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. I am interested in mathematics only as a creative art – Godfrey Harold Hardy
Recently a friend who is theoretical computer scientist was commenting to another friend (who has a degree in arts and design) about how much mathematics has in common with arts and design. He explained how some one doing mathematics creatively tries to find intricate and useful structure which may help in problem solving. He stressed how a creative combination of small ideas leads to the ‘final piece of art’. I was amused when our artistic friend gave an incredulous look. This is a common reaction. It can be frustrating to explain to others that mathematical activity is creative in its essence and not mechanical like arithmetic.
Perhaps, Sir Michael Atiyah does of a better job at explaining at least one aspect of this issue:
The art in good mathematics, and mathematics is an art, is to identify and tackle problems that are both interesting and solvable. Proof is the end product of a long interaction between creative imagination and critical reasoning. Without proof the program remains incomplete, but without the imaginative input it never gets started.
One can see here an analogy with the work of the creative artist in other fields: writer, painter, composer, or architect. The vision comes first, it develops into an idea that gets tentatively sketched out, and finally comes the long technical process of erecting the work of art. But the technique and the vision have to remain in touch, each modifying the other according to its own rules.