Isaac, son of Isaac and Hanna Newton, was born prematurely about one hour after midnight on Christmas Day 1642 in the family manor house at Woolsthorpe. He grew up displaying an unusual interest and ability in mechanical invention:

He also had considerable drawing ability. An early biographer, Dr. Stuckeley, perceived the close ties between Newton’s mechanical and drawing abilities and their abstract expression as follows:

The distinction between those who invent hypotheses and those who invent experiments, is part of a general one between different types of thinkers. Stuckeley’s remark about philosophers echoed Newton’s famous boast that he made no hypotheses, which is perhaps best understood as also a distinction made between philosophers such as Descartes and the new type of natural philosopher--scientist--of whom Newton was the exemplar.

The same pair of connected abilitie--mathematical and mechanica--was famously seen both in Archimedes and in Galileo (for whom Archimedes was a special hero). In contrast to Archimedes, Newton and Galileo had the luck to be born in societies valuing mechanical talent. Newton’s biographer, Frank Manuel, remarks on these social factors as follows:

Recall the efforts of Francis Bacon to give rise to this very type of social milieu..

This confluence of favorable physiological, psychological, and social factors led to great results with remarkable rapidity. In the summer of 1661, the eighteen year old Newton went to Cambridge. Two summers later at Stourbridge Fair he bought a book on astrology

By 1666, only three years later, he had begun the creation of dynamics; he had invented an elementary calculus and, relating Galileo’s picture of projectile motion to the orbit of the moon (illustrated in Figure 1), had conceived of universal gravitation.

Newton’s idea that that which drew the apple to Earth was the same as that which held the planets in their orbits, was independently but not uniquely arrived at. As we shall discuss further on, concepts of force and of a universal gravitation force were widespread. Similarly, his invention of the calculus was independent but not unique. Newton’s unique contributions originated in his application of calculus. It was a tool he created and used to imbed astronomical data, and concepts of projectile motion, force, and gravity, all within a single structure, linked by the logic of mathematical deduction: Newtonian dynamics. This was successful beyond anything anyone but Descartes had ever dreamed, and Descartes had only dreamed it.

Calculus was created by Newton to mathematically cope with the continuity of space and time, and the smoothness of natural curves such as trajectories. These properties, however, are inessential to the scientific worldview based on his work. Just as with Descartes’ vortex theory, we can use a discrete in place of a continuous mechanics to illustrate the concepts important to this discussion. Thus we need not deal with calculus.

Although the discrete theory used here is too simplified to be even be called an approximation, it still retains everything essential to illustrating the worldview based upon mechanics. With the mathematical difficulties, especially those often associated with calculus, having been removed, we shall see that the Newtonian world becomes little more than a board game.

Kinematics

Kinematics is the description of how things can move. In this discrete model, ‘particles’ occupying discrete cells move in a one dimensional space. Space and time are both discretized. Each will be made of discrete equally spaced points labelled with the integers: …, -2, -1, 0, +1, +2, ….

At each instant, each particle moves with a unique velocity which determines how far it will hop in the next instant. This is illustrated in Figure 2 where a portion of space is shown at two successive instants. A particle may be imagined to be at the center of the shaded box. Its velocity arrow at time T points from its position at T to its position at T+1. Its velocity arrow is two units long and represents a velocity of +2. If its velocity had been -2, the arrow would have pointed two boxes to the left and the particle would be two boxes to the left at time T+1.

Any number of such particles, each with its own position and velocity, might be placed in this model universe.

The change in a particle’s position from one instant to the next is its velocity. The corresponding change in its velocity is called its acceleration. Figure 2 shows a particle undergoing no acceleration at time T. Its velocity is +2 both at T and at T+1. We say its acceleration at T is 0 = (+2)-(+2) .

Forces cause acceleration. A body not subject to forces moves with zero acceleration: its velocity remains constant. This is the principle of inertia.

Forces arise from other particles. Thus, if there is only one particle present in the universe being described by this model, that particle just moves along forever with constant velocity.

What forces arise from other particles is determined by some sort of dynamical law. An example of a dynamical law will be given shortly but first we shall discuss how the output of such a law gets used. It is very important to understand that its use is purely mechanical. Figure 3 helps to emphasize this by describing the how a computer might proceed to calculate the motion of a system of particles on the basis of some dynamical law. It is important to remember not to think when following this diagram; you should only follow orders precisely, like a machine--this is the essence of the idea of the mechanical universe.

Start at the top of the figure with what are called the initial conditions, the starting point of a calculation. The initial conditions are the initial positions and velocities of the particles. Since the calculation starts at T it must be informed of these values. Different initial conditions lead to different subsequent paths of motion.

