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Causality and Physical Laws

Daniel J. Castellano (2009-2010)

Part I

Aristotelian Notions of Causality
Formal Cause Elucidated
Early Modern Criticisms of Substantial Forms
Mathematical Astronomy and Galileo's "New Science"
Baconian Induction
Descartes and the Geometrization of Physics
Newtonian Kinematics and the Role of Force
Laws of Nature in Classical Mechanics

Physical science purports to give explanations of physical phenomena, or more colloquially, it claims to explain why physical events occur and how physical objects work. On one level, such knowledge can have purely practical value. If we know how the natural world works, we can predict events and prepare accordingly, or more creatively, we can make use of natural principles to create devices and processes that will make our lives easier or more convenient, or fulfill various tasks we may conceive. Yet nearly all scientists are also aware of a further motivation for studying the physical world: the desire to understand why things are the way they are. We believe we have a deeper understanding of things when we can explain phenomena in terms of more fundamental principles and realities, and this intuition perhaps grasps the most generic notion of causality.

Everyone has a common sense notion of cause and effect, where one entity, the effect, is considered to be somehow the product or result of another entity, the cause. Despite the ubiquity of this intuition, the concept of causality can be difficult to define coherently. Should we speak of physical objects as causes, or rather the powers or capacities (e.g., momentum, energy) they possess? Should we speak of their effects as objects (e.g., a ball that is moved) or as events? These and many other considerations have led several eminent philosophers, such as Hume and Russell, to consider the notion of causality to be fundamentally unsound. Notwithstanding these reservations, most scientists continue to conceptualize their field of study in the language of cause and effect, and believe that they really are uncovering the why and how of physical phenomena.

In the case of physics, the most fundamental of the natural sciences, everything is explained in terms of particles and force fields. The explanatory power of physics lies in its ability to summarize all the diversity of physical activity in terms of four fundamental forces, which are characterized by mathematical equations of universal applicability. These equations are commonly called "laws of physics," following Newton's schema of "laws of motion" and a "law of gravitation." Throughout the Age of Reason and into the nineteenth century, newly discovered mathematical principles in natural science were characterized as physical "laws." Among these were the Law of Conservation of Energy (and of Momentum), Coulomb's law, Maxwell's laws of electrodynamics, the laws of thermodynamics, and so on.

The explanatory character of these "laws of physics" has been a matter of some philosophical dispute. It is not clear whether they are to be regarded as causal agents or explanatory principles, or if they are merely generalized descriptions of phenomena. The last possibility is positively contradicted by the attitude of most physicists, who insist on a nomological (law-giving) aspect to some or all of these principles. If that is the case, it begs the question: how can mathematical principles cause physical events? If these principles are not logically or metaphysically necessary, how is it determined which equations are laws of physics and which are inert abstractions? Who or what enforces or imposes these laws? These and other questions will be the subject of our present discussion, which will require us to examine the concept of causality more critically.

Aristotelian Notions of Causality

Before Galileo and Newton revolutionized the physical sciences, the study of physics was essentially a commentary on Aristotle, who gave a philosophical or metaphysical treatment of natural philosophy. While it was a grievous mistake, long ruinous to empirical science, to invoke metaphysical essences as physical causes, it was fully appropriate to treat causality as a philosophical question, so we recapitulate the Aristotelian treatment here.

Beginning with our common sense intuition of cause, Aristotle found that the term was used in a variety of ways. He used the term aition, which means "blame" in Greek, so the general Aristotelian notion of cause is something that is "to blame," or amorally speaking, accounts for the presence of some thing or state of affairs. The concept may be derived in part from the jurisprudential notion of culpability or responsibility that can apply to volitional acts. Indeed, one of the modern criticisms of the notion of causality is that attempts to project volition onto nature. For now, we will consider the four Aristotelian types of cause at least as logical possibilities.

Aristotle's notion of cause is object-oriented. We begin with an object, such as a marble statue, and consider all the principles that are responsible for that object's current state of existence. This follows our basic intuition of the concept of causality, for when we speak of a cause, we mean the cause of something, where that something is either an object or a particular determination or state of an object. In the example of the marble statue, we find four causes:

Material cause: The stuff, in this case marble, of which the statue is made. Without it, the marble statue could not exist or persist.
Formal cause: The synthesis of qualities or properties, in this case the shape or figure, that makes it one kind of thing, a statue, rather than some other kind of thing or undifferentiated matter. If it should lose the form of a statue, it would cease to be a statue.
Efficient cause: The prior agent that brought the object or state of affairs into being, in this case the sculptor. This is our modern notion of causation, but note that Aristotle identifies the agent, not the act or the event of acting, as the cause.
Final cause: The purpose, end or function of the object or effect. When the agent is an intelligent being, like a sculptor, the final cause is the intent, such as to adorn a house, or to admire. Yet even nature can have final causes, if we acknowledge that natural objects have tendencies in certain directions.

Of these four causes, only the material and efficient causes are recognized by modern science. The formal cause and final cause are regarded as superfluous or non-existent. The final cause is not of immediate concern to us, but suffice it to note that the final cause is just the form of the agent's act. The alternative to final causes is not atheism, but blind chance. If we acknowledge that an act has a certain character that makes it produce certain effects rather than others, then we have admitted a final cause. Most who deny final cause fail to understand that it is not an anthropomorphic concept. The intentional act of the sculptor is just a special case of this general concept; his intention accounts for why he made a statue rather than something else.

Although classical mechanics was interpreted in a non-teleological and even atheistic manner in the eighteenth and nineteenth centuries, it certainly was not a belief in blind chance. On the contrary, the physics of Newton was so strongly deterministic that efficient causes were deemed to sufficiently account for all effects, rendering other causes superfluous. Aristotle, by contrast, had a remarkably lax attitude regarding the natural effects, characterizing them as what happens usually, not necessarily all the time. Even Newton himself used such mild language in his laws, saying that a body in motion "tends" to stay in motion. Yet the mathematical rigidity and universality of his laws seemed to bind everything with iron clad force. What need, then, for a formal cause?

Formal Cause Elucidated

We should account for why Newtonian efficient causes were believed to supplant the more metaphysical formal causes. At first it seems strange that physics should pretend to decide a metaphysical question, but we can see the reason for this overlap once we understand the Aristotelian notion of a substantial form as applied to motion or change. A metaphysical form or essence is that which makes an entity the kind of thing it is. When an essence is considered with respect to the operations of that entity, it is the entity's nature or physis, which is its principle of motion or change. One can easily appreciate how a metaphysical concept defined as the most fundamental principle of motion might be employed to develop a theory of physical dynamics. Indeed, dynamics remained the subject of abstract philosophizing until Galileo famously proposed a "new science" of dynamics that was grounded in systematic quantitative observations, contradicting many of the predictions of Aristotelian physics.

The success of Galilean and Newtonian dynamics ultimately forced Scholastic philosophers to recognize that ratiocination about metaphysical forms could not serve as a science of dynamics. Francisco Suarez recognized that Aristotle's Physics was really a metaphysical treatment of natural phenomena, what we would today call philosophy of science rather than science itself. As Suarez and the Enlightenment philosopher Jeronimo Feijoo explicitly recognized, most of Aristotle was still sound as metaphysics. It is not our concern here to defend the Aristotelian corpus, but only to examine the notion of formal causes in light of early modern criticisms.

Consider the ancient paradox of Heraclitus, that we cannot step twice into the same river, since each time it will have different water. Parmenides by contrast, held that nothing can change, for that would involve a paradox of being becoming non-being. If we think a form (e.g., "river") is nothing more than material, then nothing persists, and all is flux, contrary to all appearances. If we consider things only as what they are defined to be (essential form), then all change is an illusion. Aristotle, being more of an empiricist, accepted the data of experience that things really do persist, yet there also really is change. The only way he saw out of the paradoxes of Heraclitus and Parmenides (the latter being far more formidable than what I have outlined) is to assert an essence with twin principles of matter and form, neither of which actually exist without the other.

Pace Heraclitus, we can step in the same river many times, for the river retains its form, and thereby persists. It matters not which water drops fill it in a given moment, as long as the quantity and location of the water is such that the river retains its form. Similarly, I remain the same person even though I am constantly replacing cells, water and oxygen. It is the being considered with respect to form that persists. Change is accounted for as the same matter may realize different forms, perfectly or imperfectly. Heraclitus and Parmenides are harmonized by recognizing that a thing may both change and not change at the same time, when it is considered with respect to its matter or form. Those who would reject Aristotelian hylomorphism and its equivalents must come up with another solution to the problems of change and permanence. Most scientists, of course, simply ignore philosophical problems altogether, and give ad hoc metaphysical interpretations without worrying about consistency.

Aristotelian matter is not the physicist's "matter" (which is actually a substance, with both matter and form); rather, it is substance abstracted from all its determinate properties. In actuality, natural substances cannot exist without having some determinate properties, so this is a virtual distinction. The reason for making this ontological abstraction is to understand which aspects of substantial existence makes possible change and permanence, as outlined above.

