Reading "The How and the Why" by David Park
Updated: Aug 17, 2021
"The How and the Why: An Essay on the Origins and Development of Physical Theory" is an impressive monograph by Professor David Park on the history of scientific endeavour since its earliest days. It is necessarily broad, covering two and a half millennia in 400 pages, but Park somehow manages to weave a coherent narrative of the development of physical ideas that at once emphasises their commonalities and divergences.
As a university graduate in physics, what leapt out to me the most was how certain ideas could have been conceived of by their creators in so different a way to that which I had been taught in lectures. Indeed, Park's narrative reveals that conceptions of the universe and its laws have changed so radically over the past centuries, such that ideas that we may today take for granted could have been blasphemous to the scientific communities of the past.
The range and scope of the book are astounding, so I shall focus primarily on those elements of physical theory that interested me due to their divergence from the physics taught in universities today. To me, they emphasise how scientific thought has changed so drastically over time, and may do so again in the future.
I have structured this review into five parts: the first four summarise those ideas which I found most interesting, divided into the categories of "motion", "atoms", "the heavens" and "methods". The fifth provides some of my general reflections on the book.
According to Aristotle (384 - 322 BC), the motion of physical bodies can be of two types: natural and violent. "Natural motion" is the movement of an object in accordance with what it is, whereas "violent motion" requires the action of an external force ("everything that moves is moved either by itself or by something else"). The idea that objects can be moved by themselves was supposed inspired by Aristotle's studies in biology, about which he "wrote far more about ... than about any other science". After all, "all of an animal's voluntary movements are by definition natural" by virtue of the fact that they need not be compelled by an external force (p.47).
However, the concept of "natural motion" could also be applied to any of the four elements that compose all matter: Earth, Water, Air and Fire. The natural motion of Earth and Water is downward (as we can see in the downward motion of a river), whereas that of Air and Fire is upward. The stars do not fit this scheme; since their motion is neither up nor down, but constantly circling above in the heavens, Aristotle ascribed to them a fifth substance, ether (p.48).
The fundamental idea behind natural motion is that "cause inheres in substance" (p.47) - a thing moves in such a way because of what it is. However, if a thing is caused to move by an external influence, it undergoes "violent motion". The key point here is that to Aristotle, violent motion "persists only as long as its cause persists" with natural motion resuming its course afterward. Why then, does a thrown ball keep travelling after it has left the hand? Aristotle "ties himself in knots" to provide the explanation: the throw "imparts motion to the air, which in turn pushes the ball with diminishing force until natural motion dominates and the ball falls to earth" (p.48).
Aristotle, following the dictum that "everything that is on motion must be moved by something", then suggests a principle which may be stated as: the velocity (V) caused by a force (dynamis, F) on an object is in proportion with the magnitude of the force, and in inverse proportion with the resistance (R) provided with the object, which is turn proportional to its bulk. In other words, V ∝ F/R (p.139).
This fundamental dictum would persist for centuries until classical mechanics was eventually constructed by the likes of Galileo and Newton. However, in the intervening period, there were a few who questioned it. For example, Byzantine philosopher John Philoponus (490 - 570 AD), doubted whether "a flying arrow can possibly be kept going by movements of the same air that also resists its motion", pointing out that in the extreme case of a vacuum, with no resistance, the arrow would be moved infinitely fast (but by what?). To get around this issue, Philoponus suggests that "it is necessary to assume that some incorporeal motive force is imparted by the projector to the projectile, and that the air set in motion contributes either nothing at all or else very little to the motion of the projectile" (p.140). In other words, once the motion is started, no external force is required to keep the object moving.
This "incorporeal motive force" was later given the name of impetus. Many years later, French philosopher John Buridan (1301-1358) took this thinking further by suggesting that "the impetus I of a moving body arises jointly from the quantity of matter it contains, M, and the velocity of its motion, V", a striking resemblance to our modern concept of momentum (p.140).
However, as late as the early seventeenth century, the shadow of Aristotle remained. In 1609, German astronomer Johannes Kepler (1571 - 1630) published his Astronomia Nova, a compendium of his decade-long studies on the motion of Mars. In it, he proposed an early form of what we would now call "Kepler's law of areas"; the line joining Mars and the Sun would always sweep out equal areas during equal intervals of time.
In what Park calls "the first time in history that anyone had ever put together a quantitative theory to account for the motion of anything" (p.176), Kepler attempted to deduce a theory of influences that could explain the patterns of planetary motion he observed (see Section III: The Heavens). This theory was also unprecedented in that it assumed the causes of planetary motion not to "lie in the planets' inherent properties ... but in a force exerted by the sun" (pp.153,154). However, Kepler's fatal error was to "regard planetary motion as violent motion, so that it had to be kept going" (p.176). From this assumption, he proposed that the sun emits a kind of motive power that, as the sun rotates on its axis, "issues from it like the spokes of a turning wheel to drive the planets along". Kepler combined this picture with his law of areas to arrive at the conclusion that this driving force must obey F ∝ 1/r, where "r" is the distance to from the sun to the planet. Kepler was displeased with the conclusion, due to the fact that the intensity of light varies inversely as the square of the distance (and therefore the motive force should as well). He would have "preferred the inverse square for his driving force, but the simple inverse was what he got" (p.176).
The final attainment of the inverse square law was the work of Isaac Newton (1643 - 1727), and it depended on two experimental observations recorded by Galileo Galilei (1564 - 1642). First was the assertion that "a ball once started rolling on a level surface would keep on indefinitely in violation of Aristotle's law, which said that without a force to keep it going it would stop" (p.202). This statement would come to be generalised by Newton as his "first law": in the absence of an external force, an object persists in its current state of motion.
The second observation relates to the acceleration of falling bodies. Aristotle, whose views remained dominant until this period, once remarked that "a little bit of earth, let loose in mid-air, moves and will not stay still, and the more there is of it the faster it moves" (p.203), a comment that was occasionally taken to mean an object's rate of fall is proportional to its mass. However, experiments involving lead balls of varying masses, carried out by Galileo as well as others, directly contradicted this notion. Moreover, Galileo pointed out an inherent logical contradiction in the idea: suppose we have a light stone and a heavy stone tied together. If we assume that the stones are separate objects, we conclude that the light stone wants to fall more slowly and the heavy stone wants to fall more quickly, therefore the lighter stone would exert a retarding force on the heavy stone and so the combination would fall more slowly than the heavy stone alone but more quickly than the light stone alone. However, if we assume the combination to be one single heavier object, then it would fall more quickly than would the heavy stone alone. Ergo, "Aristotle's supposed principle leads to a contradiction" (p.204).
