acknowledge the existence of and try to imitate real money; the intellectuals who traffic in the arbitrary deny the existence of real knowledge. For example, the authors of a standard text on scientific method have this to say about Galileo’s law of constant gravitational acceleration:

[T]he evidence for the acceleration hypothesis always remains only probable. The hypothesis is only probable on the evidence because it is always logically possible to find some other hypothesis from which all the verified propositions are consequences.21

This is offered as a bald assertion. The authors do not even suggest another “logical possibility,” much less give evidence in support of one; rather, they imply that the reader is free to dream up any “possibility” he wishes without the responsibility of citing evidence. It comes as no surprise when these authors ultimately conclude that Methods of Difference and Agreement “are not therefore capable of demonstrating any causal laws.”22

The epistemological state of a scientist is not what such skeptics would have us believe. When a scientist confronts some aspect of nature, he does not do so as a helpless newborn; he enters his investigation armed with a vast context of knowledge that precisely delimits the possibilities. A factor qualifies as relevant to his investigation only if there is some reason to suspect that it plays a causal role, a reason based on the generalizations that he has already reached, which are ultimately reducible to evidence given directly by the senses.

This brings us to the second criticism that is often brought against the Methods of Difference and Agreement. Perversely, some philosophers charge that the methods are invalid because they depend upon a prior cognitive context. For example, while discussing these methods, the above-quoted authors write: “This canon [requires] the antecedent formulation of a hypothesis concerning the possible relevant factors. The canon cannot tell us what factors should be selected for study from the innumerable circumstances present. And the canon requires that the circumstances shall have been properly analyzed and separated. We must conclude that it is not a method of discovery.”23 According to this view, the methods would qualify as methods of discovery only if they could be applied mindlessly and by rote. The need of an “antecedent hypothesis” and a “proper analysis” is what invalidates them—in other words, the need of knowledge and thought is what invalidates any discovery process.

These criticisms form a one-two punch against inductive inference. The skeptic leads with the claim that there are countless possibilities that cannot be eliminated, and therefore we cannot know any general truths (except this generalization itself, which is treated as an unquestionable absolute). When a rational man answers that the possibilities are delimited by his framework of prior conceptual knowledge, the skeptic asserts that such use of one’s conceptual framework is outside the realm of logic. His underlying assumption is that one’s conceptual framework is necessarily subjective, that is, it was not derived from sensory evidence and its elements cannot be reduced back to such evidence. Thus, ultimately, the skeptic’s attack on the validity of induction is based on his subjectivist view that concepts themselves are detached from reality.

A rational man must counter the skeptic’s first punch with the principled rejection of the arbitrary; he must counter the second with an objective theory of concepts and generalizations. All thought begins with perception; without our only direct contact to existence, there is nothing to think about. Our entire interconnected framework of concepts can be nothing else but integrations (ultimately) of percepts. This is the cognitive whole that the scientist uses to delimit the relevant factors in his investigation and guide his analysis; it is precisely what enables him to use the Methods of Difference and Agreement—and what makes his reasoning valid.

Induction as Inherent in Conceptualization

Concepts are what make induction possible and necessary.

Consider the concepts of “horizontal” and “vertical,” which played such a crucial role in the development of Galileo’s kinematics. Although relatively simple, these concepts are integrations of earlier concepts. We start with concepts of specific directions that we can indicate by pointing. Thus we begin with “up” and “down” and only reach the abstraction “vertical” at a much later stage; similarly, we start with concepts of specific horizontal directions (e.g., forward or backward, toward the sunrise or the sunset) long before we abstract to form the concept “horizontal.” It is our quest to understand the actions of bodies that gives rise to the need for these more advanced abstractions. After observing that heavy bodies fall and light bodies rise and that such spontaneous motion does not occur in other directions, we eventually recognize that “up” and “down” are similar in an important respect and fundamentally different than any perpendicular direction.

Clearly, Galileo’s discoveries would have been impossible without these wider abstractions; his law of constant acceleration is a generalization about free vertical motion and his law of constant speed is a generalization about free horizontal motion. Moreover, the formation of these concepts was itself an enormous step toward the discovery of the laws. The concepts were formed on the basis of grasping an essential difference in the way bodies move vertically versus the way they move horizontally. Armed with these and other key concepts (e.g., “friction,” “speed,” “acceleration,” “parabola”), Galileo could then ask the specific questions and formulate the quantitative answers that he found by means of his experiments.