Complementing the material principle of a thing is its formal principle, which makes a thing the kind of thing that it is. The intellect may conceive of perfectly realized forms, such as roundness and equality, even though in physical reality we may never find objects that are perfectly round or perfectly equal. Ordinarily, ideal forms are only imperfectly realized in matter. Man may be defined as the "rational animal", but birth defects or other maladies in particular men can inhibit or suppress the expression of rationality. A horse may be a quadruped, but it does not cease to be a horse if it loses a leg, nor does a mammal cease to be a mammal after a mastectomy. This is because the particular properties, such as having rationality, four legs, or teats, are consequent to the form being manifested in matter. The accidental loss of these features does not necessarily annihilate the entity, but only means the matter less perfectly realizes the form. More concretely, an animal develops certain features because it has a certain nature encoded genetically. Its loss or failure to develop certain features does not abolish its unity.

If the examples of biology are unconvincing, since no form is ever realized perfectly, and indeed, according to modern biology, there is no discrete distinction between species, this is not inconsistent with original Aristotelianism. Aristotle himself was well aware of an apparent continuity between vegetative and animal life, for example, but he did not require that all forms had to be perfectly actualized. For this reason, he defined natural principles in terms of what usually occurs, allowing for exceptions.

Newtonian physics would not allow any exceptions to its laws, so the notion of imperfectly realized forms was superfluous, since reality was believed to be fully determined by mechanical properties, leaving no room for indeterminacy or fuzziness. There seemed to be no need to appeal to the substantial form in a top-down formal causality, when everything could be fully explained from bottom-up efficient causality. Strong determinism in physical mechanics seemed to render other types of causality irrelevant.

Interestingly, in modern subatomic physics, substantial forms are perfectly realized (unless we regard the mere fact of individuation as an imperfection): an electron has precisely the same essential properties as every other electron, with the exact same rest mass and charge. This is why some of the more philosophically thoughtful physicists tend to espouse a sort of Platonism, since ideal forms do in fact seem to inform reality.

From the example of particle physics, we can see in what sense a substantial form is a "principle of motion." By "motion" we really mean physical change, not mere spatial motion. A particle responds to various forces and interactions according to the properties of its type. Simply put, it does what it does because it is what it is. This is all that is meant by making an essence or substantial form the most fundamental principle of motion for a thing. A thing is only able to act by virtue of being what it is; being is metaphysically prior to doing.

To modern eyes, this all seems terribly abstract, but this exposition will be relevant to showing how the notion of formal causation might be applied to so-called laws of physics. Before we do so, however, we should examine some of the early modern criticisms of the notion of substantial form that led most scientists in the Age of Reason to abandon formal causality as an explanation of physical phenomena.

Early Modern Criticisms of Substantial Forms

The primary reason for abandoning formal causes, as we have said, was the belief in a fully deterministic mechanics of efficient causation, rendering other causes superfluous. However, as Benjamin Hill has discussed ("Substantial Forms and the Rise of Modern Science," The Saint Anselm Journal 5.1, Fall 2007), there were also positive arguments against the notion of substantial forms, which we will review here.

As we have noted, in real life forms are rarely actualized perfectly. This is eminently the case in biology, where mutant animals can lack features that are characteristic of their species. If the substantial form is the cause of the physical activity of an animal, how can it, as an abstract universal, operate differently in different individuals? Are we to attribute these deficiencies to the "prime matter" abstracted from all form? Yet prime matter is perfectly uniform! In fact, we explain such monstrosities in terms of the efficient causation of a particular circumstance. It is not at all clear how an immutable substantial form can account for imperfect manifestations.

Hybrid animals would seem to partake of two distinct substantial forms. In principle, they could be repeatedly cross-bred to make a three-quarters breed, a seven-eighths breed, and so on, implying a continuity between substantial forms. This argument would later be buttressed by the evidence for transmutation of species or evolution, implying that there are no discrete forms for each species.

Ironically, though biology is the archetype for our notion of discrete taxonomic classification of species and genera, it is in fact offers rather poor evidence of discrete essentialism. Late Scholastics, most notably Francisco Suarez, allowed that material circumstances could prevent a seed from properly imposing the correct form. In terms of modern genetics, we would say that the DNA was not transcribed correctly or there was some error in prenatal development. This appeal to material circumstances does not make materia prima a principle of motion, for the "matter" in question is determinate matter with some form (e.g., of some chemical or cell body), so it is really a complete substance. Since biological creatures are composed of chemical substances, a circumstance (e.g., cosmic radiation) that alters these substances can prevent the formation of the creature according to its natural template. The continuity of species would seem to be much more problematic to the notion of substantial form in biology. Yet even here, at worst we would have to replace the notion of a substantial form for each species with a substantial form for each individual.

Another criticism of substantial forms came from John Locke in his Essay Concerning Human Understanding. Taking an empiricist stance, Locke noted that we never observe the substantial form directly, but only the properties from which we define the form. In fact, the only basis for classifying objects into species is their partaking of similar properties. Yet no property of a thing is intrinsically privileged, so there is no basis for choosing some properties as "essential" to a species, while regarding the others as merely "accidental." Without any empirical basis for distinguishing essential and non-essential properties, our species are just arbitrary classifications. Even if there are real substantial forms, we have no way of knowing what these are, so they are useless for classifying natural objects, much less explaining natural phenomena.

Locke's argument does not disprove the existence of substantial forms, but contends that the "real" forms, if they exist, cannot be reliably known, since we have no way of determining which properties are essential. If we cannot know which of our forms are real or merely nominal, they are useless for the investigation of nature. Locke takes it for granted that the only way to learn anything about nature is through empirical observation; this epistemological shift was a central reason for the abandonment of forms and other metaphysical accounts of nature.

Another line of argument came from alchemy, with its discovery of "mixtions," which we would call chemical compounds. Two chemicals are combined to create a new chemical, whose properties are not simply a mixture or blending of the properties of the two reagents. The compound does not seem to be a hybrid of two substantial forms, but is its own brand new form. Since submolecular physics was not understood at this time, it was supposed that the compound was simply a mixture of the two elements of which it was composed, hence the name "mixtion." This made it seem as though the substantial forms of the elements were "corrupted" into something else.

Modern molecular physics offers the solution to this problem, and perhaps provides the clearest example of substantial form in nature. The compound may consist of the same fundamental particles (protons, neutrons, and electrons), but configured in a different way so the particles are in different bound orbital states. The fundamental particles retain their intrinsic natures, in the sense that the orbital structure of the compound can be computed from the properties of electrons and nuclei, yet the behavior of the particles in their bound state is different from other bound states or from the free particle state. The compound, then, is not merely the sum of its parts, but exhibits novel properties that cannot be found in its constituents. This is not simply a large-scale property arising from the sheer number of particles (such as thermodynamic and solid-state properties), but from a fundamental alteration in the behavior of each particle due to its relation to other particles. The compound has its own substantial form, and every other molecule of that compound will exhibit the same form. Chemical compounds, it would seem, are examples of "real" substantial forms such as Locke sought. We can define essential properties in terms of the constitution and configuration of fundamental particles.

There are other possible criticisms of substantial forms, but the ones cited suffice to outline the basic limitations of this concept. Forms are difficult to know with any certainty, are of dubious utility, and it is not altogether clear how they can be responsible for change in a substance, and how we can reconcile this with efficient causation. Nevertheless, we have seen some apparent examples of substantial forms in modern physics, so the notion of formal causality may have something to contribute to understanding the nature of physical laws. Like substantial forms, physical laws are abstract idealizations that somehow are able to act as principles of motion in the real world.

Mathematical Astronomy and Galileo's "New Science"

The real problem with substantial forms is not their metaphysical plausibility, but our ability to identify them a priori and thereby explain natural phenomena. As Locke rightly pointed out, in practice our identification of substantial forms comes through observation and classification of phenomena. While Locke despaired of ever arriving at knowledge of substantial forms, he pointed at a promising way to discern which forms are physically real: systematic, empirical observation of nature. This epistemic shift would restore physical science to its empirical roots, after having long lost its way in abstract ratiocination. This is not to say that no empirical science was done before the early modern period - far from it. Technological development in the Middle Ages was aided by an understanding of simple machines and Archimedean statics, following empirically derived principles that were sometimes in tension with the predictions of natural philosophy. Astronomy (then called astrology) had developed highly sophisticated mathematical models of observed phenomena, and it was one of these models - Copernican heliocentrism - that brought into the fore the rift between observation and philosophical speculation.