The symbol for time is T, and that for the state at time T is the set of N pairs of symbols (X₁, V₁) … (X_N, V_N). The pair of symbol (X₁, V₁) represent (position of particle 1, velocity of particle 1), and and so on for subscripts 2 to N.

In this one-dimensional model, each position and velocity is a single quantity. In the real world, each position and each velocity are composed of three component quantities, one for each spatial dimension. In this model, because of discretization, position and velocity take on only integer values. In the real world, the components of position and velocity take on real number values. Conditions specific to this model aside, the concept of ‘the state of a system’ as being specified by an array of values of positions and velocities is basic to all of physics. The algorithm then shown in Figure 3 outlines how the successive states of any system are generated from the first state.

After initialization assigns initial values to the time and the state, the arrow from the box representing state initialization leads to that representing the initialization of the particle index j. It can be read as: assign the value 1 to the symbol j.

The calculation then flows to the box labelled ‘CALCULATE ACCELERATION A_j’. How this is done is specified by the dynamical law governing the system an example of which will sooin be discussed.

All of the particles’ accelerations, A_j for j=1,…,N must be calculated. They are done in turn starting with j=1 and ending with j=N. After A₁ has been calculated, the flow diagram arrow points to the question j=N?. If N=1 the answer is YES, otherwise NO. If the answer is NO, the index j is increased by one and the calculation is performed for the next particle’s acceleration. This goes on until the answer YES diverts flow to the box labelled ADVANCE TIME AND STATE.

This step assigns the values for the state at the next time. Remembering that the acceleration at T is the change in velocity between T and T +1, mutatis mutandum for velocity and position, the instructions in he box can be read as follows:

1. Assign the value T+1 to T (i.e. increase the value of T by 1).
1. Assign the value X₁+V₁ to X₁ , ..., and X_N+V_N to X_N.
2. Finally, assign the value V₁+A₁ to V₁ , ..., and V_N+A_N to V_N.

At this point, the calculation begins the iteration for the next instant.

As diagrammed above, the calculation goes on forever, but it is easy to see how one can check the value of T at this point and have it stop whenever one wishes. Now we shall discuss what goes on within the CALCULATE ACCELERATION A_j Box.

Figure 4 helps define the dynamical law used in the present model. Consider a pair of particles called A and B. The law is that if, between T and T+1, B crosses the dotted line on the left of A, B changes its acceleration by +1; if it crosses the dotted line on the right it changes its an acceleration -1; if it crosses both, it changes by the sum of the two which is 0.

Now if B crosses the dotted line on the left of A then A necessarily crosses the dotted line on he right of B and vice versa. This shows that A changes its acceleration by +1 whenever B changes by -1, and vice versa. This is the expression, within this model, of Newton’s law of action being equal and opposite to reaction

Figure 5 shows how this dynamical law is used to actually calculate acceleration A_j where j has whatever value assigned it upon entering the calculation from the one shown in Figure 3. After initializing Aj to 0, and k to 1 (k also being an index used to label particles), the algorithm is to look at each k = 1…N in turn; for each of the boundaries of k which j passes the acceleration of j is increased or decreased by 1 depending on whether it passes a left or right boundary.

Figure 6 and Figure 7 are examples for two and three particle collisions respectively.

Playing this ‘board game’ requires no intelligence, just the ability to follow rules. The rules rigidly determine every move. There is no room for free will. This presents a picture of how the universe--humans included--runs which is so insulting to both human vanity and immediate sense perception, that it would never be taken seriously were it not so successful. It was the success which established science. Prior to Newton, science claimed but had not secured, special status. Subsequent to him its status is unquestioned, but we are faced with the problem of coming to terms with it.

His gravitational theory ranged over the known universe with seemingly perfect accuracy and consistency. Not only did it connect large long known phenomena, it predicted new details too small to have been previously noticed. For example, the theory did not merely explain the elliptical orbit of Jupiter about the Sun in terms of the universal gravitation force, it correctly described very small deviations from a perfect ellipse due to the fact that the Sun was not fixed in position. The Sun is attracted by Jupiter as well as vice-versa, and it wobbles slightly causing Jupiter’s orbit to wobble.

Similarly, the theory could account for a slight decrease observed in the force of gravity near the equator due to a slight flattening of the earth which makes the equator further from the center of the earth than the poles. And it showed how this flattening came about due to the Earth’s rotation. The flattening of Earth in turn accounted for the precession of the equinoxes, known but unexplained for two thousand years. Not only that, but it predicted a slight fluctuation in this precession fifty years before it could be observed.

Thus Newtonian theory could relate the Earth’s rotation to slight changes in the force of gravity over the Earth’s surface, to the rotation over thousands of years of the constellations, all with amazing accuracy.

