What is Mathematics?

Many students enjoy doing maths due to the satisfaction of agreeing with the answer when they turn to the back of the book. A sense of achievement comes from using thought processes to go through a series of steps to determine the correct result. Mathematics is a work-out for the mind just like a visit to a gymnasium is a physical work-out for the body. Harvard physicist Lisa Randall was drawn to mathematics "because it had definite answers....I liked the certainty of knowing which answers were right". Mathematics develops mental agility, logical thinking and systematic problem solving techniques. But what actually is mathematics? Algebra, calculus and geometry are branches of mathematics which operate using a common thread. Mathematics is the manipulation of patterns according to a defined set of rules or ideas. Three classic books that discuss the nature of mathematics  are What is Mathematics by Richard Courant and Herbert Robbins, Mathematics and the Imagination by Edward Kasner and John Newman and A Mathematician's Apology by G H Hardy. The Cambridge mathematician Hardy on page 84 gives the following insights into mathematics

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.