Before Copernicus, geocentric astronomers had modeled celestial phenomena according to conceptually distinct mathematical models, as was computationally convenient. One method was that of epicycles, where observed deviations of a planet from uniform circular motion around the earth were accounted for by the mathematical supposition that the planet moved in a mini-orbit (epicycle) centered around its main orbit, resulting in a cycloid motion. The method of epicycles was generally combined with that of elliptics, which slightly shifted the center of the orbit away from the earth, in order to better comport with observation. Another, less geometrical method was the use of logarithms to model the observed motions of the planets. In none of these methods did astronomers suppose they were describing physical reality. These mathematical devices were all merely computational conveniences, while physical reality was defined by natural philosophy. In the version of Ptolemaic cosmology current in Copernicus' time, it was believed that the planets moved in concentric crystalline spheres, invisible yet solid, all centered around the earth. The rotation of these spheres accounted for the motion of the planets. Epicycles could not be taken literally, since that would involve a planet moving independently of its sphere, thereby breaking through it, and it would also involve it having two natural motions, which was considered impossible.

Most natural philosophers thought it impossible for there to be two natural motions in a planet because planets were considered to be simple natural bodies, which could have only one substantial form or principle of motion. This supposition involves twin errors, both of which would be exposed by Galileo. First, there is an assumption that the celestial bodies beyond the moon are of a different, more perfect nature than that of the earth. This error was grounded in the limitations of human observation, which was on too small a time scale to discern the mutability of celestial bodies and their motions. The second error, which is more serious, was to suppose that a high-level metaphysical abstraction, the substantial form, directly corresponded to observable natural motions. A substantial form is indeed a "principle of motion," but at only at a fundamental metaphysical level. There is no reason why the form of a thing should not entail complex or composite motions on a phenomenological level. The error of the Aristotelian natural philosophers was to think that natural phenomena were immediate manifestations of substantial form, so that once you defined the form, you specified all that a thing could do. This essentialist physics was no invention of the Scholastics, but is found in Aristotle himself. It persisted in all his commentators, be they pagan, Muslim or Christian, to the extent that they remained true to the original. Even before Galileo, however, the Mertonian Scholastics were developing physics beyond Aristotle, albeit still operating at a speculative level.

When Copernicus first published his heliocentric theory in 1543, it was actually received positively in some quarters, including some prominent Catholic universities in Spain, but this was because it was interpreted as a purely mathematical model. Copernicus offered no sound physical basis for supposing that the earth really does move. He only showed, at best, that it was computationally more elegant to posit the earth at the center of the solar system, yet even this contention was limited by his use of corrective epicycles and his failure to show substantially improved correspondence with observation. The Jesuit astronomer Christopher Clavius expressed the general sentiment of the time when he criticized heliocentrism as a physical theory, yet saw fit to use Copernicus' mathematical computations as a reference, for there was no expectation that a mathematical model ought to comport with physical reality.

The prospects of heliocentrism improved slightly when Tycho Brahe's (1546-1601) detailed observations showed that the geocentric model could not adequately account for actual celestial motions at high precision. Tycho himself, however, postulated only a quasi-heliocentric model, with the sun and moon orbiting the earth and the other planets orbiting the sun. This avoided the philosophical and theological problem of accounting for the motion of the earth, but it contradicted the notion of crystalline spheres, since there were now overlapping orbits. The Tychonian model was accepted by some astronomers, including Jesuits, who began to acknowledge that the celestial bodies were somehow able to float through the heavens as through a fluid.

Johannes Kepler (1571-1630) made use of Brahe's observations to give an elegant geometrical account of heliocentrism, positing elliptical orbits instead of Copernicus' circles. He found that he could predict celestial motions with greater accuracy than with the prevailing geocentric models, and without cumbersome ad hoc epicycles. He discerned an elegant principle whereby a planet always swept the same area of an ellipse, with respect to the sun as a focus, in a given amount of time. This meant that planets moved faster near the sun than they did when away from the sun, contradicting the philosophical supposition of uniform natural motion. Since the substantial form did not change, it was supposed that the natural motions could not change. Again, natural philosophers had tried to glean too much physical detail out of a high level metaphysical abstraction.

Despite the evident advantage of Kepler's model in its ability to model existing data and to predict future occurrences, a geometrical model alone could not give a truly physical account of celestial motion. Aristotelian natural philosophy, whatever its deficiencies, at least attempted to explain physical dynamics in terms of intelligible concepts corresponding to physical agents. In other words, it gave an account of physical causes, be they formal or efficient. Kepler's geometric model was just a generalized description of the data; a highly elegant one, no doubt, but what reason was there to believe that his model was anything more than a calculating tool? Astronomers were long accustomed to using models that contradicted natural philosophy and each other if taken in a physically literal sense, yet nonetheless gave accurate results. Why should they believe Kepler's model should be taken literally?

Consider one of Kepler's principles or laws, namely that the square of the orbital period is proportionate to the cube of the orbit's semi-major axis. What is the physical reason that celestial bodies should obey this law? Is it something inherent in their constitution, or is there some cosmic force to which they must all submit?

Kepler discovered his "laws" by seeking harmonies in nature, out of his conviction that the Creator had constructed the cosmos according to intelligible principles that the human mind, the image of God, could discern. It was no accident, then, that he called these mathematical principles "laws," for he had in mind a Lawgiver. Nonetheless, he saw the need for a physical agent to execute these laws. Kepler speculated that a power in the sun, which he characterized as an immaterial "species," moved the planets. This power was weaker at greater distances, since it was more dispersed, causing the planets to move more slowly when further from the sun. He further conjectured that the sun's rotation about its own axis might create sufficient magnetism to keep the planets in its grasp. Analogously, the earth imparted its own immaterial species onto the moon, powering that satellite's motion.

Kepler's attempt to give a physical explanation of celestial motion was highly speculative at best. Without a well-defined system of mechanics or kinematics, his theory was little better than a mathematical model. Still, Kepler insisted that his model was a literal description of the actual trajectories of the planets, even if he could not fully account for why they behaved this way.

More importantly, Kepler claimed to disprove a thesis of natural philosophy with a mathematical model of observational data. In the Ptolemaic system, planets were supposed to move in circular orbits, yet they swept equal areas in equal time with respect to an off-center point in space called the equant. Kepler found it odd that a planet should behave uniformly with respect to an arbitrary empty point in space. His own model, by contrast, had planets sweep equal areas using the sun as a vertex. This made more intuitive physical sense, as it suggested that the power of orbital motion was somehow centered on the sun, an energetic body that was the source of all light and heat to the planets. Kepler mathematically proved that no combination of circular motion and equants could account for Tycho's observed data. Therefore something had to give way, either circular orbits or uniformity of motion. The abandonment of either of these premises would contradict the philosophical thesis that the substantial form of celestial bodies imposed a single natural motion.

Although he could not conclusively discern the physical cause of celestial motion, Kepler demonstrated the power of mathematical argument grounded in empirical data to indicate facts of physical reality. It could no longer be credibly maintained that "mathematical" astronomy had nothing to say about physical reality, even if it could not yet account for physical causes. Many of Europe's leading astronomers, including those of the Jesuits' Collegio Romano, abandoned the Ptolemaic system, notwithstanding the disdain of the philosophers. However, they nearly all turned to some variant of the Tychonian system, as it seemed absurd that the earth could be moving non-uniformly without its motion being felt or affecting ballistic motion. Other physical arguments against heliocentrism included the absence of observed parallax, and the problems that the earth's rotational motion would seem to entail. Most of these objections could not be overcome without demonstrating a new theory of motion.

Galileo Galilei (1564-1642) believed that he had come up with a physico-mathematical proof of heliocentrism. His observations of sunspots and Jupiter's moons had shown palpable evidence that the celestial bodies were not of an aethereal substance, but were bound by the same physical laws as terrestrial matter. His investigations into kinematics showed how only relative motion, not absolute motion, was sensible to those in a given frame of reference. This might account for why the earth's motion could not be perceived directly. He also showed that objects fell with constant acceleration, suggesting that acceleration, not velocity, was proportionate to gravitational force, as Newton would draw out more explicitly.

Unfortunately, Galileo's arguments for heliocentrism and his nuova scienza explaining kinematics were not as fully developed as he pretended. When defending heliocentrism, Galileo argued that the rotation of the sun powered the motions of the planets. Though he attempted a physico-mathematical demonstration of this theory, it is now known to be empirically wrong. His critics, mostly Dominican professors of natural philosophy, were justified in their skepticism toward arguments based on mathematical models. The agreement of a mathematical model with observation is no proof that the model gives an account of real physical causes.

In order to silence the troublesome Galileo, his critics accused him of heresy before the Congregation of the Index, on the grounds that he had presumed to interpret Scripture according to his particular heliocentric theory. The head of the Index, Cardinal Robert Bellarmine, was knowledgeable about astronomy, having trained under the aforementioned Clavius. He consequently gave a cautious and balanced opinion regarding heliocentrism, allowing that, in the event that the theory was conclusively proved, theologians (and by implication philosophers) would have to revise their interpretations accordingly. However, until that time, heliocentrism could only be advanced as a hypothetical supposition. Theology and natural philosophy "trumped" mathematical astronomy, so to speak, and the burden of proof was on the latter to overturn any thesis held by the higher sciences.