Instead, Galileo arrived by experiment at the following principle: "the straight-line acceleration of heavy bodies takes place according to the odd numbers, starting from one. Choose whatever equal intervals of time you wish. Then if the moving body, starting from rest, travels a distance of 1 ell, say, during the first interval, it will travel 3 ells during the second, then 5, then 7 and so on, continuing with the successive odd numbers". As Pythagoras pointed out, the sum of the first N odd numbers is the square of N (p.16), therefore, the distance traversed in this observed motion increase as the square of time. It had earlier been discovered by Merton College scholars in the Middle Ages that in motion that is "uniformly difform" (we would say "uniformly accelerated"), the distance traversed also increases as the square of the time. Galileo therefore concluded that a falling body gains velocity at a constant rate (p.205).
Newton knew about both of these observations. On their account, one should not ask 'what keeps the planets moving?', but rather, 'what keeps the planets orbiting the earth rather than flying off into space?'. This is the question Newton sought to answer, and he speculated that, just like an apple falling from a tree, the planets may be drawn to the earth on account of their weight. Employing Kepler's "harmonic law" and an analysis of centripetal forces, Newton arrived at the inverse square law that Kepler himself sought but never reached (p.180,182).
Newton had arrived at a description of what we call the "force of gravity", but he was still unsatisfied as to what the physical causes that process were. In a very famous letter from around 1691, Newton makes this following assertion: "that gravity should be innate, inherent and essential to Matter, so that one Body may act upon another at a Distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical matters a competent Faculty of thinking, can ever fall into it" (p.188).
Young students of physics are often made to take for granted the fact that objects can influence each other at a distance - after all, this is "Newton's law of gravity" - without discovering that Newton himself expressed great discomfort at the idea. To deal with this, he developed several ideas pertaining to a kind of "ether" that pervades space, and that could perhaps transmit the influences that result in gravity (p.188).
Though Newton occupied most of his Principia with questions of astronomy, there are also included those three laws of motion that today bear his name. In fact, however, only the third law ("action = reaction") was truly novel and revolutionary. The first law ("a body persists at rest or in uniform motion in a straight line unless acted upon by an externally applied force") was a generalisation of observations first reported by Galileo concerning the principle of inertia. The second law ("F = ma" in modern terms) was also inspired heavily by Galileo, both his experiments concerning falling bodies (showing that gravity produces acceleration), and his "principle of relativity". The latter states that physical laws are the same in all inertial frames of reference: i.e., there is no privileged frame of reference from which to measure "absolute motion", therefore all motion is relative (Virgil once said, "Forth from the harbor we sail; the city and harbor move backward") (p.145). According to this point of view, velocity cannot be a response to a force because velocity is relative depending on which inertial frame of reference one occupies. Only acceleration is of physical significance.
The edifice that Newton built would continue to dominate natural philosophy for the next few centuries, and it would not be until the twentieth century that relativity theory and quantum mechanics would supplant it with radically different views of nature. However, twentieth-century physics is covered amply in the current landscape of popular science literature, so I shall stop there with the topic of "motion".
The first figure to have reputedly suggested that the world is made of "atoms" was the Greek philosopher Leucippus, who lived in the fifth century BC (all we know about him is through the words of his pupil, Democritus). Leucippus was purportedly responding, in a way, to the ideas of an earlier philosopher, Parmenides (born 515 BC). One core idea of Parmenides' thought appears to be the assertion that what exists is all that can exist, and therefore nothing can come from nothing. This immediately presented a metaphysical problem concerning motion and change: how can the world be constantly changing if it is impossible for things to come into existence from non-existence? (p.22,23).
Leucippus and Democritus supposedly solved this problem by asserting that nothing in fact comes into existence from nothing: rather, all change that happens in the world is simply the re-arrangement of pre-existing elements that have always been and will always be there. All kinds of "motion", whether the movement of an object from one place to another, or the transformation of a seed into a plant, can ultimately be ascribed to the re-arrangement of these "atoms" into new combinations (p.27)).
The atomic worldview that stemmed from Democritus has features both familiar and foreign to modern physics. According to this tradition, atoms "move around, or else in solid bodies they are hitched together, and in this position they vibrate. Atoms of Fire are spheres (since Fire never adheres to itself or anything else), but since flames deposit soot or moisture not all the atoms in a flame are sphere. The soul is also composed of spherical atoms, so small that they can wander through the body ... how else can soul move all our muscles?" (p.27).
Perhaps the most radical import of this tradition was the idea of a "void", which directly contradicted Parmenides: "emptiness is non-existence; something that is physically nonexistent is logically nonexistent and so cannot enter an argument concerning things that exist (p.26).
Democritus apparently attempted to use this idea of atoms to explain how we perceive the world. He made a distinction between the "primary" and "secondary" properties of an object. Primary properties inhere in an object; how the object is in reality. Secondary properties are the ones that "register in our minds", such as taste and appearance: "Sweet exists by convention, sour by convention, color by convention; atoms and void exist in reality". The gap between what exists "by convention" and what exists" in reality" is crossed by atoms: as Park interprets Democritus, "taste is explained by the shapes of the atoms tasted; sound, apparently, by the passage of atoms of sound through those of air" (p.28).
Plato added an intriguing spin on this idea by drawing a correspondence between the structure of particles and the "four elements" (an idea first introduced by Empedocles (p.25)). In his Timaeus, Plato introduces what are now known as the five "Platonic solids": those three-dimensional shapes that are bounded by regular polygons (polygons all of whose sides and all of whose angles are equal). He supposed that all matter is made from these polygons: "fire is the tetrahedron because its sharp points and angles will sever the connections between other atoms. Earth is a cube because of its structural stability" (p.36). Air is an octahedron and water is an icosahedron, whereas the dodecahedron was used to construct the heavens.