Newtonian dynamics was subsequently applied beyond gravity to all known phenomena with similar success. Limits were reached only with the advent of relativity and quantum mechanics in the early 20th century whereupon it became absorbed into more comprehensive physical theories. When this occurred, its basic form was also absorbed. All basic physical theories still embrace the concept of a state made up of (generalized) positions and velocities, which is initially arbitrarily assignable, but whose subsequent behavior is strictly deterministic. They reduce in principle to strictly deterministic ‘board games’.

How can we reconcile our subjective sense of the world with a board game? How can we be reconciled with a worldview based on a mindless game?

In Newtonian mechanics, the acceleration of a particle is the net force upon it divided by its mass. In the discrete model just discussed, all particle masses were, for simplicity, set to unity. Thus the acceleration each underwent equalled the total force upon it; the dynamical law determining the accelerations of the particles at each instant was equivalently, a force law.

Force is a key component of both Newtonian dynamics and of our subjective sensations. It connects immediate human sensation with mathematical theory. The history of the force concept is one of transcendence of the naive metaphysics born of immediate sensation.

When we are born, everything is unexpected; every occurrence is a surprise to the wide-eyed infant. This cannot last for long; the brain cannot remain in a continuously surprised state. Its job is to learn what to expect. Learning in this sense is a continual process of reclassification of phenomena from the category of the unexpected to the expected.

Being natural is close to being expected. How else do we learn what is natural? Even when behavior has unpredictable details, like the erratic flutter of leaves, it is called natural if flutter is what we have come to expect of leaves. We have statistical expectations of leaves: an expected range of speeds, and of amplitudes of motion to and fro. Things are behaving naturally if behaving expectedly, even if only in a statistical sense.

The association between the expected and the natural is made even for phenomena of miraculous complexity such as life. An animal’s innumerable parts–every atom, molecule, cell and organ of it–act together to form each and every organized action it takes. Every movement requires the coordinated motion of >>10²⁷ atoms. Yet the same person who will say that such organized behavior is miraculous finds it completely natural. We are willing to call the most miraculous kind of organized activity natural as long as it is something we have learned to expect.

Composing a symphony is not called a natural process even though symphonies are incomparably less complex than any kind of life. Composing is an art, not normally said to be a natural process. If however, the same symphony somehow kept cropping up, independently composed by different people in different places and times, this judgement would change. It would soon become expected, and its beauty would be considered to be of the perfectly natural type, perhaps like that of a rose.

As we learn to expect something, it appears more natural.

Being natural is also associated with being unforced, and conversely, we guess something is subject to force when it acts unnaturally, or unexpectedly. This association is created in infancy, with the help of sensors of pressure and tension. The infant learns to change its immediate environment by either moving itself or other things, and while doing so it senses the pressure and tension in association with changing its environment. Unexpected changes are associated with force.

With an infant’s increasing mobility and consequent variety of sensation, it also individuates: it learns that it exists, it becomes conscious of having a will. The infant distinguishes its self from its mother, and its self and its mother from its remaining universe. Certain motions in the environment (the mother moves, the bottle appears) appear to be consequent to motions of mother. The infant can then draw an unconscious analogy between its self and its mother: like its self, mother has her own self with a will which also generates forces. The infant associates will with the generation of force which changes the position and velocity of things from what would have been their unforced, natural and expected state.

With increasing experience increasingly many phenomena become expected. The shining sun, the supporting floor, and so on, become classified as natural, and are gradually conceived mechanically, that is, as being governed by natural law. Some people eventually reach a view in which everything–the universe, life, consciousness–is either expected, or if unexpected, is at least a posteori to be expected; to them. the whole world is perfectly natural.

The basic scenario outlined here is supported by clinical psychology, and many will find it not in disagreement with introspection. Piaget made the following observation:

Here he is saying that the infant projects its association between will, force, and change onto the environment somewhat before its complete individuation: its full consciousness of self (or of the ego). As that consciousness develops, the infant becomes increasingly able to classify what occurs ‘mechanically’. Piaget is assuming an association of mechanical with natural that science, in the stage in history we are now at in this discussion, was still working towards.

The history of religion and of natural philosophy reflects that of the child’s psychology, and the history of the concept of force, testifies to the tenacity of infantile ideas. Max Jammer begins tracing this history as follows:

Thus the pre-scientific mind, like that of the infant hypothesizes that all forces are generated by wills. The most primitive version of this hypothesis imbues even individual stones with such an animistic force. At increasingly higher level the force is attributed only to amulets, and with increasing abstraction, a belief in spirits, and then gods (which are less local and specific than spirits). Neither is monotheism wholely immune from this association so long as it retains anthropomorphisms such as God’s ‘will’.