Cardinal Bellarmine's conception of the issue offers insight for us even today, for it forces us to face the question of whether our mathematical accounts of physical phenomena are just generalized descriptions that "save the appearances," or if the mathematical objects in our equations truly correspond to physically real entities. What do we mean by "momentum," "force," and "energy," not to speak of the more arcane quantities of particle physics? Which of these are real physical objects, and which are merely mathematical constructs? Few, if any, physicists, can give a metaphysically coherent account of relativity or quantum mechanics, yet they believe in these theories because the calculations yield the correct results. In other words, they "save the appearances." Yet how do we know, for example, if the general relativistic account of gravity is really a physical explanation, or if so-called "curved spacetime" is just a geometrical model of phenomena?

Such questions, to the extent they were understood by Galileo, were answered by an appeal to the superior precision with which his theory could predict actual observations. If he could not prove that his theory was right, he could at least show that other theories were wrong to the extent that they contradicted newer, more precise measurements. Even the Scholastics admitted contra factum argumentum non est: against a fact there is no argument. The "new science" initiated by Galileo did not pretend to discover ultimate metaphysical truth, but only the best possible theory that explained all actual observations. This empiricist epistemology would be much more formally articulated by Sir Francis Bacon.

Baconian Induction

Sir Francis Bacon (1561-1626) regarded the Scholastic learning of his day as "contentious" verbal quibbling over trifles, aimed more at winning arguments than learning anything of substance. Thus he characterized Scholastic natural philosophy as "sophistical," as if it were mere wordplay. This is an unfair characterization of Scholastic thought, grounded as it was in intelligible concepts linked in a logical structure, not merely a semantic or syntactic game. Nonetheless, Bacon was accurate in his judgment that most natural philosophy relied too much on abstract speculation, grounded only in cursory observations of nature. He instead proposed that useful knowledge of nature should emphasize meticulous, detailed observation of the world, with little or no reliance on abstract generalizations and deductions.

Bacon evinced a supreme distrust in the mental faculties of man, and proposed systematic observation of nature as a corrective to the deceptions of sense and intellect. Instead of judging nature by how it appears to human senses, he suggested that sophisticated instruments and methods of observation be employed in order to see natural objects as they really are. Human intellect, according to Bacon, was too inclined to discern patterns where there were none, constructing correspondences between objects that have no real physical connection. Worst of all, it was too quick to make general judgments without first gradually accumulating sufficient observations to support such judgments. Bacon's aversion to generalization sets him apart from most early modern scientists, but this idiosyncrasy is still enlightening, since it is the basis of his theory of induction, which would have lasting influence on the methodology of modern science.

Bacon had great disdain for deductive logic and the Scholastic custom of expressing all arguments in syllogistic form. This attitude is ironic, considering that the success of modern science in large part depends on mathematics, which is utterly dependent on deductive logic. He also had little regard for what he called "classic induction." According to Bacon's characterization, the induction used by logicians inferred general propositions from just a few observations, and then deduced more specific propositions. For example, one might observe that the stars in the sky twinkle, and infer that all celestial objects twinkle. If we later discover a new kind of celestial object, we deduce a priori that it must twinkle. Naturally, this is a highly hazardous inference if our initial observations were few in number or limited in scope.

What Bacon calls "classic induction" resembles the "inductive syllogism" of Aristotelianism, which simply infers that what is true of every member of a class is true of the class. The inductive syllogism, Bacon correctly observes, cannot yield reliable scientific knowledge unless we were to observe every instance of a class, at which point generalization would not add much to our knowledge. This is why Bacon emphasizes a new kind of induction, which does not generalize beyond actually observed data. This empirical induction was known to Aristotle and the Scholastics, and explored by the Franciscan Roger Bacon in the thirteenth century, but it was Sir Francis Bacon who first posited it as the central methodology of natural philosophy.

Baconian induction differs from classic induction not merely in the number of observations needed to establish a general principle, but in its building up of propositions from the more specific to the more general, rather than the other way around. Bacon never clearly indicated how much observation was needed to establish a natural principle or law, and indeed his own reasoning suggests that it is impossible to arrive at knowledge of any natural principle from a less than exhaustive observation of particulars. This difficulty is the so-called "problem of induction," namely that it is not logically valid to infer knowledge of a universal from knowledge of some particulars. Any group of particulars, however large, is infinitesimal compared to all possible instances of the class, so it is doubtful that Baconian induction could even demonstrate that a natural principle is more probable. This limitation did not seem to bother Bacon, since he insisted that we should never generalize beyond the set of actual observations.

Most early modern scientists did not share Bacon's contempt for generalization, but on the contrary believed that a finite amount of observations could confirm general principles that were often derived from a priori mathematical deductions or intuitive guesses or thought experiments. This was the case with Galileo's principle of uniform acceleration, Kepler's laws of orbital motion, and almost every other great scientific discovery of the early modern era. As William Harvey astutely observed, Bacon's theory of induction did not accurately model how scientists operated in practice. In real life, they relied on intuitive guesses or hypotheses, often substantiated by mathematical deductions or theories of physical agency, and then confirmed these hypotheses by observation. The strength of a hypothesis, however, did not depend solely on observational confirmation, but also on its physico-mathematical theoretical foundation. This is just as well, since no general principle could possibly be established by induction from observation alone; at most one could prove it "saves the appearances."

We must conclude that the Baconian emphasis on experimental induction is only one aspect of the theoretical basis of Galileo's nuova scienza. Another equally important aspect is the notion that mathematical analysis can generate a theory of physical principles, as Galileo tried to show in his dubious proofs of heliocentrism. This latter aspect of the new physics would be more fully developed by Sir Isaac Newton, who applied a new differential calculus to Galileo's principles of motion and relativity of reference frame, producing a mathematically elegant theory of mechanics that agreed with observations to an astonishing degree of accuracy. Yet a mathematical model, we have seen, might merely "save the appearances," as Cardinal Bellarmine and various natural philosophers correctly noted. To this criticism we may add Bacon's warning that a single new observation could easily undermine a supposedly universal principle. This admonition would prove prophetic when applied to Newtonian mechanics, as the supposedly universal laws of kinematics would break down at velocities approaching the speed of light and distances much greater than the terrestrial scale. Both experimental induction and mathematical hypothesizing fail to establish natural principles reliably, so perhaps Bacon's aversion to generalizing beyond the data was justified.

The problem with a truly Baconian physical science, however, is that, in the absence of general principles, we cannot propose physical causes for phenomena. Each particular phenomenon is sui generis, and we dare not classify a group of instances as being of the same kind until we have exhausted the range of possible observations. To suggest that the same kind of agent can cause many different instances of phenomena, we would need to admit the possibility of inferring a general principle from particular instances, but Baconianism all but despairs of this possibility. It would seem, then, that there is little prospect of arriving at a theory of physical causes with a purely empirical science, so that this science is not truly a physics at all, but only a systematized description of phenomena.

Of course, few scientists would accept such an intellectually barren characterization of their work. Most who pursue a career in the natural sciences do so out of a desire to truly understand the "how" and "why" of natural phenomena. This was especially the case in the early modern period, when much science was done by men of independent means, as there was practically no government or commercial support of basic research. These men were motivated primarily by the desire to understand, seeking knowledge either for its own sake or for the betterment of mankind.

In practice, scientists developed theories of natural principles from abstract speculations of a philosophical character, such as those of Descartes, or of a mathematical and logical thought experiment, such as those of Galileo and Harvey. Experimentation served to corroborate or refute the hypothesis. What seemed most convincing about a hypothesis was not its ability to account for existing observations, but its ability to predict the result of an observation not yet made, such as that of a controlled experiment. It is one thing to fit a model to existing data, but to predict future data suggests that the model really does tell us something about reality. However, even here the model may simply be a generalized mathematical description of reality, and the terms of an equation need not correspond to real physical agents.

Descartes and the Geometrization of Physics

Before the acceptance of Newtonianism, René Descartes (1596-1650) was the leading systematizing theoretician of the new physics. Like the other mechanists of his day, Descartes rejected substantial forms as causative principles, so he sought an alternative metaphysical foundation for physics. An amateur philosopher with no technical expertise, he employed his natural genius to develop a crude metaphysics where matter was simply spatial extension, and motion was explained in terms of contact or interaction between corporeal bodies. Heat and cold, moisture and dryness, indeed every known physical quality, according to Descartes (following Galileo and Gassendi), was reducible to the motion, size, shape and arrangement of corpuscular particles. In a word, all physical reality could be explained by geometric properties.

Accordingly, much of Descartes' work on physics was highly mathematical, yet his explanations of fundamental physical causes relied instead on metaphysical theses. Like the Scholastics, Descartes recognized that mathematical accounts can only tell us the correct quantitative relationships among physical entities, but natural philosophy is necessary to give us real understanding of nature. Indeed, he criticized Galileo for relying too much on mathematical arguments to explain a narrow range of effects, rather than establishing the "first causes of nature." In contrast with Bacon's prescription, Descartes began with the most universal truths, from which more specific discoveries could then be understood.