However, Plato's figures are not "atoms" in the sense that whereas "atomos designates something that cannot be cut", these shapes can be composed into triangles. The faces of Air, Water and Fire are equilateral triangles, which can be cut further into six right-angled triangles. The face of Earth is a square, which can be cut into four right-angled triangles (which are, however, different to the right-angled triangles of the other elements). Thus, "the eight triangular faces of the body of Air can rearrange to form two bodies of Fire, and we know that a fire will not burn without air. Earth, however, cannot participate in these exchanges because its triangles are of the wrong shape. This explains why Earth is incombustible" (p.38).
Despite the emergence of atomic theory in antiquity, the idea makes little mark throughout the Middle Ages. Isidore of Seville (560 - 636 AD), a Spanish scholar and cleric, suggests that this may be partly due to the Church's low opinion of the atomists - followers of Democritus such as Epicurus: "Epicurus ... [was] a lover of vanity and not of wisdom, ... wallow[ing] in carnal filth and declaring that bodily pleasures were the highest good ... He assigned the origins of things to atoms ... from whose chance combination all things arise and have arisen. He said that God does nothing, that everything is made of particles, and that in this respect the soul is no different from the body ... The same material is used and the same errors are repeated over and over by heretics and philosophers" (p.97).
Nevertheless, he seems to have taken the idea of atomism to surprising lengths: "There are atoms in a material body and in time and in numbers and in letters":
"You can divide a body like a stone into parts, and the parts into grains like sand, and again the grains of sand into finest dust, continuing, if you could, until you come to some little particle that you cannot divide or cut. This is an atom of the body."
"An atom of time means this: You divide a year, for example, into months, the months into days, the days into hours, and you can still divide the hours until you come to an instant, a droplet of time as it were, so short that it cannot be even slightly lengthened [sic], and therefore cannot be divided. This is an atom of time".
"There are atoms of numbers: for example, eight is divided into fours, four into twos, and twos into ones. One is an atom because it cannot be divided".
"It is the same with written language, for one can divide speech into words, words into syllables, and syllables into letters. The letter, the smallest part, is the atom, cannot be divided. The atom is therefore what cannot be divided, like the point in geometry". (p.96,97).
In his formulation, the idea of the "atom" is taken in its most general possible sense, as the fundamental, indivisible unit of something, whatever it may be - not necessarily just matter. (Curiously, the concept of "atoms of time" may have re-emerged in some form in the modern notion of "Planck time", the smallest possible amount of time that can be measured due to the indeterminacy present in Heisenberg's uncertainty principle, about forty-four degrees of magnitude below unity).
Later in the book, Park discusses some elaborations on the idea of atomism by thinkers such as Descartes and Newton, but I won't go into them here. Instead, I would like to discuss some speculations carrying a flavour of atomism in the thinking of James Clerk Maxwell (1831 - 1879).
Even though Maxwell's theory of electromagnetism remains eminently influential today - having influenced the development of fields such as relativity theory and quantum mechanics - Maxwell's approach to it carried some assumptions that we would nowadays regard as completely foreign. At the time, phenomena concerning electricity and magnetism were well understood experimentally, but there was want of a theory to explain them. The concept of "action at a distance" was treated with as much suspicion as it had been by Newton more than a century before, and so it was assumed that electromagnetic forces were transmitted by a medium, some kind of "ether".
In the emerging science of electromagnetism, the word "field" was beginning to be used to "denote the state of things in the apparently empty space surrounding a magnet or a charged body" (p.268). These "fields" were usually taken to be "a manifestation of some state of tension or motion in the ether that would not be there if the agent producing it were taken away".
From this starting point, Maxwell constructed a truly fantastical theoretical model of electromagnetic interactions in the ether: "I suppose that the "magnetic medium" is divided into small portions or cells, the divisions or cell wells being composed of a single stratum of spherical particles, these particles being "electricity". The substance of the cells I suppose to be highly elastic, both with respect to compression and distortion and I suppose the connection between the cells and the particles in the cell walls to be such that there is perfect rolling without slipping between them and that they act on each other tangentially" (p.268).
This "ethereal machine" was constructed and described by Maxwell in excruciating detail, and its primary achievement was that it accounted for all the known electromagnetic phenomena, even the generation of electromagnetic waves (the model was characterised by a set of partial differential equations, which could produce a class of solutions whereby the electric and magnetic fields "detach themselves from the charges and currents that produce them and travel off into space in the form of electromagnetic waves, as far as you please") (p.269).
However, the strangest thing about the theory was that "although it was based on a detailed model of the ether almost no numbers relating to the model ended up in the equations: nothing about the sizes of the cells or the masses of the particles forming the walls - nothing but two experimentally measured numbers, k and μ, which tell how strong an electric field is produced by a given electric charge and how a strong magnetic field is produced by a given current" (p.269). The 1865 paper in which Maxwell announced these results "sets the stage with remarks about the ether; then the equations begin and ether is left behind". It turns out the relevant actors in the theory are the fields, not the ether in which they are supposed to exist. As such, the theory could say nothing about what the ether actually is.
Astonishingly, this aspect of Maxwell's theory was strongly criticised at the time: "to most physicists of the time the great task of physics was to unify the different laws into a single theory. A good many people thought that not only was ether a fundamental substance; it might be the fundamental substance ... The flaw in Maxwell's theory was that though it undoubtedly squared with the facts it contributed little to this great project" (pp.270,271). Only gradually would physicists begin to think of the "electromagnetic field" in its own right.
III: The Heavens
The world above ours has always exhibited such regularity and apparent order that we may appreciate why astronomy is the oldest of the natural sciences. Ever since distant antiquity, there have been those who have attempted not only to describe the order of the heavens, but to explain it, and to do so they constructed models that reflect their belief that underneath the order lies some kind of structure - a machine.
Anaxagoras, a pre-Socratic philosopher, appears to have been the first to teach that the moon is not self-luminous, but reflects the light of the sun. He gave a proper explanation for the moon's phases and the phenomena of solar eclipses. However, he assumed the earth to be flat, and so had more trouble explaining lunar eclipses (pp.55,56). Park suspects that by the time of Parmenides, Plato and Aristotle, the idea of a round earth was commonplace.
