John D. Barrow (1988), The World Within the World, Oxford.
‘Our experience shows that only a small part of the physical Universe needs to be studied in order to elucidate its underlying themes and patterns of behaviour. At root this is what it means for there to exist laws of Nature, and it is why they are invaluable to us. They may allow an understanding of the whole Universe to be built up from the study of small selected parts of it.’
Science, it is widely agreed, originated from two main sources. One was the need to develop practical knowledge and to pass it from generation to generation. The other was a more spiritual concern with the nature and origin of the world. Common to both of these well-springs of science was an appreciation of the regularity of Nature. The way to build an arch that would not fall down today was to build it in much the same way as an arch that had not fallen down yesterday. The way to predict the waxing and waning of the Moon this month was to assume that it would follow much the same course as the waxing and waning that had been observed last month and the month before.
The observation of regularity in Nature allows predictions to be made concerning the future course of particular events. In many primitive societies these regularities were ascribed to the activities of gods or other mystical spirits. However, gradually, over a long period of time, there emerged the notion that the behaviour of the world was guided by a set of natural laws that were themselves regular, in the sense that identical situations could be expected to have identical outcomes.
One of the first scientists to make frequent use of the concept of a law of Nature, in the sense that we now use that term, was the Franciscan friar and scholar Roger Bacon (c. 1214–1292).
Bacon is traditionally credited with the invention of the magnifying glass, but he is best remembered as an effective advocate of the scientific method and a follower of the maxim ‘Cease to be ruled by dogmas and authorities; look at the world!’ He lived at a time when the commonly accepted view of the world was fundamentally religious, and the Catholic church to which he belonged was coming to embrace the authority of the ancient Greek philosopher Aristotle on matters pertaining to physics and astronomy. Bacon’s independence of mind brought him into conflict with the church, and he suffered fifteen years of imprisonment for heresy. Nonetheless, he helped to prepare the way for those who, irrespective of their own religious beliefs, insisted that the scientific investigation of Nature should be rooted in experiment and conducted on a purely rational basis, without reference to dogmatic authority. 1
Laws of Nature are now a central part of science. Carefully defined concepts, often expressed in mathematical terms, are related by natural laws which are themselves often expressed in a mathematical form. Just what those laws are is a central concern of physicists, who see their branch of science as the one most directly concerned with discovering and applying the fundamental laws of Nature. Improvements in our knowledge of natural laws have repeatedly led to a broadening and a deepening of our understanding of the physical world and hence to a change in the scientific world-view. However, the fundamental requirement that the laws should be rational and rooted in experiment has survived all changes to the detailed content of those laws.
Roger Bacon once said ‘Mathematics is the door and the key to the sciences’. This statement aptly summarizes the role of mathematics in science, particularly in physics, and it is not hard to see why.
Much of physics is concerned with things that can be measured and quantified, that is, expressed as numbers, multiplied by an appropriate unit of measurement such as a metre or a second. It is natural to turn to mathematics to try to reveal patterns underlying such measured data. This is more than a matter of arithmetic. By Roger Bacon’s time the basic ideas of algebra had been developed, mainly by Arabic mathematicians and astronomers. The idea of representing a quantity by a symbol, such as x or t is extremely powerful because it allows us to express general relationships in a very compact way.
Mathematics has been an immensely effective part of the scientist’s toolkit throughout history. It was the increased use of mathematics in the sixteenth and seventeenth centuries, in the hands of individuals such as Galileo Galilei (1564–1642) and Isaac Newton (1642–1727), that opened a new era of physics and marked one of the greatest flowerings of science. Galileo and Newton, it should be noted, were both, at key times in their careers, professors of mathematics. In both cases they brought mathematical precision and rigour to the study of science, and in Newton’s case made major breakthroughs in mathematics in the process. The types of mathematics used in physics are extremely varied. Practically every branch of mathematics that has developed over the centuries has been used within physics. Sometimes physics has provided direct inspiration for new mathematical concepts, sometimes abstract mathematical theories have found completely unexpected uses in physics, years after their introduction as products of pure thought.
It is probably fair to say that no single individual has had a greater influence on the scientific view of the world than Isaac Newton. The main reason for Newton's prominence was his own intrinsic genius, but another important factor was the particular state of knowledge when he was, in his own phrase, 'in the prime of my age for invention'.
In 1543, a century before Newton's birth, Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favour of a heliocentric view in which the Earth moved round the Sun. By removing the Earth, and with it humankind, from the centre of creation, Copernicus had set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers. The most famous of these must surely have been Galileo, who was summoned to appear before the Inquisition in 1633, on a charge of heresy, for supporting Copernicus' ideas. As a result Galileo was 'shown the instruments of torture', and invited to renounce his declared opinion that the Earth moves around the Sun. This he did, though tradition has it that at the end of his renunciation he muttered 'Eppur si muove' ('And yet it moves').
In the Protestant countries of Northern Europe, thought on astronomical matters was more free, and it was there in the early seventeenth century, that the German-born astronomer Johannes Kepler (1571-1630) devised a modified form of Copernicanism that was in good agreement with the best observational data available at the time. According to Kepler, the planets did move around the Sun, but their orbital paths were ellipses rather than collections of circles. This discovery, first published in 1609 in Kepler's book Astronomia Nova (The New Astronomy), was essentially an observational result. Kepler had no real reason to expect that the planets would move in ellipses, though he did speculate that they might be impelled by some kind of magnetic influence emanating from the Sun.