Aristotle had reasoned that space was but an accident of corporeal substances, without which it could not exist. In this view, it was incoherent for there to be empty space without substance, for that would be to speak of space as if it were a substance. Descartes collapsed this metaphysical distinction and identified corporeal substance - "matter" in the modern sense of the term - as identical with spatial extension. In other words, matter was nothing more than chunks of space. Descartes further reasoned that, since space in principle could be divided infinitesimally by divine power, there could be no true atoms or indivisible units of matter. Descartes' lack of philosophical training betrayed him here, as his analysis confused distinctions in reason with distinctions in possibility or actuality, an error that no competent Scholastic would have made. Although Descartes considered these theses to be "clear and distinct" facts about matter, they proved to be shaky hypotheses leading to untenable interpretations of celestial phenomena.

In Cartesian physics, an object could move only by pushing some other object out of the way, so all motion was to be explained by direct physical contact. Motion itself had further principles, which Descartes inferred from Galileo's discoveries. The first two Cartesian laws of motion summarized the principle of inertia: "each thing, as far as is in its power, always remains in the same state; and that consequently, when it is once moved, it always continues to move," and "all movement is, of itself, along straight lines." These tendencies to resist motion and to move along straight lines are presented as primitive facts, having no further explanation save the will of God. Descartes agreed with modern mechanists against the Aristotelians that a cause did not need to continually operate to keep an object in motion once it is in motion, though he also believed that inertial motion was distinguishable from rest.

In Aristotelian physics, local motion, like any other physical change, required the operation of some causal agent. Continuing motion would require continuing operation of the cause of motion. Inertial mechanics seems to dispense with this metaphysical requirement, as an object, once moved, will continue to move on its own even without continuing operation of the cause that first moved it. In fact, no metaphysical upheaval is necessary, for as Aristotle noted in the Physics (VII, 2), local motion may be caused by the mover pushing an object while moving along with it or by pushing the object away without following it. The latter kind of action characterizes motion at constant velocity, while Scholastic philosophers erroneously thought this required constant pushing, since practically all terrestrial kinematic phenomena involve friction, air resistance or other forms of damping that prevent observation of continuous inertial motion. This is why the first empirical evidence of inertial mechanics was found in astronomy. Inertial motion still requires a mover, but the power of motion, once transferred, remains in the moved object.

The retention of motion by a moved object is described in another Cartesian law of motion: "that a body, upon coming in contact with a stronger one, loses none of its motion; but that, upon coming in contact with a weaker one, it loses as much as it transfers to that weaker body". This is an imperfect formulation of what would eventually become the law of conservation of momentum. Descartes measured the amount of motion by multiplying the speed of an object by the amount of matter it contained. Since matter was extension for Descartes, this meant multiplying speed by geometric volume rather than mass. Further, he did not consistently treat velocity as a vectorial quantity involving direction, so that only parallel components of linear momentum are transferred.

Descartes rejected substantial forms on the grounds that they entailed teleology or goal-directedness in inanimate objects, but he was not altogether consistent on this point. The Aristotelian account of gravity would have every massive body tend toward the center of the earth. Descartes found this problematic, since it seemed to entail that each body should have "knowledge" of where the earth's center is. This aversion to attributing any sort of "knowledge" to inorganic matter accounts for why modern physicists have always tried to explain everything in terms of contact interactions, rather than action at a distance. However, Descartes, like his successors, somewhat inconsistently allowed matter to have intrinsic "tendencies," such as the tendency to follow a straight line and the tendency to resist motion. Descartes even admitted the classical quality of flexibility as an intrinsic property of matter. Still, while he could not explain these properties or tendencies or internal forces, Descartes refused to characterize them as quality-bearing forms of corpuscles, instead insisting that they were all modes of matter or motion.

Cartesian mechanics sought to explain all physical phenomena in terms of purely geometric properties of corpuscles that moved each other through direct contact. This theory had great conceptual appeal to most mechanists, so Cartesianism became the dominant explanation of physics, and it was even able to survive the challenge of Newtonianism for a while, since the latter had to resort to an occult gravitational force acting at a distance in order to explain celestial motion. Cartesianism, by contrast, explained the motions of the planets in terms of direct interactions with the matter that filled space.

In Cartesian kinematics, circular motion resulted in a centrifugal force driving the object away from the center. The circular motions of the planets around the sun, in this view, ought to drive them further and further away unless they are counteracted by another force. Descartes postulated that space was filled with bands of particles that moved in circles, having nowhere else to move, and created vortices with local centrifugal forces of varying degrees of strength. When a planet ascended among vortices just strong enough to counterbalance the centrifugal force from its movement about the sun, it settled into a stable orbit. Terrestrial gravity was similarly explained by vortices of corpuscles surrounding the earth.

Descartes' vortices were every bit as ad hoc as Ptolemaic epicycles, and they could not be directly observed, even though they effectively acted like the crystalline spheres as palpable conveyors of celestial bodies. Despite its contrived, non-empirical nature, Descartes' theory was regarded by many mechanists as superior to Newton's, since it postulated a corporeal physical agent acting through direct contact. The mechanists, no less than the Scholastic philosophers, appreciated that a truly physical theory must postulate real physical agents as causes, instead of merely giving a mathematical description of phenomena.

However, a theory of physical causes is not sustainable if it cannot accurately account for observed phenomena. Ultimately, Cartesianism could not compete with Newtonianism's ability to predict future astronomical observations with astonishing accuracy. At a bare minimum, a physical theory must account for observations, but is this enough to give us physical causation, or are we just giving mathematical descriptions of phenomena?

Newtonian Kinematics and the Role of Force

Sir Isaac Newton's Principia Mathematica (1687) lends an almost Platonic air to the scientific discourse of the seventeenth century, supposedly an age of ascendant empiricism. Much like Kepler, Newton was motivated by a religiously inspired conviction that there is a harmonious order in the universe, and he avidly sought evidence of such order, in apparent dismissal of Bacon's warning that the human mind tends to impose order where there is none in nature. Yet Newton held distinctly Baconian attitudes, as expressed by his famous declaration, "I form no hypotheses" (Hypotheses non fingo). The strength of his mathematical theory lay in its incredibly precise predictions of experimental results. Its simple elegance made it clear that it was not just an amalgam of fudge factors, but somehow related real physical quantities to each other. This would raise important questions, such as: what were these quantities, and which of them represented causal agents? Or perhaps the relations themselves, the "laws of mechanics," were causal in some sense?

Newton famously gave us his laws of motion or kinematics and a gravitational force law as the most fundamental principles he could discern in nature. Like Descartes, Newton did not interpret Galilean relativity in a way that would abolish an objective distinction between inertial motion and rest. Newton believed in absolute space and time, which implied absolute motion, and indeed one of Newton's criticisms of Cartesian mechanics was that it sometimes failed to distinguish between real and apparent motion. Newton believed he was giving an account of an objectively real physical activity when he explained motion; his was not a relationist view of motion.

Newton posited space and time as existing independently of all objects. Metaphysically speaking, Newton's "space" functioned as a substance or substratum upholding all corporeal substances. It differed from Aristotelian space in that it was homogeneous, so there was no special place or center of the universe; space was infinite and uniform in all directions. The Newtonian laws of mechanics did not in any way depend on absolute location, and thus seemed to corroborate the view that space is uniform. Newton also differed from Aristotle by effectively regarding space as a substance rather than an accident or property of substances. His illustrious contemporary Gottfried Leibniz (1646-1716), in contrast, developed a modified Aristotelian view, contending that space and time were only relations between physical objects.

Since Newton distinguished bodily objects from the space they inhabited, he had to assign them an additional attribute beyond Cartesian extension. Matter required impenetrability in addition to extension. Impenetrability, according to Newton, was not a property of matter itself, but the result of a repulsive force, and force was a consequence of motion. God created an impenetrable body by taking a region of space and making it move so as to generate a repulsive force and become able to occupy different spaces in succession. This motion, like all physical motion, was governed by divinely ordained laws, the laws of motion.

Newton was unable, however, to explain the most potent mechanical force - gravitation - in terms of kinematics. It could not even be explained as a property of matter, since it seemed to depend on the presence of two objects, so there could be no gravity where there is only one object. Newtonian gravity appeared to be a purely relational force, involving action at a distance across the void of space. It was thoroughly unmechanical, as it could move objects without direct contact, and seemed to involve one object's effective "knowledge" of the location and mass of the other, without any medium to convey mechanical contact. Small wonder, then, that even the mathematical potency of Newton's gravitational law was not able to win over some of the most eminent physics theorists, most notably Leibniz.

