1 The lawful Universe

1.1 Science and regularity

‘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.’
John D. Barrow (1988), The World Within the World, Oxford.

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).
Roger Bacon Figure 1.1 Roger Bacon
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.

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.

1.2 Mathematics and quantification

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. For example, in the equation
equation 1.1
the symbol h represents the height fallen by an object that has been dropped from rest, the symbol t represents the time the object has been falling, and g is a constant with a known value (g = 9.81 metres per second per second). Equation 1.1 encapsulates a wealth of information about falling objects, information that is precise and useful. The tools of algebra allow us to go further. For example, the above equation can be rearranged to read
equation 1.2
so now, if we know the height fallen by an object, we can work out how long it has taken to fall.

Mathematics provides a natural medium for rational argument. Given an equation that relates various quantities, the rules of mathematics allow that equation to be re-expressed in a number of different but logically equivalent ways, all of which are valid if the original equation was valid. Given two equations, mathematical reasoning allows them to be combined to produce new equations which are again valid if the original equations were valid. Long chains of reasoning can be put together in this way, all of which are guaranteed to be correct provided that the starting points are correct and no mathematical rules are transgressed. Quite often these arguments are so long and detailed that it would be impossible to follow them in ordinary language, even if it were possible to express them at all.

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.

Despite its power, physics students often find the extensive use of mathematics troublesome and some think of mathematics as providing a barrier to understanding. Do not let this happen to you. From the outset, you should regard mathematics as a friend rather than a foe. As the course progresses, you may meet some mathematical ideas that are new to you, or you may need to improve your ability to use methods you have met before. These are not distractions from trying to understand physics, but are the tools needed to make that understanding possible. It is only through using mathematics that a secure understanding can be achieved. When you see an equation, welcome its concision and clarity and try to ‘read’ the equation just as you would the large number of words it replaces. Learn to get beneath the squiggles and the equals sign and to understand the quantitative assertion that is being made.
"I see through your squiggles" Fig 1.2 "I see through your squiggles."
Later, you will see how graphs can be used to visualize an equation and how consideration of special cases and trends can help unpack its meaning.