Piaget says that During the later stages [of its development],… dynamism is ousted from the child’s conception of the world by a more mechanical way of thinking; nonetheless, as Jammer says, this infantile dynamism remains at the heart of adult beliefs: occultism, magic, and pagan mythologies. These two statements are not in contradiction for the following reason. We know that patterns of thought are never erased from the brain, only censored. Just as the physical brain is built from an ancient and primitive core overlayed with successive evolutionary additions, so also is its psychic content built up in layers, by education and experience, starting from a primitive and infantile core. Indeed, much of traditional psychotherapy is concerned with reaching down, past censors, to these primitive emotional thought patterns in the hope of helping a persons more recent emotional overlays come to terms with them. So infantile dynamism remains in the adult brain regardless of more sophisticated overlays. It is therefore not surprising that when a person digs down to express their most deeply held beliefs, their religion and philosophy, they discover their intuitions within this infant core. If education’s subsequent psychic layers have not been integrated with this core, they are easily pealed off without a trace during times of stress. So even if a person may know that a stone is inanimate, the infant within remembers that it can exert animate force.

The progression of psychological thought noted by Piaget and religious thought noted by Jammer is also reflected in the history of natural philosophy starting from that of ancient Greece. Their early cosmologies were animistic, they imbued the basic stuff out of which all matter was created with life. Force at some subsequent point became abstracted from matter into ‘spirit’ Eventually,

In his Dialogues, , Plato says:

and elsewhere,

Taken together these associate any motion with life because common observation is that the natural state of matter is non-motion. If you throw a rock, you give it a certain spirit; while the spirit is within it, it keeps going; when the spirit dies away or leaves it, the rock stops. On the other hand, since heavenly bodies move forever, the contain a living spirit forever (i.e., they are immortal).

Aristotle does not associate all motion with spirit and life; he begins to distinguish animate motion from inanimate motion, and he also introduces the idea of forced and natural motion:

That is, everything has a natural state of motion (up for smoke, uniform circular motion for planets, down for rocks, …) and in addition can respond to forced motion. When a force is present, the motion becomes unnatural. However:

Thus here, the concept of a natural force does not yet exist, and natural motion has separated somewhat from that of will and is due to an ambiguous unforced internal ‘tendency’.

Skipping to the mid fourteenth century, John Buridan separated natural motion from spirits (intelligences) even further,

He is saying that the celestial spheres to which the heavenly bodies are attached do not need to be continually pushed by ‘intelligences’; they were given an initial impetus (now called a momentum) by God which now suffices to keep them moving without further efforts required on His part. The reason the spheres keep turning is the absence in heaven of ‘corruption’ (which we nowadays call dissipation, most commonly due to friction). There is not only no force associated with natural motion, there is not even an Aristotelian ‘tendency’. A naturally moving object is given an inanimate ‘impetus’ by God at the begining of time which it would keep forever in the absence of external forces of dissipation.

In the mid fifteenth century, Cusanus was able to use a model globe of the Earth to illustrate this point:

Yet even with all this insight, deeper, more animistic layers of consciousness incongruously assert themselves as he describes this motion further:

We see here that after all, motion has only incompletely separated from spirit.

The view of William Gilbert, the contemporary of Galileo and Kepler, when discussing the force of gravity, is:

This brings us full circle as Gilbert connects, even though only by analogy, the force of gravity with the psychology of the infant’s desire to return to the mother’s body. The naive metaphysical layer of the infant is just below the surface.

Kepler made perhaps the most important break in the ancient and deeply rooted connection we have been following. In annotations to the second edition of his Mysterium Cosmographicum published in 1621, after telling the reader to substitute the word ‘force’ in place of ‘soul’ (which is used here as yet another designation of mind, intelligence, will, spirit, and so on) he goes on to say:

He then tells us that the key to his conversion was his discovery of a mathematical relation obeyed by soul. Between the first and second editions of this book, he had discovered his second law of planetary motion. As each planet’s distance from the Sun periodically increases and decreases along its elliptical orbit its velocity decreases and increases so that it is always sweeping out area at a constant rate. What this means is illustrated by Figure 8 which shows a planetary orbit over two equal periods of time. When the planet is closer it swings around at higher speed than when it is farther. It does this in such a way that the shaded areas are equal.

Like those before him, Kepler initially thought that when something was set in motion, it gained a certain amount of some soul-like quality from the mover. The planets had gained such a quality from God when He originally set them into motion. The possession of velocity implied the possession of this soul. This belief was undermined, however, when he discovered that the planets periodically lost and gained velocity and hence soul with mathematical regularity. This mechanical variation, its dependence on geometric distance, and the fact that life is usually dissipated to, not ingathered from, its environment, were not consonant with soul-like qualities.