Leibniz maintained that gravitational force was the product of fluid vortices, like those of Descartes. He held that, prior to any empirical science, we must know at least the nature of motion and the nature of bodies in order to render our empirical observations intelligible. The nature of motion, according to Leibniz, is expressed in the principle of inertia, while the nature of bodies is that they can be moved only by something adjacent to them. Newton's gravity law evidently contradicted the nature of bodies, so it could not be interpreted as a physical agent.

Newton, by contrast, while admitting that he could not discern a physical explanation or cause of gravity, nonetheless insisted that he had proved that gravity itself was causal. In a certain respect, this was a Baconian line of thinking: we do not need to prove the most general thing in order to know a more specific thing. Thus we do not need to know what more general principle is behind gravity in order to know that gravity itself is causal. Leibniz had instead demanded that we must know the most universal principles of matter and motion before discerning more specific principles. In this, like Descartes before him, he followed the classical Aristotelian approach to hypothesizing.

For Newton, a hypothesis was not a basis for interpreting results, but just a tentative guess for the purpose of raising questions for future research. When he declared that he did not form hypotheses, he especially meant those that invoked the existence of entities that could not be observed directly. Newton, the great mathematizer and discerner of divine order in nature, rejected metaphysical accounts of physics, yet this was not all. "For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy." Even a mechanical property could not be hypothesized a priori; it must be deduced from observation.

Newton, then, would have us not worry about what physical properties are behind gravity. It suffices to know from observation that the gravitational force law really does describe actual behavior for us to be content that gravitational force is a real causal agent. This almost instrumentalist notion of science, while having the advantage of steering us away from unreliable speculation, nonetheless seems to offer a weak basis for demonstrating causality. We are faced with the problem of induction, where no amount of observations suffices to identify a universal principle, and with the ghost of Cardinal Bellarmine, who reminds us that a mathematical model may save the appearances without telling us what the causal agent is. What is the agent in Newtonian gravitation? Is it the bodies themselves or some property of the bodies? Is it their relation to each other? Is the law itself a causal agent? For Newton, it would seem that the force is the immediate agent, since this is what can be experimentally observed. Yet what is force?

Newton conceived of efficient causes kinematically; that is, objects induced other objects to move by direct contact, and transferring their motion to the moved object. Force is a consequence of motion, being nothing more than the time derivative of momentum. Newtonian momentum, which is the product of mass and velocity, serves as a measure of the strength of the motion. Newton's laws of motion are fundamental to his physics, and all Newtonian forces, with the notable exception of gravity, are expressed mathematically in kinematic terms. Newton identified force as the product of mass and acceleration, and understood this to mean that force was nothing more than the acceleration of a given mass. We should pause here and consider when a mathematical equation gives a physical identity.

There are some mathematical equations that evidently represent true physical identities. For example, in v = dx/dt, velocity is truly identical with the time derivative (rate of change) of position; there is more than a mere equation of quantities. However, velocity is something more than a simple composite of position and time, for we may specify the position and time of an object without knowing its velocity. Thus velocity might be considered a real property or "quality" of an object or of its motion. The same may be said for the second-order time derivative of position, acceleration, which Galileo still referred to as a "quality."

In general, however, the mathematical equations used in physics need not be physical identities. Consider, for example, Newton's third law expressed as F = dp/dt; that is, force equals the time derivative of momentum. This does not necessarily mean that force is a change in momentum. It only means that the magnitude of a force is proportionate to a change in momentum. I say proportionate rather than equal, since the equality is contingent upon our definition of units. Without a special choice of units, the equation would be F = k dp/dt, where k is some constant, so F is only proportionate to dp/dt. To show that force truly is a change in momentum, we would have to show, at the very least, that there is no force where there is no change in momentum, and that there is always a force where there is change in momentum, and vice versa. Even this would not strictly prove identity, but only practical equivalence.

The concept of force (vis), Newton's efficient causal agent in mechanics, was borrowed from Scholastic physics, with some modification. In Scholasticism, the Latin term vis ("strength, violence") was used for an external agency that forces an object to diverge from its "natural" or intrinsic motion. Newton acknowledged the existence of these violent forces, and indeed the major portion of his mechanics is devoted to explaining how they quantitatively affect motion. However, in addition to violent force, he also spoke of a vis insita or vis inertiae, which we might call an inertial force. This inertial force is not a vis in the Scholastic sense, for it is not violently imposed from without, but is intrinsic to the object. Thus force, for Newton, is more than violence, but rather it is any agent that causes motion, extrinsic or intrinsic.

To better understand Newtonian force, which is a change in momentum, we should briefly examine the predecessor of momentum, the "impetus" of the Scholastics. The Arab philosopher Ibn Sina first postulated that moving objects have an "inclination" (mayl) to motion that is proportional to mass times velocity, and this inclination is not dissipated except by external forces of resistance. Thus a moving object of a given mass would continue to move at the same velocity, were it not for resistance in the air or other external forces. A moving object retains its own motion until something alters it. The Scholastic Jean Buridan elaborated Ibn Sina's hypothesis into a full-blown quantitative theory of "impetus" (impetu), an inclination to motion that can be imparted from one object to another, which is proportional to the velocity of an object. Buridan agreed with Ibn Sina that impetus is not dissipated, except by external forces of resistance. He added that gravity imparts a downward impetus on projectiles. His successors, the Mertonian Scholastics, associated impetus even more explicitly with violent motion. When an agent moved an object by violence, it was said to impart both an actual force or motive power, and an impetus or inclination. In Mertonian physics, both force and impetus are things imparted by an agent that moves an object violently.

Impetus is nothing other than what we call momentum, and in fact Galileo referred to the product of mass and velocity as impetus. Buridan believed there could also be a circular impetus that accounted for celestial motions. We would modify this notion somewhat, yielding what we now call angular momentum. It is intriguing that a correct concept of linear momentum was achieved by medieval Scholastics primarily by abstract ratiocination, and that this philosophical physics yielded a correct account of pendulum motion through thought experiment. Galileo and Newton were heavily indebted to the Scholastics, who gave them the concept of momentum and some of its basic mechanics ready made.

In modern discussions of Scholasticism, impetus theory is often mischaracterized as entailing the thesis that projectiles do not retain their motion. While such a thesis may be derived from Aristotle's Physics, it was not characteristic of the Mertonian school. Most Scholastics in the Thomist tradition did believe that projectiles could lose their motion without some external resistance, and they attributed this to an inertia, or tendency to rest, that was intrinsic to massive bodies. Here we see how the concept of an "inertial force" would be intelligible in Newton's time. Newton's inertial force differed from that of the Scholastics in that it was not a resistance to motion, but a resistance to change in velocity.

A force can be considered at least two ways: it is a change in momentum, or it is that which changes momentum. Newton seems to have considered force in the latter sense, since he explicitly assigns it causal agency. The second sense need not be exclusive of the first, for one might attribute agency to change itself, since action is but a type of change. Thus action itself, not some actor (e.g., the corporeal body), is the principal agent in mechanics. Indeed Newtonian mechanics may be conceived as forces pushing and pulling dumb bodies this way and that. Corporeality or matter seems to be purely passive with respect to violent force. The only force Newton postulates that is intrinsic to matter is not the power to move, but a resistance to change in its velocity.

The medieval Scholastics, being more philosophically exact, would only say at most that impetus is proportional to mass times velocity, not that it is equal (since that requires arbitrary definition of units), much less identical. We have a sense of what mass is (the amount of stuff or bulk) and a sense of what velocity is, and impetus appears to be an altogether distinct quantity. It is not bulky or slow or fast, so we cannot consider it a simple composition of mass and velocity. What sort of physical combination could correspond to the multiplication of quantities of two qualities (mass and velocity)? Modern physicists do not worry about this; if they think of the question at all, they will say that momentum is mass times velocity. We can multiply length by length and get area, and a physicist will simply treat any quantitative property as an extensive measure and see the product as a mathematical object in some phase space. This is not physical analysis, but deals purely with mathematical fictions. The modern physicist thinks the proof that these mathematical objects are real is their empirical predictive power, but this only proves that the mathematical theory "saves the appearances."

Length by length gives area because length and length are measures of the same thing: one-dimensional space. We consider each point along one dimension as having length along another dimension, and thus we get an area. Can we apply this to mass times velocity? Can we say that each value of velocity has a mass? Hardly; but we might say that each piece of mass has velocity. This at least is physically intelligible. Thus the total momentum of an object is the sum of the motions of all the mass in it. Momentum, in this view, is the total motion of an object, if we think of motion as some actualization or energy that can be cumulative.

Early modern physicists did not always clearly distinguish momentum, force, and energy. In Aristotelian philosophy, the term energeia referred to the actualization of being, not what we would call physical energy, which we intuitively conceive as the power to do physical work. Both momentum and force seemed to give objects the capacity to move other objects, so they were conceived in a sense that we might now call energetic. In Aristotelian jargon, the power to do something, considered as a potential, was called dynamis. Energeia was the actual realization of a power (dynamis) in action. The distinction between potentiality and actuality was essential to Aristotle's metaphysical analysis of change and motion, which rejected the paradoxes of Parmenides (change must be an illusion, for being cannot become non-being) and Heraclitus (entities do not persist, but are constantly replaced). Our notions of potential and kinetic energy come from this intellectual tradition, thanks in part to the work of Gottfried Leibniz.

