by George A. W. Boehm

Never before have so many people applied such abstract mathematics to so great a variety of problems. To meet the demands of industry, technology, and other sciences, mathematicians have had to invent new branches of mathematics and expand old ones. They have built a superstructure of fresh ideas that people trained in the classical branches of the subject would hardly recognise as mathematics at all.

Applied mathematicians have been grappling successfully with the world's problems at a time, curiously enough, when pure mathematicians seem almost to have lost touch with the real world. Mathematics has always been abstract, but pure mathematicians are pushing abstraction to new limits. To them mathematics is an art they pursue for art's sake, and they don't much care whether it will ever have any practical use.

Yet the very abstractness of mathematics makes it useful. By applying its concepts to worldly problems the mathematician can often brush away the obscuring details and reveal simple patterns. Celestial mechanics, for example, enables astronomers to calculate the positions of the planets at any time in the past or future and to predict the comings and goings of comets. Now this ancient and abstruse branch of mathematics has suddenly become impressively practical for calculating orbits of earth satellites.

Even mathematical puzzles may have important applications. Mathematicians are still trying to find a general rule for calculating the number of ways a particle can travel from one corner of a rectangular net to another corner without

Now that they have electronic computers, mathematicians are solving problems they would not have dared tackle a few years ago. In a matter of minutes they can get an answer that previously would have required months or even years of calculation. In designing computers and programming them to carry out instructions, furthermore, mathematicians have had to develop new techniques. While computers have as yet contributed little to pure mathematical theory, they have been used to test certain relationships among numbers. It now seems possible that a computer some day will discover and prove a brand-new mathematical theorem.

(from The New World of Mathematics, Faber and Faber, London, 1959)