I should teach the world the following passage, from Galileo Galilei's 1623 work The Assayer, in which he explains that the book of nature is written in the language of mathematics: 'Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.'

The truth uncovered by Galileo transcends his particular time or individual insight. But it is appropriate to teach this truth through his words. Galileo was one of science's founding fathers. Moreover, he wrote in the vernacular. The historical peculiarities of the text only deepen our appreciation of his achievement - for his readership, the contrast between the book of nature and the Holy Book would be immediate.

Mathematics has three great roles in science. The first of these is precision in describing experiment, through counting and measuring. Galileo is lauding mathematics not as pure logic, but as the key to deciphering empirical evidence. Galileo's science is driven by the great dialectic between theory and experiment. The first role of mathematics is to quantify observations.

Without measurement, observations often remain vague. Patterns remain at the level of rough associations. With measurement, patterns are sharply revealed as laws. Through statistics, subtle relationships emerge.

The second great role of mathematics in science is rigorous chains of reasoning. Mathematics allows us to derive consequences from laws. It allows us to understand how consequences follow from laws. This is key both to the practical utility of science, and to deriving predictions used to test laws experimentally.

Science is built upon networks of inferences. Many chains of evidence weave together, into a web of explanations. Mathematical reasoning shows exactly what preconditions are necessary and sufficient, for a consequence to follow. Mathematical reasoning shows which points in an argument depend upon which other points, and it does so exactly. Mathematics is our best attempt at watertight reasoning. By following mathematical arguments, we are able to discard those assumptions that are irrelevant to the problem at hand, focusing upon those assumptions that are essential to the subject matter. We also become familiar with surprising counter-examples, that can slip through our reasoning by violating unstated assumptions.

The third great role of mathematics in science is the formulation of abstract concepts. Scientific knowledge requires that we work with idealised concepts. Galileo's 'geometric figures' are mathematically perfect. They are not found in the real world. The application of mathematics requires stepping back from immediate experience, to operate with concepts that are abstracted from the world. It is only through abstract concepts that we obtain scientific knowledge. Not all concepts are mathematical, but mathematics provides us with our clearest examples.

In these three roles, mathematics forms the foundation stone of science. It used to be thought that physics was uniquely mathematical. Some saw physics as a model of 'hard' science for others to emulate; others warned that the distinctiveness of other sciences made such a project misguided. Today, biology has taken on a leading role in the sciences. Biology has retained some of its distinctiveness, but it is notable that it has also become far more mathematical, in all of its aspects.

For Albert Einstein, 'one may say the eternal mystery of the world is its comprehensibility'. The comprehensibility of the world is by no means obvious. It is an astonishing discovery, not to be taken for granted, that science is possible at all. The role of mathematics is at the heart of this empirical fact - that the world is comprehensible.