Kepler’s conversion was not as simple as the quote might suggest. He did not simply drop his previous ideas upon discovery of his law. For example, he found his second law in December of 1601 and yet was still discussing forces as souls in March of 1605; but eventually he came around to writing that:

Though they did not mention it, this viewpoint had to have influenced later thinkers like Descartes and Newton. In this way, Kepler contributed much more than his laws of planetary motion to Newton’s final synthesis. Newton mathematically defined velocity and force. As Kepler had realized, once it could be mathematically defined, all life went out of it.

Paintings of only a few centuries commonly featured spirits pushing, pulling, and blowing objects in the environment in order to make the move. Spirits then filled the air, conveying force and motion. Today, waves and fields¾ electromagnetic and gravitational¾ fill the air. They perform the same functions as spirits but with lifeless, mathematical exactitude. The story outlined here has been of this transition in sensibility, a transition from a naive to a more sophisticated metaphysics. It illustrates how the naive metaphysics born of immediate sensation is transcended by the connections formed by science between immediate human sensation and mathematical theory. This process continues today.

Newton’s theory required that the only force particles felt when separated by astronomic distances was that due to gravity. In order that this be true, nothing which could exert an ordinary mechanical force could occupy interplanetary space. If there was matter in the space the planets had to traverse, they would be slowed down by it. Interplanetary space could not contain matter; it was a vacuum. This was exactly in opposition to the Cartesian view in which space was matter.

In the years since the meeting of Beekman and Descartes the mechanical worldview had become pre-eminent. Newton himself had been taught it in college. He would have liked to understand gravity from this eminently reasonable viewpoint¾ to explain gravity in terms of microscopic mechanical forces. Descartes had asserted that microscopic mechanical forces were exerted between bits of matter/space moving in vortices, but he had failed to convincingly explain gravity using this idea and, in fact, Newton was able to show in his Principia that Cartesian vortices could not explain gravity.

Although no one knew how to deduce the force of gravity from mechanical forces, Newton knew how to deduce all of astronomy from gravity, that is by simply postulating that the gravitational force of attraction on a particle of mass m due to a particle of mass M separated by a distance R, was of the form Fµ mM/R² . He could not explain that force¾ how it always happened to appear at the right place, at the right time, and with the right value. He just showed that if it did, the phenomena could then be accurately deduced.

Figure 9 shows the Sun far from a planet with nothing between them, and yet exactly the right force somehow appeared at the planet. At each instant, that force somehow appears out of nowhere to acts at a distance far from the sun.

How does this force, which acts at the planet¾ i.e. is at the planet)¾ ‘know’ the Sun’s mass and where the Sun is? How does the information get from the Sun to the planet if between them there is nothing but empty space? How does the force, or the planetm ‘know’ that, millions of miles away, the sun has not suddenly moved? Nothing was in interplanetary space to transport the information about this sudden movement and yet it appears instantaneously at the planet¾ conveyed instantaneously by nothing.

Another view of this problem is provided by Figure 10. At the top a magnet has attracted a piece of iron which lies directly beneath it on a smooth surface. Between them lies empty space (or air whose presence makes no difference in this case). Suddenly the magnet is moved to the right. The iron will follow almost immediately. But how does the iron know which way to go? There is nothing in the vacuum between the magnet and the iron to convey this information to the iron.

The mechanical worldview explained that the sun sends force to each planet like the hand sends force to nail via hammer; like we interchange information via sound waves in air; like force is transmitted via water pressure. Hammer, air and water all establish physical connections for the mechanical transmission of information.

Another way of stating the problem of action-at-a-distance force is that it does not satisfy continuity, the condition which requires that between cause and effect there be a continuous connection of intermediate causes and effects. This is illustrated by Figure 11 which shows pictures of a disturbance propagating along a stretched string. One end is wiggled ( initial cause); the other end eventually wiggles (final effect); between them, in space and time, a disturbance travels down the rope. At any intermediate time, the disturbance, having reached an intermediate point, can be thought of as both the effect of a prior disturbance (intermediate effect) and the cause of a later disturbance (intermediate cause).

The chain of cause and effect explains how you, if you are wiggling the string, are the prime mover behind the final effect. Each intermediated disturbance is the agent–the direct cause–of the next disturbance. A large wiggle at one end produces a large wiggle at the other, the direction of the wiggle–right or left, up or down–is similarly transmitted, and so on. All this is like information telegraphed down a wire. In the Cartesian view of gravity, the role of the string is played by matter/space.Despite all the failings of Cartesian physics, its mechanistic underpinnings easily explained these important properties and the scientific community was loathe to let go of them.