The idea that the earth sat at the centre of the universe, surrounded by a sphere of fixed stars, "may well be prehistoric" (p.59), though it came in many varieties. It was thought that the sphere of fixed stars rotated about two fixed poles, and that the equatorial line lay midway between the poles. The sun moved in a plane not parallel to the equatorial plane, for it was known that it spent half the year to the north of the equator and half the year to the south. As to why winters are cold, some believed that when the sun is below the equator, it is in fact under water. This is because "the spherical earth, which has to be supported somehow, was often thought to be floating on water" (p.57), an idea apparently shared by Thales (according to Aristotle).
However, the sky is not merely the sun and stars; there is the moon to worry about, and those wanderers (planetes) that move against the stellar background. It was thought that the whole celestial sphere rotated once a day, which is why the stars make one full circuit throughout this duration. The sun, which circles the earth along with the sphere, also charts its own loop within the sphere itself, and it makes this journey once a year (hence the seasons). The moon revolves around the earth once a month, hence the observable lunar phases.
There are nevertheless a few complications with this model. The sun does not move against the stars at a perfectly uniform rate, and the moon's orbit in fact "wobbles" with period of about eighteen years (p.59). Most troublesome are those planets, which undergo a strange kind of "retrograde motion" that frustrates any effort to ascribe them a simple path within the celestial sphere.
Eudoxus (b. ~408 BC) was the first to construct a theory of the planets that explained the phenomena reasonably well. In his model we are to imagine that "the stars are fixed to an immense dark spherical shell ... rotating once a day around an axis", which cuts the earth's north and south poles. Within this sphere "are fixed two pivots that serve as axes of a second, smaller sphere, made of transparent substance, that rotates within the first. The sun is mounted on the equatorial plane of this inner sphere". It is then possible to turn "the outer sphere uniformly at one revolution per day and the inner one uniformly at one revolution per year", such that we reproduce the sun's motion to a reasonable degree (p.62). Overall, in addition to the sphere of fixed stars, the sun required three extra spheres to explain its both alongside the fixed stars, as well as its annual motion and periodic variations in its position in the sky.
The moon's motion is explained similarly - the "wobble" of the moon's orbit, just like the annual alterations in that of the sun, required the existence of three spheres. Then, in order to explain the motions of the other five planets (Mercury, Venus, Mars, Jupiter and Saturn), one needed four extra spheres for each. Twenty planetary spheres, three spheres for the moon and sun each, plus one for the fixed stars, brings us to a total of twenty-seven spheres.
To account for errors in the Eudoxan scheme, Callippus (370 - 300 BC) brought the number of spheres up to thirty-three. This is the model that Aristotle encountered; yet he found it still lacking in that it could not explain how some of the inner spheres, which are meant to be pegged to the outermost sphere, could in fact "know" how the outermost sphere is moving. He thus added an extra twenty-two spheres, not meant to carry any planets, but rather to convey the motion of the outermost sphere to the planetary spheres. There were now fifty-five spheres (p.63).
Park speculates that "for Aristotle the essential purpose of this vast complexity was to be material and efficient cause of events everywhere, emanating from final causes residing in the prime mover" (see Section IV: "Methods"). Yet, no matter how successful it was, there was one set of phenomena that could not be explained by a theory of concentric spheres around the earth. It was known that the brightnesses of the planets - in particular, Venus and Jupiter - change over time. It was also known that the moon's "size" in the sky changes relative to that of the sun, as can be evidenced in solar eclipses. The only way to explain these phenomena within the concentric sphere model was to assume that the inherent sizes and/or brightnesses of the celestial objects vary over time, a proposition "unthinkable to Aristotle" (p.64).
Notwithstanding modifications by later thinkers such as Heraclides of Pontus (387 - 312 BC) and Pliny the Elder (23/24 - 79 AD), this system of ideas was that which Europe "inherited in the thirteenth century when Latin translations of [Aristotle's] Metaphysics and On the Heavens, as well as Arabic works based on them, became available for the first time". This system also included modifications proposed by the Roman scholar Ptolemy (100 - 170 AD).
In order to explain phenomena such as retrograde motion and changes in apparent brightness, Ptolemy considered that planets moved on a sort of "two-track" system. Echoing Apollonius of Perga (240 - 190 BC) and Hipparchus of Rhodes (190 - 120 BC), he "mounted the centers of the planetary orbits on rotating circles called eccentrics, centered on the earth, and on the planetary circles themselves [the three thinkers] mounted other rotating circles called epicycles" (pp.71,72). Using this model, Ptolemy was able to construct tables of planetary motion that were "so accurate that they were used with only minor numerical changes until Copernicus".
Ptolemy's calculations for the least and greatest distances of the celestial bodies appeared to show that the moon's greatest distance equals Mercury's least, that Mercury's greatest distance equals Venus' least, and so on. This marvellous correspondence inspired Ptolemy to imagine a system of "planetary spheres considered as thick spherical shells with the planetary mechanisms [epicycles etc.] inside them, lubricated in between by thin layers of ether, with the stars not far beyond the outermost sphere" (p.73). The model, the "Ptolemaic System", "dominated astronomical theory during the later Classical phase, the Arabic period, the Middle Ages, and in the seventeenth century Johannes Kepler's first book was an elaboration of it" (p.73).
This picture would not be definitively overthrown until the work of Prussian astronomer Nicolaus Copernicus (1473 - 1543), who showed convincingly that the precise astronomical predications of Ptolemy could be reproduced using a much simpler model, postulating that "the earth is a planet and that all the planets revolve on circles and epicycles around the sun, that the apparent daily motion of the stars results from the earth's rotation, and that the regressions of the planetary orbits are easily understood as consequences of the earth's motion" (p.143). Moreover, he argued that his heliocentric model was not merely a convenient mathematical construction - it held physical truth as well. He gave clear physical arguments against the geocentric view, asserting, among other things, that "the immense sphere of the heavens, rotating once a day at colossal speed, would suffer a huge centrifugal force; yet outside there ... there is "no body, no space, no void, absolutely nothing" to contain it" (p.147). His view was not accepted easily, but after seventy-five years after his death, "the heliocentric theory was almost universally accepted among astronomers, and [Copernicus' great work] De revolutionibus was firmly in place in the church's index of forbidden books" (p.149).
