I. Mathematics is an exact language for description, calculation, deduction, modeling, and prediction — more a systematic way of thinking than a set of rules.
Using a legal analogy, mathematics is a language for writing contracts with Nature that Nature accepts as legally binding.
II. The practical importance of mathematics lies in its ability to describe the real world.
The real world consists of what matters. The word “matter” as a noun is used for what the physical world is made of. But if we ask, “What’s the matter with Anne?” we may be asking about a physical ailment, or we may be asking about an idea that is causing Anne to behave strangely. Ideas matter.
The whole point of mathematical education is to make ideas real for students, ideas that were not real for them before. Ideas like fractions, for example. The fact that 2/3 is smaller than 3/4 matters in the real world.
III. Mathematically educated people are stem cells of a technologically advanced society. Because of the universality of mathematics, mathematicians and well educated users of mathematics are flexible in applying and inventing tools for work in technological environments which never existed before.
IV. Learning mathematics involves the profound assimilation of intellectual and aesthetic criteria as well as practically orientated ones. The very difficulty in learning mathematics makes it a personality-enhancing experience.
[With contributions and borrowings from David Corfield, Tony Gardiner, Michael Gromov, Frank Quinn, David Pierce.]