The immediate scientific reaction to the Principia was therefore mixed. Huygens, for example, accepted Newton’s force law between planets and the consequences deduced from it:

This ‘remarkable property of gravity’ was needed to obtain Newtonian theory’s vast and incontrovertible agreement with data. But he did not believe that Newton had ‘explained’ gravity. Why? This is what Leibniz wrote to Huygens after reading Newton’s Principia:

Huygens was a Cartesian and had put forward his own explanation of gravity in terms of vortices. Although it was clearly wrong, it explained [gravity] … plausibly, that is, mechanically. Leibniz’s use of the Aristotelean expression virtue signaled that he believed Newton was regressing towards ancient and discredited forms of explanation which were equivalent to having spirits move planets. Spirits were not only agents of motions but were ‘intelligences’ carrying information.

Newton’s thoughts on all this are revealed In a letter written in 1693 to a clergyman:

Thus Newton agreed with his critics. Nevertheless, toward the end of his Principia, he states:

This merits careful reading for it is an expression of an important part of the modern scientific worldview, the culmination of a long development we have been following.

He has succeeded in explaining–that is, mathematically reducing–a large amount of observational data to his theory of gravitation.

He had not explained the force of gravity, meaning that he had not solving the problem of how one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another. He could find nothing in observed phenomena which could suggest what such an explanation of gravity might be. It is crucial that the suggestion had to come from phenomena as opposed to metaphysical presuppositions. As he goes on to say:

What does "deduced from the phenomena" mean? The phrase cannot be taken literally. With respect to ‘deduced’, it is impossible to deduce theory from phenomena in the literal sense of logical deduction. The phrase must be taken to mean that theory is closely tied to, suggested by, directly inspired by phenomena. With respect to ‘phenomena’, in Newton’s case, the most basic phenomena were in the reduced form of Kepler’s laws.

The methods used by mathematical astronomers during the development which led to Kepler’s laws are analogous to those used by Newton and physicists who have followed him. As we have seen, Babylonian astronomers found that organizing observations in tables which displayed differences in positions of repeating events, and then differences of differences, revealed an underlying simplicity which could be used to predict future events. This underlying simplicity was an example of a phenomenological law derived directly from data. The more geometrically minded Greeks organized and extended the Babylonian data via the concept of uniform circular motion which is clearly suggested by direct observation of the heavenly motion now understood to be due to the Earth’s daily rotation. When extending these ideas to include planetary motion, Greek and subsequent astronomers relied on simple variations of uniform circular motion: they displaced the center of circles from the sun, they allowed constructions involving circles upon circles (epicycles) and so on. The heliocentric theory revived and extended by Copernicus was, again, both a mathematically simple step (of course, not simple in other respects) and a striving towards simplicity. Mathematically, it amounted to merely a change in in origin from the diurnally rotating center of the Earth to the non-rotating center of the Sun. And it strove towards and attained the almost complete elimination of epicycles (molehills were made out of mountains) and the long range order in planetary motion this revealed. These two simplifications were essential in order that Kepler could see his way to elliptical orbits. And it is important to remember that in principle, elliptical orbits are themselves a choice, made on the basis of simplicity, out of an infinite number of mathematical curves which could have been equally well fit to the data. Ellipses are conic sections and as such are simple generalizations of circles studied since antiquity. As a mathematician who had studied the properties of conic sections in great detail, Kepler would naturally think of ellipses to use to fit the data; it would be exceedingly unnatural for him to try to think of something else out of the blue. Who purposely multiplies work?

In sum then, mathematical astronomers created phenomenological theory by fitting observation to simple mathematical forms in a succession small conservative steps, re-using old ideas whenever possible, and making variations when necessary. They combined pragmatism with vision. They had a clear goal which was the simplest possible representation¾ the greatest reduction¾ of data, and they looked for what worked to achieve that goal. That was the pragmatism. On the other hand, those astronomers who advocated heliocentrism chose to strategically overlook its many apparent defects such as the physical implausibility of a spinning Earth. These defects made their more sensible colleagues reject heliocentrism. That was the vision part.

The remarks of Newton being discussed here apply particularly to the pragmatism of the scientific method he was defining. His interest is focussed on explaining the data. When he says …hypotheses … have no place in experimental philosophy... , he reads out of science as defined by his experimental philosophy, all statements which either cannot now, or can never be, tested by data, whether they be from Hermetic occultism or from metaphysical prejudice (mechanism). He reads into the broader realm of physics the pragmatism of the mathematical astronomers:

To be graduated from mere hypothesis, it is sufficient (to us it is enough) for a law to account for all the phenomena in some large domain of phenomena (for all the motions of the celestial bodies), and this means that the data in a large domain of phenomena (chemistry, optics, electricity,…) must be reduced to theory.