Key concepts played a similar role in Newton’s optics. The question that led to his first major discovery was: Is blue light refracted more than red light? Obviously, the question is impossible without the concept “refraction.” This concept is an integration of all cases in which light bends at an interface between two materials. Such an action can depend on only the nature of the light and the materials, which are also identified in conceptual terms (e.g., all light of a particular color, all glass of a given type). Thus when Newton varied only the color of light and saw the subsequent change in the angle of refraction, he simply identified his observation conceptually in order to reach the generalization that blue light is refracted more than red light.

Or consider his investigation of why bodies appear colored. Newton knew that when sunlight shines on a body, the light is reflected, absorbed, or transmitted. Without conceptualizing the various possible actions of light, he could not have understood how colors are separated by reflection. With these concepts, his experiments led inexorably to the conclusion that colors arise when part of the spectrum is reflected more strongly than the rest, which is absorbed or transmitted.

When we have a properly formed concept, one that unites concretes by clearly defined essentials, we are in a position to know at once when an attribute discovered by study of some instances is applicable to all instances. Using Ayn Rand’s analogy between a concept and a file folder, we can say that such generalization about the referents of a concept is implied in the very act of placing each new item of knowledge in the file folder. By doing so, one is claiming: “This is now part of my knowledge of X; in other words, this is true of all Xs—including the vast majority of them that I will never encounter.”

If a person refrained from induction, his words would not designate concepts at all; they would be mere sounds. In the case of first-level concepts based on perceptually given similarities, he could apply a name to some referent he encountered, but without induction he could not apply any of his prior knowledge about such referents. So the name would not serve any cognitive function; he would remain in the state of an ignoramus confronting each new object from scratch.

To illustrate this point, consider an infant who begins with the implicit definition of man as “a thing that moves and makes sounds.” By further observation of specific men, he eventually discovers that when men make sounds, they are communicating messages to each other, and that when they move, they are doing it purposefully, in order to satisfy various desires. Now this child goes to the next block and sees more men who move and make sounds. However, when he is asked: “Why do you think they are moving and making sounds?” he answers: “I have no idea; I have never seen these particular men before.” This would be an example of holding a concept minus induction.

Imagine the other bizarre behavior of such a child. His parents would continually hear responses such as: “I didn’t know that this particular glass would fall when I dropped it”; “I didn’t know that this particular fire would burn me”; “I didn’t know that this particular water would quench my thirst”; and so on.

We would not conclude that this child was cautious about leaping to generalizations—rather, we would conclude that he suffered from some incapacitating mental disease. He would be the missing link between animals and men, able to apply a word, but uselessly, because he could not apply the knowledge earlier gained about the things named by the word. Thus his words would be nothing more than concrete symbols associated with a few observed particulars, and therefore he would entirely lack the human capacity for thought.

A concept is a commandment to go from some to all—it is a “green light” to induction. The rules of the road mandate that we move forward through a green light; the rules of human cognition mandate that we generalize among the referents of our concepts. When we do so, we move forward by means of a unique mechanism (the conceptual faculty) not possessed by other animals, an integrating mechanism designed to take us from particular instances to universal generalizations.

When Newton discovered that the Sun, Earth, Moon, apples, and comets all exert a specific type of attractive force (“gravity”), he was logically compelled to ascribe this force to all bodies. By doing so, he integrated astronomy and mechanics and ushered in the modern scientific era; he was able to explain the planetary orbits, the fall of terrestrial bodies, the ocean tides, the motion of comets, the shape of the Earth and the motion of its spin axis—in short, he was able to present an intelligible, integrated universe for the first time. His generalizations followed precisely because he was able to relate the new concept of “gravity” to the whole framework of prior knowledge.

In optics, we have seen how Newton discovered that ordinary white light is a mixture of colors that form a “spectrum”—that is, an ordered array made up of red, orange, yellow, green, blue, and violet. Like “gravity” in mechanics, the concept “spectrum” is a key integration that made possible many further discoveries in optics. For example, when scientists found that heat existed beyond the red end of the spectrum and photographic paper was blackened beyond the violet end, they were compelled to extend the concept to include non-visible infrared and ultraviolet light. This, in turn, was a key step toward the discovery that light is an electromagnetic wave, a discovery that achieved the grand-scale integration of optics with electromagnetism. Such is the power of concepts and the inductive reasoning that they necessitate.

Of course, the truth of our generalizations depends on the validity of our concepts. An invalid concept is a “red light” to induction; it stops the discovery process or actively leads to false generalizations.

