What can pure mathematics offer to society? (Q2892577)

This paper is based on a talk (the Albert Einstein Memorial Lecture) given by the author to a general public in 2001. It tries to explain, what (pure) mathematics is, and how it is connected to ``the real world'', in particular to science. After an overview on three ``traditional attitudes'' to pure mathematics (Platonist, logicist, formalist), the author presents three examples of pure mathematics and their (sometimes unexpected) connections to ``the world beyond mathematics'': the prime number theorem and its connection to quantum chaos (via the zeros of the Riemann zeta function); Brouwer's fixed-point theorem and its connection to game theory and thus economics; the four-color theorem and its connections to theoretical computer science and to ``real-world'' problems (given in terms of interval graphs). Moreover, some (then) unsolved problems of pure mathematics are described. Based on these examples, the author tries to explain the work of pure mathematicians. They prove theorems or try to find new proofs for known theorems, because ``they love their subject, they are curious about things mathematical, and they may also want to get on their careers -- nevertheless, the collective effect of their work is very different from the sum of all their individual motivations.'' The work of (individual) pure mathematicians contributes to a ``general mathematical culture'', to knowledge that then, often unexpectedly, can be used by scientists and has an impact on the ``real world''.NEWLINENEWLINEFor the entire collection see [Zbl 1237.00021].