Yet, Copernicus' model was far from the final word - it still included now-outdated notions such as epicycles, and it did not explain the mechanisms that drive planetary motion. The next great revolution was ushered by Johannes Kepler (1571 - 1630). After graduating from the University of Tübingen, Kepler found work teaching mathematics and astronomy at Graz, Austria, securing his reputation by published an "annual astrological forecast ... [that] correctly predicted a cold winter and an invasion by the Turks" (p.149). At Graz, he published his first book, elaborating on the Copernican model by suggesting if one nests Platonic solids within each other, one can reproduce the orbital radii of the planets given by Copernicus.
After these unsuccessful explorations, he moved to Prague to work as an assistant to Tycho Brahe (1546 - 1601), the "supreme observational astronomer" (p.149). It is largely thanks for Brahe's comprehensive positional data (and some observational results given by Copernicus) that Kepler was able to arrive at his stunning conclusions. Firstly, he discovered that the sun lies exactly in the plane of each planetary orbit (a fact not noticed by Copernicus). Then, studying the orbit of Mars, he discovered that the planet travelled faster closer to the sun and slower when it was further, such that the line connecting Mars and the sun sweeps out equal areas in equal times (see Section I: Motion; this is now known as "Kepler's second law"). He further discovered that the orbits of Mars, and thus all planets, follows the mathematical form of an ellipse ("Kepler's first law"; as Park notes, "discovered" may be a strong word insofar as Kepler's method consisted of drawing conclusions about a specific case then proposing that they extend to the general case, "Kepler guessed the whole truth about the solar system from two points of a single orbit"(p.152)).
Kepler made other discoveries that described, not just single planets, but the solar system as an interconnected whole. For instance, he discovered that - in modern terms - the "cubes of the mean orbital radii of the planets equal the squares of their periods" (p.153; "Kepler's third law"). He also recorded the minimum and maximum angular velocities of the three outer planets (Mars, Jupiter and Saturn) and found that when one found their ratios, one found that they approximated the frequency ratios of certain musical intervals (p.155):
These "harmonic" relations (truly in the Pythagorean spirit) are fascinating, but they ultimately had no effect on future physics. Today, we have "forgotten these harmonies but for Kepler they were his supreme discovery ... for they revealed more clearly than any other evidence that the Creator had designed the world according to principles of mathematical beauty" (p.155).
In the years afterward, great strides in astronomy and cosmology would be made by the likes of Galileo, Hubble and Einstein... though I won't go into them here.
Throughout the ages, methods of inquiry into the nature of the world have changed. This may involve the types of acceptable kind of explanation, or the tools used to generate theories and predictions. In some way, these methods are the foundation of physics, for it is through them that theories can be made.
Park notes that Parmenides (b. 515 BC) was the first to proclaim that "to arrive at any true statement about the world one must argue it, logically" (p.21), whereas others before him had been "content to invent and proclaim some version of universal truth". Aside from the insistence on logic as a method of inquiry, Parmenides' great contribution was to conceive of an abstract world, perhaps the world as it really is, that is different to the world of the senses. It is not possible to attain truths in this abstract world through experience, only through thinking logically about ideas already present in the mind; as Park explains, "insofar as the ideas in our mind are logically arrived at, they have their roots in other ideas, not in experience" (pp.22,23).
Park summarises Parmenides' thought in eight key points:
The object of thought is thought (truths come logically from ideas, not experience).
What exists can be thought about.
Nothing can be thought about what does not exist.
The world exists. It is what exists. It is One.
The world is what it is and does not become something else.
Neither the world nor any part of it came into being or will pass away (from 1,3 and 5).
The world is a timeless whole (from 6; see Section II: Atoms)
The world is not the one we perceive with our senses (the perceived world is a world of change).
In a way, Parmenides' ideas can be thought of as constructing a program that would define the entirety of natural philosophy down to the present. He imagines that the "world" as it really is, is an abstract world not available to the senses, perhaps in the same way that physicists imagine physical law to be timeless and unchanging. It is this timeless order that underlies the changing world of the senses. Indeed, even the word "timeless" is misleading, for "the world of Parmenides was not so much an unchanging world as a world in which change and fixity are undefined" (p.23), leading to questions about motion and how it can exist if the world if "change" could not exist.
Plato (~425 - ~348 BC) founded his theory of Ideas/Forms on an interpretation of Parmenides that "restrict[s] the range of possible discourse to a conceptual world with which we have no sensory contact at all" (p.23) - perhaps in the same way that mathematics, say, can only talk about itself. Indeed, in his Timaeus, Plato provides an answer to the question of his Parmenides' timeless world is related to the world of experience: "we believe on the basis of experience, but we do not know truth: "As being is to becoming, so is truth to belief". And our world, the world of belief, was made by a Divine Craftsman as a moving model of the world of timeless being and truth" (p.34).
Furthermore, Plato argues that "the nature of the ideal being was everlasting, but to bestow this attribute in its fullness on a creature was impossible. Wherefore he resolved to have a moving image of eternity, and when he set in order the heaven, he made this image eternal but moving according to number, while eternity itself rests in unity, and this image we call time" (p.35). With these words, Plato expresses a principle that has provided the foundation of natural philosophy for the following two and a half millennia: by logic and inquiry, we aim not to understand the world of experience, but rather, the timeless eternity of Parmenides, of which the world of experience is merely a moving image.
Park summarises Plato's theory of Ideas as such: "the Ideas live timelessly in Parmenidean abstraction: Justice, Love, the Good, and so on, which the prepared mind, after much study, can perceive all at once in a single act of understanding" (p.39). To Plato, these Ideas are real, they are what is real: "the subjects of the verb "exist" and the objects of the verb "know"". And this is where Plato might have diverged from his pupil Aristotle: "for Plato, Justice is real; for Aristotle it is also real, but its reality resides in the specifics that led us to the idea in the first place" (p.40). For Plato, "the real world is the world of ideas ... while the world of opinion [of experience] depends on it; it is a copy of it with flaws inherent in the nature of the copy", whereas for Aristotle, "the real world is the one we experience and nothing else" (p.41). This distinction divided the thinkers of the Middle Ages, and it is so fundamental that "perhaps each of us inclines naturally a little bit one way or the other" (p.40).
Park notes that Plato's ideal Forms have stayed with us long into the era of modern physics as mathematical forms, even though, compared to the Forms discussed by Plato, "our own are incomparably more complex and make contact with experience at many more points" (p.42). Park argues that "it is hard to see how these structures can be derived from experiment", before quoting a bit of Einstein: "It seems that the human mind has first to construct forms independently before we can find them in things". The idea of the Platonic Form thus gives rise to the"theory".