The early modern physicists did not speak in terms of potential and kinetic energy, but in terms of passive and active force. In Scholastic philosophy, the distinction between passive and active is not quite the same as that between potential and actual. A passive entity is a recipient of an act, while an active entity imparts an act. By contrast, potentiality is the capacity to perform an act, while actuality is the realization of the act. Leibniz called the product mv (mass times velocity, or momentum) the vis mortua (dead force), identifying it as a measure of when force is "passive," while mv2 was the measure of the vis viva (living force) or active force. Despite his use of the terms "passive" and "active," it is clear from context that Leibniz was referring more to potentiality and actuality. Momentum, for Leibniz, is when force is inert, while the vis viva, which is proportionate to modern kinetic energy, is when force is actually being used. To modern eyes, it seems dimensionally wrong for Leibniz to characterize momentum and energy as kinds of force, yet if we consider force as motive power, then this identity seems conceptually correct. An object with momentum has the power to impart motion onto another, even if there is no force in the Newtonian sense of mass times acceleration. Similarly, kinetic energy measures the power of an object to move itself, and force is but a derivative of energy.

Newton dealt with momentum's force-like behavior by characterizing momentum conservation as the result of an "inertial force" or intrinsic resistance to change in velocity. In this way, he was able to remain dimensionally consistent, with force being proportionate to mass times acceleration, as the inertial force is a resistance to acceleration. Momentum is the basis of both violent and inertial force, the "stuff" whose change causes motion in other objects.

Although Newton believed in a real distinction between rest and motion, his forces were not dependent on absolute velocity, but only the relative linear velocities of interacting objects. The force experienced by an object is due to a relative change in momentum, and this relativity is essential to explaining, for example, why we may not feel the earth's motion. Inertial mechanics, were motion at constant velocity does not generate force, overcomes this conceptual obstacle to heliocentrism, and makes possible a stronger distinction between motion and motive force.

In medieval physics, it was widely believed that any motion must be attributed either to the intrinsic "natural" motion of a body or to an impetus imparted by a violent force. Impetus proved to be a sound concept, identical with modern momentum, but in place of "natural motion" modern physics proposed an intrinsic resistance to change in motion. Once a body was set in motion, its nature resisted change in velocity, until acted upon by a violent force. Thus a body could continue to move indefinitely at constant velocity even without receiving additional motive force. One could have an infinite amount of motion with only a finite motive force. This seeming contradiction was resolved by recognizing that motive force is proportionate to acceleration rather than velocity.

If force is the only efficient cause in physics, then inertial mechanics means that no external agent is needed to account for the continued motion of an object at constant velocity, once it is initially accelerated to that velocity. However, there is an efficient cause at work, namely Newton's "inertial force" which is intrinsic to the object.

Newton treated force as a causal agent, in contrast with the Aristotelians who saw force as the act of some agent. In Scholastic physics, impetus was conceived as active motive force, in contrast with the potentiality or capacity of dynamis. Newton's force metaphysically resembles impetus, insofar as it is active. Yet where is force when it is not acting, but exists only in potentia? An adequate answer would not arise until the nineteenth century development of a theory of energy. In the absence of such a theory, Newton would have to let his "inertial force" serve as a passive potentiality. Leibniz similarly postulated momentum, the stuff that inertial force conserves, as "dead force."

If we take seriously Newton's contention that forces are causal agents, we get strange results when we consider the force of gravitation, which is not derivable from kinematic interactions of direct contact, but instead is described by a force law: F = GmM/r2, where m and M are the masses of two objects, r is the distance between them, and G is a universal constant. Some forces, it seems, are defined in terms of a universal law, and somehow this law, through force, is able to effect real physical events.

In the discussion thus far, we have tacitly assumed that force is a real physical entity. In practice, we often do not measure force directly, but compute it from measured values of properties such as mass and velocity. We should consider the possibility that at least some of our so-called forces might be just mathematical constructs that help us compute changes in momentum and kinetic energy. For now, however, let us examine the notion of Newtonian force laws as causal or at least explanatory entities.

Laws of Nature in Classical Mechanics

The age of mechanism, in which Newton was ultimately triumphant, gave rise to the notion that the universe is governed by "laws of nature," and that these laws can be given mathematical form. It is not too surprising that Newton and other Englishmen should use such a metaphor, as Elizabethan laymen took much interest in legal theory. As political philosophizing would become more popular in England and France as the seventeenth century yielded to the eighteenth, the idea that nature, no less than men, was ruled by law, conformed with the spirit of the age. Newton himself had no qualms about identifying God as the Lawgiver, and the same was true of other Christian men of science. Interestingly, as secularism grew popular among the intellectual elites of the eighteenth and nineteenth centuries, the concept of physical law would remain, even as it was made independent, and even opposed to, the concept of a divine Lawgiver.

The "laws of nature" or "laws of physics" can be interpreted in several different ways. In the strongest sense, they may be considered as real causal agents that effect or prohibit certain classes of events. In a somewhat milder sense, they are not efficient causes of physical events, but nonetheless act in a nomological way, somehow confining or constricting the way things are. This sense is similar to the formal cause of Aristotelianism. In a yet milder sense, we might consider the law as an explanatory principle, that renders intelligible why things are the way they are. Finally, we might consider that the law is purely descriptive, a mathematical shorthand for summarizing classes of phenomena, but carrying no import as to how or why things happen.

Where on this spectrum should Newton's laws be placed? First, we may consider his famous three laws of motion:

  1. Galilean inertia: An object in uniform (linear) motion tends to remain in that motion until an outside force acts upon it.
  2. An applied (violent) force of vector quantity F to a body of mass m causes an acceleration of vector quantity a in the object such that F = ma.
  3. For every action there is an equal and opposite reaction.

There certainly seems to be causality among entities within the laws as formulated above. In the second law, this is most explicit, as a force causes change in an object's velocity; or, to take a more Scholastic interpretation, the agent that applies the force causes this change. In the first and third laws, causality is less obvious. Does each action cause an equal and opposite reaction, as the term "reaction" seems to imply? What causes an object to remain in uniform motion?

Newton understood the first law in terms of an inertial force, so he would say that the inertial force of an object causes it to remain in motion, until some external violent force causes it to deviate from that motion. As for the third law, Newton understood action in terms of force communicating motion by direct contact. He explained in his Principia:

If some body impinging upon another body changes the motion of that body in any way by its own force, then, by the force of the other body (because of the equality of their mutual pressures), it also will in turn undergo the same change in its own motion in the opposite direction.

Although Newton indicates that the "opposite reaction" is subsequent to the initial "action," this explanation is not as enlightening as it seems. Body X touches body Y, so that the force of X changes the motion of Y, and then the force of Y causes body X to change its motion by the same amount in the opposite direction. The question remains: why is there a force in Y equal and opposite to the force in X? Newton answers "because of the equality of their mutual pressures," which is intuitive enough, but hardly any more so than the idea of equal and opposite reaction. Indeed, "pressure" is nothing but force per unit area, so this explanation basically says the reaction is equal because the forces are equal, but gives no insight into why the forces are equal.

We might try to supply this deficiency by positing the laws themselves as organizing principles or even causes of nature. In this view, the laws themselves are fundamental realities of nature, universal principles that constrain the behavior of particular objects. For reasons known only to the Creator, it has been ordained that every action should have an equal and opposite reaction. We may observe in particular instances that this is the case, and by induction arrive at the belief that this is a universal law. Yet in the order of causality, the universal law is what mandates that the particular interactions of bodies conform to a common principle.

The origin of each physical law is mysterious. It is a brute fact inferred from observation, and is not deducible from some more fundamental principle, as far as we know. This is why Newton's attempts to explain the first and third laws end up being circular. He explains inertia in terms of an inertial force, and the equality of reaction in terms of the equality of pressures. Yet the operation of each physical law is no less mysterious than its origin. Given that the third law of motion is an ordained principle of nature, how does it impose itself on particular events? Is it an efficient cause, or perhaps a formal cause?

Adding to the mystery is the dependence relationships among the laws themselves. We may mathematically derive other principles from physical laws. Are these corollaries to be considered effects or consequences of the laws of physics? Consider, for example, conservation of momentum. Newton regarded this principle as a corollary to his laws of motion. However, if we wanted to take conservation of momentum as our fundamental postulate, we could derive from this the principle of inertia. There is no way to tell, from mathematics alone, which physical principle is more fundamental. The order in which we discover physical principles need not reflect their actual order of physical or metaphysical dependence. The order of physical or metaphysical dependence need not be the same as that of logical or mathematical dependence.