This pragmatic attitude manifests itself in method, as the following example illustrates. When Newton first started thinking about gravity and mechanics in the latter part of the 1660’s neither he nor anyone else had a clear idea of the meaning of force, mass, acceleration, or inertia. Like everyone, he had a sensation of force associated with causing a body to start moving or with stopping an already moving body, and he knew that this force increased with the speed and the size of the body. In his notes, Newton started with a verbal description of force: Force is ye pressure or crouding of one body upon another. The first thing we should note is that spirit is not present in his force concept. Our previous discussion shows that even this simple idea required a long metaphysical development. Also important to note is that he already understands force to be an effect of one body upon another as opposed to being a property of a body. His early notes also contain the assertion,

which shows him beginning to quantitatively relate force with change in quantity of motion, that is, change in velocity.

What is crucial is how he soon began to go beyond such verbal statements. In order to do so he understood that he needed to develop a measure of force; the word had to be related to quantity through measure. Until that was done, force could not be said to be scientifically defined. And this really means, well-defined¾ defined with the rigor that science requires.

Newton had to guess at a definition which was on the one hand in agreement with the equivocal verbal descriptions he and his predecessors had sketched out, and on the other hand was univocal. He also did this in a specific context which was that of uniform circular motion. He asked for the force exerted by an object undergoing uniform circular motion on that which holds it to that motion¾ that which prevents it from flying off in a straight line as Galileo had shown it would. That force is, for example, the string tension constraining a whirling object.

Neither of these actions¾ the guess and the specific context¾ appear to be profound, and yet in a sense they are; they characteristize methods embodying the pragmatism of theoretical physics and very definitely do not characterize philosophic thought. Modest as they are and appear to be, the separate physics from philsophy.

Newton guesses at a useful definition of force; the guess will be validated by its usefulness. The guess could be precisely stated because he knew everything there was to know about uniform circular motion; the properties of circles were thoroughly understood and uniform motion is the simplest of all motions.

The guess, illustrated in Figure 12, was that force is to be measured by the length of the line BD for reasons which will be explained. The figure shows a small circular arc from A to D which is being traversed by a particle undergoing uniform circular motion around the point C a distance R away. R can be taken to be the length of a string constraining the particle to move along the circular path.

The arc is traversed in the time t at speed v. If the particle had not been attached to the string when it started moving with speed v at point A, it would have moved in a straight line to B during the time t. Therefore BD is a measure of its deflection due to the force exerted on it by the string over the time t, and this is why it was chosen as a measure of the force by Newton.

It is immediately apparent that this measure has a problem in that it also depends on the time t. It is not purely a measure of the force deflecting the particle; the larger t, the larger the deflection. This measure of force is defined precisely but not accurately. Newton was naturally aware of this and we shall shortly see how the problem was dealt with.

The length of the arc is the distance traversed in time t at aspeed v which is vt. The angle ACB in radiancs is the arc length divided by the radius: vt/R. Some elementary geometrical reasoning then shows that the angle DAB is half of this, or (1/2)(vt/R). A portion of Figure 12 is expanded in Figure 13 to indicate that BD is approximately equal to the length of the arc BE cut by the angle DAB on a circle centered at A and of radius vt. This arc length may be written as (vt)(1/2)(vt/R) = (1/2)(a)(t²) where v²/R could be identified as the acceleration a thanks to the work of Galileo who had derived the expression (1/2)(a)(t²) as the distance an object drops in a time t when subject to constant acceleration.

Newton was still 20 years away from precisely identifying (a) as an instantaneous acceleration. He had not explicitly formulated the idea of instantaneous quantities. He had not yet distinguished force from what would today be called impulse (force exerted over a finite time and measured as force x time). But he did know a lot about limits and therefore knew that as t got smaller, the length of BD approached that of BE so that his formula became increasingly accurate. He knew that the idea of force also would also have to include a concept like that which we now call mass. At this stage, his measure did not; thus the length of BD he was using as a measure of force was only a measure in the sense of proportionality. Since the time dependence of his measure of force manifest by its factor of (t²) was also a proportionality, the inaccuracy in the definition of force which it represented was easily accounted for. One could, for example, always choose to use t=1 in some units, which is in fact what Newton often implicitly did.

Thus far the discussion has shown that based on the general idea of force which had by then developed, Newton guessed at a precise definition of a valid measure of the force constraining a object in uniform circular motion known as the centrifugal force,, and found that it should be proportional to v²/R. The context of uniform circular motion was important for two reasons. First because the mathematics involved in such motion was completely understood, and second because such motion approximates that of the planets around the sun according to the Copernican system. Indeed, most of Newton’s predecessors and competitors had also concentrated on uniform circular motion for just these reasons.