Recall the concepts of “natural” and “violent” motion in Greek physics. On the one hand, these concepts made a fundamental distinction between motions that are in fact similar; for example, smoke rising in air is regarded as “natural,” but wood rising in water is regarded as “violent,” and a ball swung around on the end of a rope is moving “violently,” but the moon orbiting the Earth is moving “naturally.” On the other hand, they group together motions that are very different; for example, rising smoke and falling rocks are both moving “naturally,” and a ball swung in circles and another ball with constant horizontal velocity are both said to move “violently.”

Such concepts cannot be reduced back to observed similarities and differences. They are juxtapositions rather than valid integrations, and therefore it is impossible to reach true generalizations among their dissimilar referents. When the Greeks tried to generalize, they were led into a series of falsehoods, such as: “The violent horizontal motion of a projectile is caused by the air pushing on it”; “Celestial bodies are made of an unearthly material called ‘ether’ that moves naturally in circles around the Earth”; “In the absence of external forces, all heavy bodies move toward their natural place at the center of the Earth”; and so on. The science of physics was stopped at this red light until the concepts of natural and violent motion were rejected.

The concept “impetus” provides an excellent example of a more complex, mixed case. As we saw, this concept was based on a false premise, but it was nevertheless a first attempt at a valid and important integration. Buridan was correct in thinking that something about a freely moving body remains the same in the absence of frictional forces, and dissipates as a result of such forces. However, because he thought that a force is necessary to cause motion, he misidentified the nature of the conserved property. He proposed an intrinsic attribute of the body that supplies the internal force propelling it, and he called that attribute “impetus.” Since no such attribute exists, all generalizations referring to it are false. Yet physicists found that the facts regarding motion could not be integrated without some such idea, and therefore “impetus” eventually had to be reformed and replaced rather than simply rejected outright. After Galileo identified and eliminated the underlying false premise, Newton was able to grasp the concept of “momentum” that had been out of Buridan’s reach.

Although a valid conceptual framework does not guarantee the truth of subsequent generalizations, the errors of generalization committed by scientists can usually be traced to some inadequacy in their conceptual framework. When scientists overgeneralize (i.e., extend their conclusion beyond its legitimate range of validity), it is often because they lack the concepts necessary to identify important distinctions. Galileo struggled with this problem in several cases. His claim that a circular pendulum is isochronal for all amplitudes was an unlucky guess on an issue that he could not resolve without the concepts of “infinitesimal,” “limit,” and “cycloid”; when he mistakenly extended his analysis of frictionless motion to rolling balls, it was because he lacked the dynamical and mathematical concepts necessary to grasp the effects of rotation; and when he assumed that his law of constant vertical acceleration applied even at large distances from the Earth’s surface, it was because he lacked the concept “gravity.” But such missteps are corrected in the normal course of pursuing science by means of a rational method. Once the requisite concepts were formed, scientists were immediately able to qualify Galileo’s conclusions in ways that he could not.

A concept can function as a green light to induction only if it is defined precisely—and, in physical science, the required precision is mathematical. Galileo had to define the concepts of “speed” and “acceleration” in mathematical terms before he could arrive at his theory of motion. A similar development can be seen in optics. Prior to Newton, the topic of colors had been treated in a way that was almost entirely qualitative, and as a result there was very little progress. An essential aspect of Newton’s achievement was that it transformed the subject into a quantitative science. He began by measuring the different angles of refraction for each color, and he ended triumphantly by associating each color with a precisely calculated wavelength in his analysis of the famous “rings” experiment. The cognitive integration necessary to validate a high-level generalization in physics is made possible only because the discoveries and laws are formulated in quantitative terms. Thus progress requires that the key concepts be defined in terms susceptible to numerical measurement. Such measurement is both the primary concern of the mathematician and the primary activity of the experimentalist.

Thus induction in physics is essentially dependent on two specialized methods. Experimentation provides the entrance into mathematics, and mathematics is the language of physical science.

Author’s note: The next chapter of the book focuses on the work of Johannes Kepler and the role of mathematics in scientific induction. Following that, I describe the step-by-step process by which Newton discovered the laws of motion and gravitation, and thereby revealed the nature and power of the inductive method in its full glory.

About David Harriman

Mr. Harriman is the editor of Journals of Ayn Rand and a senior writer for the Ayn Rand Institute. He has worked as a physicist for the U.S. Department of Defense and he has taught philosophy at California State University San Bernadino. He has lectured extensively on the history and philosophy of physics. He is currently writing two books: one demonstrating the influence of philosophy on modern physics (The Anti-Copernican Revolution), and another presenting Dr. Leonard Peikoff’s theory of induction (The Inductive Method in Physics).

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