The mechanists themselves, especially Newton, appear to have conceived of the laws of physics as divine ordinances imposed on nature. Though some laws were regarded as corollaries of others, at the base there was a set of laws that could not be reduced to any other laws. In this view, the "laws of nature" were truly nomological, being edicts of the Divine Sovereign. These laws were expressions of God's dominion over nature, so it was by divine power that natural bodies were subjected to these principles rather than others. This did not mean that natural bodies were just extensions of the Deity, but rather the divine decrees called "laws of nature" defined the powers of natural bodies and the rules by which they may be exercised. This view of the relationship between God and the "laws of nature" was a staple of early modern thought, especially in England, which saw the rise of "natural theology."

A theistic account of the laws of nature was most consistent with their apparent universality and eternity. Although the discoveries of Galileo and other astronomers proved that even the celestial bodies were corruptible, at the same time they discovered physical principles that unerringly foretold events with remarkable precision, and held true throughout all space and all time without the slightest deviation. They could hardly be blamed for referring these principles ultimately to God. Indeed, in the absence of a unifying Intelligence in the cosmos, there would be little reason to expect, a priori, that all physical phenomena should be subject to intelligible universal principles. We might just as well expect each local phenomenon to be sui generis, so that it tells us nothing about what is true in other places and times. The scientist's expectation that experimental results should be repeatable makes sense only if we assume that the world is governed by universal laws. Otherwise, there is no reason to expect the same results in different laboratories at different times. All modern scientists, including the most strident atheists, operate on an epistemic assumption that aligns with theism.

Although Newton and the other early mechanists were theists, they generally believed that natural bodies had real powers of their own, and were not merely puppets of the Deity. The "laws of nature" defined these natural powers, which enabled corporeal bodies to act as real agents. Since the mechanists believed that corporeal bodies (or their associated forces) were the only causal agents in nature, it followed that the laws of physics defined principles of local interaction, even though they were of universal applicability. Although God Himself could perform any action, including the imposition of universal and eternal laws, natural bodies were believed to be able to act only through direct contact. There is some tension here (though no contradiction) between the belief that physical laws must be universal and that physical interactions must always be local.

This tension was most pronounced in the Newtonian account of gravitation. Newton was convinced that gravity was a real physical force that acted on bodies, yet he could not explain it in terms of the direct contact model of mechanics. Two bodies, far removed from each other in space, nonetheless were able to attract each other, effectively applying an external force on the other body. It was not clear if gravitation should be regarded as an intrinsic or extrinsic force. On the one hand, celestial motions seemed to be "natural motions," which suggested that gravity was an intrinsic property of matter. This indeed is suggested by the name itself, which means "heaviness," and by the fact that it is proportionate to mass. However, there could be no gravitational force in a single body. Gravitational force could be experienced only by interaction with some other massive body (or by another part of the same body). Thus it was always experienced as an applied or external force. Further, the magnitude was also proportionate to (the inverse square of) the distance, so the force seemed to really depend on the bodies being apart. This entailed not only action at a distance, but the apparent ability of a body to effectively "know" how far away another body was. At any rate, it was not at all clear how gravitation could be explained in terms of local action alone.

If gravitation is an external force, imposed from without, then celestial motions would be violent motions rather than natural motions. However, the gravitational force law is proportionate to the product of both masses, including that of the object moved, so it would seem that gravitation also has the character of an intrinsic force. The form of the force law is strange when compared with Newton's second law. It reads: F = GmM/r2, where m and M are the two masses, and r is the distance between them. Instead of mass times acceleration, we have mass squared over distance squared, a totally different dimensionality for force. We can balance the dimensions by defining G in ad hoc units, but this is without justification since G, as far as we know, is just an arbitrary constant, whose value depends on the units we choose for mass, distance and time (force units are derived from these). The empirically measured value of G, for a given set of units, may be accepted as a brute fact, just like the form of the force law.

The force law of gravitation seems to entail a different kind of physical causation from that of kinematics, since it defines an attractive force that depends on the distance between objects, and is not the result of a direct transfer of momentum by contact. There is a way to restore locality to universal force laws such as gravitation, and that is with the notion of a force field, which we will discuss later.

The power of Newton's law of gravitation seemed to derive from its mathematical elegance and universality of application, as though the force relation itself was a real physical entity that bound substances to act in certain ways. Thus mechanists spoke of "gravity" or "gravitational force" as if it were some palpable thing acting directly on an object, and it became common to invoke mathematical equations as physical explanations of phenomena. To the question, "Why does the earth move in such a way?" an appeal to the "law of gravitation" was deemed a sufficient answer. Yet we have already noted the problematic nature of regarding a mathematical equation as a physical agent. Furthermore, in all other matters, Newtonians required direct contact between bodies in order to exert force on each other. Thus the forces themselves were agents, not the mathematical formulae that described them. Yet with gravitation, a relational force law seemed to act as a causal agent, reaching across the vacuum of space between celestial bodies. What are we to make of this?

The law of gravitation is a relational form, since it does not define the qualities of a substance, but rather the interaction between substances. Most subsequent laws of physics and chemistry discovered in the seventeenth and eighteenth centuries would share this relational character. The question of concern to us is whether relational laws of physics can serve as causal agents or at least as organizing principles in nature. In ordinary speech, it is common to speak of the "laws" of physics or chemistry as explanatory principles. In answer to "Why does X occur?", a physicist or chemist may consider an adequate response to be, "Because of the law of Y". On closer inspection, however, it seems impossible that a mathematical law can cause or prohibit any physical event. After all, there are many mathematical equations that do not correspond to physical reality, yet they are no less mathematically valid. Why should one mathematical abstraction rather than another have the power to affect physical reality? Is it not more likely, as Paley held, that "laws of nature" are impotent, being nothing more than a statement of sequences of events actually observed?

Still, the universality, precision, and astonishing predicting power of mathematical laws of physics make it tempting to attribute some real causal agency to them. They cannot be efficient causes if efficient causation is attributed to the forces themselves, but they might define the relational forms of forces and other physical entities. In this way, the "laws of physics" may be expressions of formal causes. Unlike the Aristotelian substantial forms, however, a "law of mechanics" is not the form of a specific kind of substance, but the form of a necessary relation between physical entities of certain kinds.

It may seem superfluous to invoke formal causes when efficient causes suffice to account for physical phenomena, but this is a modern prejudice without sound philosophical basis. The existence of one type of cause need not eliminate others. This is true not only of formal causes, but even of final causes. Leibniz believed that all corporeal phenomena can be derived from efficient cause, yet this did not prevent him from also recognizing final causes ("higher reasons"). In other words, just because we know that A caused B mechanically, that does not preclude that there could be a deeper reason why something happened. For example, I can explain the building of a house mechanically: the carpenter drove this nail, then that one, etc., but that does not mean there was not a deeper reason or purpose why the house was built.

This same error pervades modern science, and especially evolutionary theory. Philosophical materialists think that by giving the naturalistic efficient cause they have given everything, and have somehow proven that there is no higher purpose. By this reasoning, since human beings evolved from "chance" accidents, they cannot possibly be the product of a higher purpose. If such an argument were valid, the fact that my parents met by accident would prove that I am just a meaningless accident. Such an inference, of course, is silly, since it could still be the case that the seeming accidents and chance occurrences are governed by some fate or providence, which is precisely the question at issue. Efficient cause, even when it is strongly deterministic, need not abolish final cause. Nonetheless, we will restrict our focus to efficient and formal causes, since modern physical science has nothing to say about final causes, due to its self-imposed epistemic constraints.

In standard Newtonian mechanics, force is imparted by direct contact between objects, so the power of motion is transferred from one object to another. In this model, force was an agent of efficient causation. An Aristotelian might say that, properly speaking, it is not the force but rather the object imparting the force that is the true agent. Force, then, might just be a mathematical fiction that is useful for quantifying the motion that is imparted. It need not be a real natural object or physical "stuff." At most, it is an accident or property of matter, and a relative accident at that, since its magnitude depends on reference frame.

With Newtonian gravitation, there is no interaction by direct contact, so it is not clear at all that there exists some stuff called "gravity" that acts as an efficient cause of motion. Newton, nevertheless, insisted that, although he could not understand what gravity was, empirical induction sufficed to show that gravity was a true efficient cause. Skeptics might say the law of gravitation is just a useful mathematical fiction that saves the appearances. There does not seem to be any mechanism of efficient causation in Newtonian gravitation. What is the agent? Is it the massive bodies acting on each other from a distance without a medium? It would seem, perhaps, that the mathematical form of the gravitational law dictates how objects behave. This might make gravitation at least a formal cause if not an efficient cause.

Newton's theory of gravitation ultimately triumphed because the competing theory of vortices failed to match its accurate predictions. In empirical science, a theory is commonly judged to be "right" or "wrong" based on its ability to accurately predict determinate classes of phenomena. However, can we be confident that a theory judged "right" by such a standard really gives physical causes, or perhaps is it simply a quantitative description that "saves the appearances"?

Continue to Part II


© 2010 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org

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