In the context of planetary motion it was therefore natural that Newton should substitute the expression he had obtained for the centrifugal force into Kepler’s law relating planetary periods and radii. Kepler had found that the square of a planet’s period Y (that is, its year) was proportional to the cube of its distance from the Sun: Y^{2 µ} R³. Now the year is the time it takes the planet to circle the and when in uniform circular motion, this is related to its velocity v, by vY = 2p R. Combining these relations one finds

v²/R = (2p R/Y)²/R µ R/Y² µ R/R³ µ 1/R²

The force constraining the planets to their orbit varies inversely with the square of their distance from the Sun.

At this point, physical intuition would have suggested to Newton that his guess for the measure of force had been an important step in the right direction. He had replaced a property of many particular planetray orbits by a force apparently exerted by the Sun on any body in space. All varieties of gravitational motion¾ of the planets, of projectiles, of tides,...¾ might be understood in terms of a simple force law. It would result in a dramatic reduction of astronomical data; the data would be understood in terms of the force concept, and a simple 1/R² force law.

This is the way the inverse square law for gravitational attraction was ‘deduced from the phenomena’. This is the sense in which Newton understood this word in the context of physics, and it is the way new results are generally deduced in physics. It is done by combining intuitions such as that of force, with precise mathematical definitions and with phenomenological laws such as that of Kepler.

Many further steps, mostly small steps, were necessary to solidify the argument, to make it approximate deduction in the mathematical sense of necessity. Each step reduced more data by explaining it in terms of the same force law. As this was done, the force concept became perfected; its dependence on mass and direction was worked out in the context of these applications. The concept of force itself (instantaneous force) was distinguished from that of force exerted over time (impulse). And an understanding of inertia had to slowly emerge. In all this we see how concepts were ‘deduced from phenomena’.

Newton, the physicist, deduced concepts from phenomena; in much the same way one can say that Descartes, the philosopher, deduced concepts from metaphysics. Newton was willing to work with a concept of force he did not completely understand. As he worked with it, as he studied phenomena using it, it became a more refined (or accurate) theoretical tool. Even when it was inaccurately define such as when it was confused with impact, mathematics required that the concept of force be precisely defined.

Descartes, on the other hand, would only work with concepts he clearly understood. This meant in practice that he sought the most basic concepts of which he was aware¾ thinking, being, extension, motion, …¾ formulated statements about them he could not doubt, and made these statement the irreducible basis for his philosophy, natural pohilosophy included. These basic concepts and the relations between them which he could not doubt (I think therefore I am, the essence of matter is extension ,God must exist, ...) were highly refined statements of the naive metaphysics all of us develop in the early years of life.

These difference between Philosophy and Physics (and science in general) are characteristic. The goal of the scientist is centered on understanding the phenomena, that of the philosopher is centered on understanding thought. These are not incompatible but they are different and the methods of one are not those of the other.

Comparing Newton’s private letter to the clergyman to his publication, we see that his attitude is that metaphysical beliefs, however strong, are not part of public science. This is the reverse of Descartes who held that metaphysics is, as its name implies, the very foundation of science. Newton’s attitude was not an aesthetic he wished to impose on science. His removal of metaphysics’ position of primacy was an existential imperative, a forced recognition of the power of mathematical physics, forced upon Newton as much as on everyone else. The mathematical analysis of nature had led to results, so impressive and so useful, as to determine, and not be determined by, metaphysical intuition. Not even implausibility could interfere with such utility. This overriding pragmatism in Newton’s experimental philosophy is basic to modern scientific practice.

Until the existence of overwhelmingly convincing objectively verifiable data reduction on a massive scale was first revealed by Newton, this imperative, as well as the consequent separation between science and philosophy it represented, could not have been clear. After Newton it was and mere plausible explanation on the level promulgated by Descartes rapidly became unacceptable as science. Plausibility in the sense of metaphysics¾ is it reasonable to suppose that A can influence B from a distance with no connection between them? Or to suppose that there can be a wave with nothing waving? Or that something can be both a particle and a field? …¾ became secondary to Newton and to physics whereas to Descartes and to philosophy it has remained primary.

This attitude of science was a great letting go of Descartes’ simple and easy to know ideas as the sure bases of true knowledge. It was a great, perhaps the greatest juncture, in the history of ideas. Letting go set humanity adrift from the impression of certainty its intuitions had given it. Scientific intuition now grows from experience gained in dealing with scientific reality, and the latter is a super-sensible reality which our mind creates out of mathematical logic and experimental fact.