Kepler's ideas were underpinned by new discoveries in mathematics. Chief among these was the realization, by René Descartes, that problems in geometry can be recast as problems in algebra. Like most revolutionary ideas, the concept is disarmingly simple.
Newton's good fortune was to be active in physics (or 'natural philosophy' as it would then have been called) at a time when the cause of Kepler's ellipses was still unexplained and the tools of geometry were ripe for exploitation. The physics of Aristotle was clearly inadequate, and all other attempts seemed unconvincing. The new astronomy called for a new physics which Newton had the ability and the opportunity to devise. He was the right man, in the right place, at the right time.
For years before Newton, people had been trying to understand the world from a scientific perspective, discovering laws that would help explain why things happen in the way that they do. Bits of knowledge were assembled, but there was no clear idea how these bits related to one another; understanding was fragmentary. Newton's great achievement was to provide a synthesis of scientific knowledge. He did not claim to have all the answers, but he discovered a convincing quantitative framework that seemed to underlie everything else. For the first time, scientists felt they understood the fundamentals, and it seemed that future advances would merely fill in the details of Newtonís grand vision. Before Newton, few could have imagined that such a world-view would be possible. Later generations looked back with envy at Newton's good fortune. As the great Italian-French scientist Joseph Lagrange remarked:
'There is only one Universe... It can happen to only one man in the world's history to be the interpreter of its laws.'
At the core of Newton's world-view is the belief that all the motion we see around us can be explained in terms of a single set of laws.
1. Newton concentrated not so much on motion, as on deviation from steady motion - deviation that occurs, for example, when an object speeds up, or slows down, or veers off in a new direction.
2. Wherever deviation from steady motion occurred, Newton looked for a cause. Slowing down, for example, might be caused by braking. He described such a cause as a force. We are all familiar with the idea of applying a force, whenever we use our muscles to push or pull anything.
3. Finally Newton produced a quantitative link between force and deviation from steady motion and, at least in the case of gravity, quantified the force by proposing his famous law of universal gravitation.
In keeping with his grand vision, Newton proposed just one law for gravity - a law that worked for every scrap of matter in the Universe, for moons and planets as well as for apples and the Earth. By combining this law with his general laws of motion, Newton was able to demonstrate mathematically that a single planet would move around the Sun in an elliptical orbit, just as Kepler claimed each of the planets did. Moreover, thanks to the understanding that gravity was the cause of planetary motion, Newtonian physics was able to predict that gravitational attractions between the planets would cause small departures from the purely elliptical motion that Kepler had described. In this way, Newton was able to explain Kepler's results and to go beyond them.
In the hands of Newton's successors, notably the French scientist Pierre Simon Laplace (1749-1827), Newtonís discoveries became the basis for a detailed and comprehensive study of mechanics (the study of force and motion). The upshot of all this was a mechanical world-view that regarded the Universe as something that unfolded according to mathematical laws with all the precision and inevitability of a well-made clock. The detailed character of the Newtonian laws was such that once this majestic clockwork had been set in motion, its future development was, in principle, entirely predictable. This property of Newtonian mechanics is called determinism. It had an enormously important implication. Given an accurate description of the character, position and velocity of every particle in the Universe at some particular moment (i.e. the initial condition of the Universe), and an understanding of the forces that operated between those particles, the subsequent development of the Universe could be predicted with as much accuracy as desired.
Needless to say, obtaining a completely detailed description of the entire Universe at any one time was not a realistic undertaking, nor was solving all the equations required to predict its future course. But that wasn't the point. It was enough that the future was ordained. If you accepted the proposition that humans were entirely physical systems, composed of particles of matter obeying physical laws of motion, then in principle, every future human action would be already determined by the past. For some this was the ultimate indication of God: where there was a design there must be a Designer, where there was a clock there must have been a Clockmaker. For others it was just the opposite, a denial of the doctrine of free will which asserts that human beings are free to determine their own actions. Even for those without religious convictions, the notion that our every thought and action was pre-determined in principle, even if unpredictable in practice, made the Newtonian Universe seem strangely discordant with our everyday experience of the vagaries of human life.
Newtonian mechanics is concerned with explaining motion, yet it contains within it the much simpler idea that some things never change. Take the concept of mass, for example, which appears throughout Newtonian mechanics, including the law of gravitation. In Newtonian mechanics, mass is conserved. This means that the mass of the Universe is constant and the mass of any specified collection of particles is constant, no matter how much rearrangement occurs within the system. A chemist might take one kilogram of hydrogen and let it react with eight kilograms of oxygen to produce water. According to the law of conservation of mass , nine kilograms of water will be produced, the same as the total mass of the ingredients (1 kg + 8 kg = 9 kg). You may think this is trivial, but it is not. Conservation laws are rare and wonderful things. There is no general law of conservation of volume for example. The initial volume of the hydrogen and oxygen is far greater than the final volume of the water. The fact that mass is conserved really is a deep discovery about the checks and balances that exist in our Universe.
Newtonian mechanics introduced several other important conservation laws, including the celebrated law of conservation of energy. Not too surprisingly, this law states that the total energy of the Universe is constant and the total energy of an isolated system of particles is constant. But the full meaning of these words will only become apparent once the concept of energy has been properly defined.