Living with a New Mathematical Species

The history of mathematics can be viewed as a counterpoint between the finite and the infinite, between the discrete and the continuous. Although rooted in geometry, Greek mathematics was primarily finite, concrete, and specific. Modern mathematics, in contrast , is infinite, abstract, and general. Aristotle in-veighed against the actual infinite, reflecting the Greek cultural distaste for the incomplete form. Centuries later, Leibniz and Newton overcame Aristotelean scruples in proposing methods of calculating with in-finitesimals. Now, after three hundred years of New-tonian mathematics, computers are forcing a return to mathematical preferences of the pre-Newtonian age-to the finite, the specific, and the concrete This return to our roots is a natural consequence of increasing mathematical maturity. Weierstrass resolved the paradox of infinitesimals by reducing analysis to arithmetic; he showed how to interpret the infinite concepts of calculus in terms of the finite structures of arithmetic. Twentieth century mathematics has been dominated by the Weierstrass synthesis-a working intellectual compromise between the finite limitations of human mental processes and the infinite visions of human imagination.