              The Restless universe
Introduction to The restless Universe

1 The lawful Universe

1.1 Science and regularity 2/2

» 1.2 Mathematics and quantification 1/2

1.2 Mathematics and quantification 2/2

2 The clockwork Universe

3 The irreversible Universe

4 The intangible Universe

5 The uncertain Universe

6 Closing items

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Other titles in the Physical World series

Describing motion

Predicting motion

Classical physics of matter

Static fields and potentials

Dynamic fields and waves

Quantum physics: an introduction

Quantum physics of matter

1 The lawful Universe

1.2 Mathematics and quantification

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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 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 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.
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