First: the notion of “spectrum,” wh

First: the notion of “spectrum,” which is nowadays used all over science, and which most people believe to have originated in physics. The truth lies otherwise. The notion of spectrum was introduced by David Hilbert, one of the great mathematicians of this century, around 1910. It is not clear what Hilbert’s motivation was in developing his theory of spectrum, but it certainly did not come from physics. Some twenty years later, the discoverers of quantum mechanics found that Hilbert’s special theory was what the doctor ordered, and proceeded to appropriate it and to make good use of it. Soon afterward, the idea of spectrum spread all over science.
Second: the discovery of infinite fields (you do not need to know what finite fields are), which was made dy the American mathematician E. H. Moore at the beginning oh this century. At the time, an eminent European scientist came out with the trouncing statement, “At last we have an instance of a mathematical idea for which we can be sure never to expect any applications whatsoever.” Some fifty years later, courses in finite fields are being required of undergraduates in every electrical engineering department. Anyone who makes a telephone call or who uses an ATM machine is benefiting from codes that would be impossible without the application of the theory of finite fields.
Third: Brownian motion, or, to put it in deliberately abstruse terms, “integration in function space” (again, you do not need to know what these terms refer to), the work of MIT mathematician Norbert Wiener in the 1920s and 1930s. When Wiener’s firts articles on this subject were published, they were ridiculed for hitting “the utmost limit of abstraction,” and even some mathematicians joined the chorus of condemnation. Wiener’s insistence on a rigorous foundation of theory (which even Einstein before him had not thought to be necessary) was dismissed as a mathematician’s foible, as another instance of that morbid delectation in axiomatics that mathematicians are acussed of wallowing in.
Nowadays, Wiener’s foundations of Brownian motion, was impotance was at first cavalierly dismissed, has paid off. It has led to the theory of stochastic integrals, which is taught – would you believe it? – in economics departments and business schools. Specialists in stochastic integrals are handsomely rewarded on Wall Street.
Fourth: when the senior author was an undergraduate at Princeton, there was a lone mathematics professor on the faculty whose specialty was the mathematical theory of knots and links. He was not taken seriously because his research was thought to be extravagant. Today, the mathematical theory of knots is the talk of the town. In physics, it is believed to be the key to the innermost secrets of nature, hidden in what is now known as topological quantum field theory. In morecular biology, our understanding of the folding of morecules and proteins is essential in unraveling the enigmas of the genetic code. And as chemistry becomes more precise thanks to computer stimulation, the most amazing knots among atoms and molecules are disclosed to the human eye.