One of Aristotle's criticisms of Plato's theory of Forms was that it "does not explain how form makes anything happen" (p.46). One on the other hand, cause rests "at the center of Aristotle's theory of nature". For every process that occurs in the world, Aristotle stated that four kinds of cause are at play: the formal cause, material cause, efficient cause and final cause. Park gives the example of a silversmith making a bowl out of silver:
"The formal cause is the design for the bowl that determines its final form".
"The material cause is the silver."
"The efficient cause is the silversmith".
"The final cause is the purpose for which the bowl was made".
Abstractly speaking, the formal cause is "one whose operation we recognize when we see the new form after the change is complete". The material cause "must exist in order that there can be a formal cause" (i.e., there must be a material in which the form is to be embodied). The efficient cause is "whatever initiates the change". The final cause is the "reason why the efficient cause operates, guiding the world one step further toward a final, divinely determined actuality". For Aristotle, the final cause is a "device for escaping from necessity" or at least "remov[ing] the appearance of necessity". This is because all events are pulled toward a future that we cannot see; they do not follow necessarily from what has already transpired (pp.46,47).
Arguably, out of Aristotle's four causes, the one that finds its place most secure in modern physics is the efficient cause; processes are explained in terms of whatever initiates them. This way of thinking perhaps found its apogee in the "mechanical philosophy" associated with thinkers such as René Descartes (1596 - 1650). Descartes divided the world into two categories: res extensa, that which is extended, and res cogitans, that which thinks. The first refers to "material substance that occupies space ... includ[ing] both inorganic and organic matter ... [which] functions like a machine" (p.217). Even animals are considered "mechanisms[s] consisting of particles of matter interacting and moving according to mathematical laws of motion". On the other hand, res cogitans "is not material but comprises pattern, function, dynamism", including the soul. In Descartes' philosophy, when talking about res extensa, we are "asking about physical mechanisms, and the answers must invoke only the kinds of cause that Aristotle called efficient" (p.218). Hence, the invocation of any kind of final cause is invalid, such as when one says that a kettle is boiling "because you are going to have a cup of tea". This 'Cartesian dualism', Park notes, is a "basic ingredient of the modern point of view". It faced no great challenge until the advent of certain discoveries in optics and quantum mechanics.
Aristotle and Descartes illustrated how one could explain a physical process, while Parmenides and Plato defined what it is that can be talked about. But what of scientific techniques, such as mathematics and experimentation?
Aristotle was surprisingly loathe to verify his ideas on motion by experiment (see Section I: Motion). As Park points out, "the experiments that Galileo performed two thousand years later to refute [Aristotle's ideas on violent motion] could have been done by Aristotle, had he cared to, using available technology, and much trouble would have been saved" (p.48). Why didn't he do so? Park explains that, from the point of view of one for whom natural and violent motion are completely different things, the experimenter's only tactic is to manipulate violent motion. If the overwhelming majority of the world's motion was natural, then experiment can "touch only an irrelevant fringe of everything that happens".
Aristotle's attitude toward mathematics was equally skeptical. His physics is a "science of causes built on the axiom that when the causes of a phenomenon have been established the phenomenon itself can be explained by a syllogistic argument as a necessary effect of its causes" (p.101). In the course of investigation, a mathematician "eliminates all the sensible qualities [i.e., perceptible qualities of external objects] e.g. weight and lightness, hardness and its contrary, an also heat and cold and other sensible contrarities, and leaves only the ... attributes of things [as] quantitative and continuous, and does not consider them in any respect".
This criticism of mathematics in natural philosophy - that it reduces the qualities of nature to number - is compounded by what Aristotle perceives to be a glaring omission in mathematics: the notion of "beauty". For example, "one cannot prove from geometry ... whether the straight line or the curve is the most beautiful of lines" (p.101). Similarly, one could not prove using mathematics that "rotatory motion is prior to rectilinear motion, because it is more simple and complete", a fact that is fundamental to his theory of planetary motions (see Section III: The Heavens).
Indeed, these criticisms may contain some element of truth. Park cites the example of Archimedes of Syracuse (287 - 212 BC), who used mathematics to derive the law of the lever. Despite Archimedes' rigour in formulating postulates and then, from them, deriving laws, Park points out that he still had to make assumptions about the physical world that could not come from the mathematics. The situation is "exactly as Aristotle had said it is: to derive even a simple physical law, mathematics is not enough" (p.102). This is because mathematics "talks only about itself", and so in order to describe the physical world, mathematics requires the help of two other kinds of statement: "nonmathematical statements, expressed in words, that tell how the mathematical symbols are to be related to our sensory experience of the world around us, and mathematical propositions derived from experience as to how the world actually seems to behave".
This dogma was challenged several centuries later by the English philosopher/theologian Robert Grosseteste (1168 - 1253). He noted that there are three paths to what might be called "well-founded opinion": logical deduction, experimental observation and mathematics. Aristotle gave "absolute primacy" to the first, for while the latter two can only provide a demonstration that something is so (demonstratio quia), logical deduction could provide a demonstration that something is necessarily so (demonstratio propter quid). Any credit given to experiment or mathematics "detracts a little from that given to logical necessity" (p.103). The Aristotelian program was dominant throughout the Middle Ages, for "God thinks; he does not need to perform experiments to know the truth, and so the philosophers of the Classical world and the Middle Ages sought to create a science that consisted of necessary conclusions drawn ultimately from premises that could not be otherwise" (p.133).
Grosseteste challenged this view by claiming that under favourable circumstances mathematics can "provide a demonstration propter quid that has as much authority and explanatory power as a syllogism" (p.103). He believed that insight into the causes of things could be found in mathematics: "There is immense usefulness in the consideration of lines, angles, and figures, because without them natural philosophy cannot be understood ... For all causes of natural effects can be discovered by lines, angles, and figures, and in no other way can the reason for their action possibly be known" (p.103). Later, Grosseteste went further and claimed that instead of simply elucidating the cause, mathematics is the cause. The underlying mathematical structure is "causally responsible" for, say, the motion of objects (Park speculates that Grosseteste could be reflecting on remarks by Saint Augustine (354 - 430 AD) to the effect that "God reasons and acts mathematically"). As Park notes, this would be a radical statement even today, with the current dominant status of mathematics in the physical sciences.
This call for the increased use of mathematics was echoed by Grosseteste's student, Roger Bacon (1214 - 1294). In his book Opus Majus (the Greater Work), Bacon, who had an interest in educational reform, argued for the necessity of mathematics in the sciences with these following points (p.134):
"Change always involves some augmentation or diminution, and these must be considered quantitatively"
"The comprehension of mathematical truths is innate in us".
"Of all parts of philosophy mathematics is the oldest, and therefore it should be studied first".
"It is not beyond the grasp of anyone" ("for the people at large and those wholly illiterate know how to draw figures and compute and sing, all of which are mathematical operations").
"The clergy, even the most ignorant, are able to grasp mathematical truths, even though they are unable to attain the other sciences".
"Every doubt gives place to certainty and every error is cleared away if a subject can be reduced to mathematical proof".
"Therefore if other sciences are to be rendered free from error they must be founded on mathematical principles".
Bacon gives an equally vigorous defence of the use of experimentation in science: "I now wish to unfold the principles of experimental science, since without experience nothing can be sufficiently known. For there are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience ... For if a man who has never seen fire should prove by adequate reasoning that fire injures things and destroys them, his mind would not be satisfied thereby, nor would he avoid fire, until he placed his hand or some combustible material in the fire, so that he might prove by experience that which reasoning taught" (pp.134,135). The increasing regard for experiment in natural philosophy was ushered in by Bacon, as well as influential figures of the church who agreed with his sentiments, including Albertus Magnus (1200 - 1280) and Thomas Aquinas (1225 - 1274). In the hands of future thinkers such as Galileo, experimentation would propel natural philosophy very far indeed.
Isaac Newton (1643 - 1727) would later articulate his own rules as to how experimentation might eventually lead to an understanding of physical law (pp.214,215):
"We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances" ("Nature is pleased with simplicity, and affects not the pomp of superfluous causes").
"Therefore to the same natural effects we must as far as possible assign the same causes". New explanations must not simply be conjured for every new phenomenon.
"The qualities of bodies which admit neither intensification nor remission of degrees and which are found to belong to all bodies within the reach of our experiments are to be esteemed the universal qualities of all bodies whatsoever". That is to say, if there are some characteristics that apply equally to all observed bodies (i.e. they obey the laws of motion), then they we should assume them to apply to all bodies in the universe.
"In experimental philosophy we are to look on propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions". In other words, hypotheses are to be generated through induction from experimental observations: the observation establishes the specific case, induction expands it to the general case. He expressed a dislike of wild shots in the dark (though was not immune to them himself).
Newton's Principia was a true achievement of mathematics, demonstrating how mathematical work could lead to a grasp of quantitative physical law. Yet, one drawback was in the kind of mathematics that he used. As Park notes, all of Newton's derivations follow the methods of Euclidean geometry, which were "tedious because they were poorly adapted to the work he was doing (p.210). Tedious also in the sense that Newton was forced to construct entirely new geometrical figures for each problem that he faced ("a figure that is useful for one geometrical proof is not very useful for another; every theorem demands a fresh figure, a fresh exercise of ingenuity" (p.245)). It in rather ironic that despite developing the method of calculus, there is "no sign of it in his explanation of mechanics", even though it would have made things much simpler (and is used universally today for mechanics). Thus, despite being a towering achievement, the Principia was also a "disaster for British science, for it sanctified archaic methods already rendered obsolete by the rapid development of calculus on the Continent" (p.210).
Ascribing an algebraic structure to Newton's mechanics was the achievement of Continental thinkers such as the Swiss mathematician Leonhard Euler (1707 - 1783). Unlike Newton, Euler expressed physical quantities in symbols, and wrote formulae much in the same way we would do today - for example, Newton's second law is dp = fdt, where dp is a small increment in momentum, f is an applied force, and dt is the short time interval (tempusculum) over which the force is applied. Park notes that "for Newton, as for the ancients, the idea of a symbol representing pure numerical quantity did not exist ... in the Principia the lines on a diagram represent only what they actually show, distances and directions, and nothing else" [emphasis added] (p.245).
Euler was not the first to use calculus to solve problems in mechanics; after all, Jean Bernoulli (1667 - 1748) had used it in 1710 to derive Kepler's law of ellipses from Newton's laws of motion. Euler was, however, the first to arrive at the idea of expressing time explicitly in an equation, an idea that was "so radical that after the Mechanica [his book on mechanics] Euler needed another fourteen years to arrive at it" (p.246). On the contrary, Newton's geometrical forms were frozen in time - they showed "distances and directions" and how they related to one another. No explicit time dimension was required.
Euler was also the first to find how to "represent directed quantities in terms of three orthogonal components and to use coordinates as an arbitrarily defined system of place-markers in the background space (p.247)". It is therefore thanks to Euler that we can express Newton's second law of motion succinctly as "F = ma", with 'F' and 'a' representing vectors within a Cartesian coordinate system. I found it rather surprising in retrospect that despite Euler's invaluable contributions to Newtonian theory, Euler's name was rarely mentioned alongside that of Newton in the classes in which I was taught Newtonian mechanics.
Another major milestone was marked by Joseph-Louis Lagrange (1736 - 1813), Euler's successor as director of mathematics at the Prussian Academy of Sciences. His Mécanique analytique (1788) was unique in that it eschewed the geometrical method completely for one that depended solely on algebra and analysis. In the preface he writes, "There are no diagrams in this book. The methods I explain need no constructions, no geometrical or mechanical arguments, but only the processes of algebra [we would say calculus], performed in regular sequence. Those who love Analysis will see with pleasure that mechanics has now become part of it and will thank me for extending its domain" (p.248).
Lagrange's innovation was to develop a general method for solving dynamical equations. This differed from the style of Newton, who "states a proposition, draws an appropriate diagram, perhaps fudges something, and announces the result", the result being that his theory "contains truth but not a method" (p.248). Lagrange's Mécanique, however, contains a method: find whether the system of analysis has some kind of symmetry, and if so, determine the dynamical quantity that remains constant as a result of this symmetry (using Lagrange's equations). As an example: in the system composed of the sun and one planet, "the sun is (nearly) a perfect sphere and the attractive force it exerts is the same in every direction" (pp.248,249). The consequence of this symmetry is that one dynamical quantity is conserved: angular momentum. From the conservation of angular momentum one can derive Kepler's law of areas.
Lagrange's key contribution was to provide a methodology for solving problems of mechanics that could be applied in the general case, without the need for constructing geometrical proof after proof for each specific problem. His methods are applicable even to those situations in which we would be hard-pressed to create geometrical constructions at all, for they are at present "the most fruitful line of approach to the quantum theory" (p.248). The explosion of scientific knowledge over the past few centuries was thus accompanied - and enabled - by meteoric advances in scientific technique.
The most valuable contribution of David Park's book, in my view, is to spur us to question those dogmas that education often pushes us to take for granted. When students are taught physics in school, there is often a right answer: the right answer. This is true of those exam questions that require numerical calculation or algebraic proof, but it also true of "explain that" questions, which the examiner often marks in accordance with a specific mark scheme. "Mark schemes" are an integral part of our system of examination, but in my view they completely contradict the spirit of science, which is to question established dogma and find different ways of looking at the world.
The book achieves this, I think, by making apparent just how different these ways of looking at the world can be. Aristotle's causes are totally foreign to the physics taught in schools, as well as the connections between theology and natural philosophy that inspired figures such as Kepler and Newton. Yet, they are a large part of how we got to where we are.
It is understandable, at the same time, why they are often left out of physics education. Various topics must compete for space within the physics curriculum, which itself compete with other academic subjects for the student's educational time, which itself must complete with the student's time overall. Why spend this bandwidth on ideas that find no place in modern physics?
At the same time, I agree wholeheartedly with Park's assertion that "one does not step into the middle of a new art form or scientific structure and profit from it at once. One has to know something, to see it against the background of history" (p.386). If we do not know the history of a science, how could we know where we are?
Interesting also are Park's remarks on the concept of "necessity". Given what we know about the world, is it possible to formulate theories that follow necessarily from the facts? Park quotes the Andalusian polymath Ibn Rushd (1126 - 1198), who, in the process of criticising Ptolemy's planetary model, complains that "mathematicians propose the existence of these orbits as if they were principles and then deduce conclusions from them which the senses can ascertain. In no way do they demonstrate by such results that the assumptions they have employed as principles are, conversely, necessities" (p.76). Park remarks that he "know[s] of no model or theory in fundamental science that would satisfy him".
Ibn Rushd's complaint raises the problem that given any set of facts, there may be more than one theory that can satisfy them - so how are we to judge from experiment which theory is correct? Park gives a hypothetical example: "suppose that our astronomical observations were not very good - it is enough to suppose that the earth's climate might always be a little cloudy - so that the small discrepancy in Mercury's orbit and the other relativistic effects like the bending of starlight by the sun could not be observed. We would then, in the twentieth century [when this book was written], have two experimentally indistinguishable but conceptually unreconcilable theories of gravity" (pp.397,398). Of course, we have indeed recorded such relativistic effects as the bending of starlight - but what can guarantee that we will, in the course of experimental endeavour, discover every relevant fact required for assessing our theories of nature?
This somewhat pessimistic attitude was the one to which I subscribed going into this book. However, Park provides a new, interesting spin. Persisting with the topic of gravity, he notes. that "there is no way to combine special relativity, Newtonian gravity, and [Einstein's] principle of equivalence into a consistent whole, so that ... sooner or later someone would have to find a theory of gravitation that is in harmony with the rest of physics" (p.398). In other words, "the physics at which we eventually arrive is likely to form a consistent whole in which there is little room to change parts of it. As to whether there could exist a completely different consistent whole, agreeing with the same experiments (but not necessarily with all possible experiments), obviously this is a logical possibility, but nothing we know about the world gives any hint as to what it might be".
Lastly, the book touches somewhat tentatively on question of "why?" In the introduction, Park remarks that"there is a cliché to the effect that science tells how but not why, a cliché so powerful that it is sometimes used to define the boundaries of science: "'Why' does not begin a scientific question". But most physicists are very interested in questions of why" (p.xxi). Indeed, for many thinkers, it was the question. As we have seen, Aristotle sought to base his theories on notions of beauty and simplicity, the "why" that mathematics could never recover. For Kepler it was much the same - the mathematical beauty found in the harmony of the planets revealed the design of the Creator. We find the inspiration for Newton's natural philosophy not all too different: "This most beautiful system of the sun, planets, and comets could only proceed from the counsel and dominion of an intelligent and powerful Being ... As a blind man has no idea of colors, so we have no idea of the manner by which the all-wise God perceives and understands all things ... We know him only by his most wise and excellent contrivances of things ... we admire him for his perfections" (pp.215,216).
Indeed, for as long as humanity has thought about such questions, there has been a conviction, borne out of no practical or experimental proof, that the world exhibits order (pp.387,388):
King Solomon (990 - 931 BC): "Thou has created all things in measure and number and weight".
Anaximander (610 - 546 BC): "The source from which existing things derive their existence is also that to which they return at their destruction, for they pay penalty and retribution to each other for their injustice according to the assessment of Time".
Heraclitus (535 - 475 BC): "The sun will not transgress his measures; otherwise the Furies, ministers of justice, will find him out".
Saint Augustine (354 - 430 AD): "Behind the heaven, the earth, the sea"; all that is bright in them or above them; all that creep or fly or swim; all have forms because all have number. Take away number and they will be nothing ... Examine the beauty of bodily form, and you will find that everything is in its place by number. Examine the beauty of bodily motion and you will find everything in its due time by number".
Dante Alighieri (1265 - 1321): ""The elements // of all things," she began, "whatever their mode, // observe an inner order // that makes then universe resemble God.""
We have already seen, indeed, the conviction of Robert Grosseteste (1175 - 1253) that mathematics is the cause of all physical process, at a time when only two mathematical laws were known - the law of reflection and the law of the lever (p.104). Closer to our own time, Albert Einstein (1879 - 1955) questioned whether physical law could be necessitated by principles of simplicity: "What I'm really interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all" (p.397). Hence, the question of "why" things are the way they are has always remained at the forefront of natural philosophy.
This review covers a mere fraction of what this book has to offer (I didn't even delve into its chapters on modern physics!). I would highly recommend it to anyone interested in the history of science, and hope that its contents will be taught more readily to students of physics in the future.