Computers are both the creature and the creator of mathematics. They are, in the apt phrase of Seymour Papert, “mathematics-speaking beings”. More recently J. David Bolter in his stimulating book Turing'sMan [4] calls computers “embodied mathematics”. Computers shape and enhance the power of mathematics, while mathematics shapes and enhances the power of computers. Each forces the other to grow and change, creating, in Thomas Kuhn's language, a new mathematical paradigm.

Until recently, mathematics was a strictly human endeavor. But suddenly, in a brief instant on the time scale of mathematics, a new species has entered the mathematical ecosystem. Computers speak mathematics, but in a dialect that is difficult for some humans to understand. Their number systems are finite rather than infinite; their addition is not commutative; and they don't really understand “zero”, not to speak of “infinity”. Nonetheless, they do embody mathematics.

The core of mathematics is changing under the ecological onslaught of mathematics-speaking computers. New specialties in computational complexity, theory of algorithms, graph theory, and formal logic attest to the impact that computing is having on mathematical research. As Arthur Jaffe has argued so well (in [12]), the computer revolution is a mathematical revolution.

NewMathematicsforaNewAge

Computers are discrete, finite machines. Unlike a Turing machine with an infinite tape, real machines have limits of both time and space. Theirs is not an idealistic Platonic mathematics, but a mathematics of limited resources. The goal is not just to get a result, but to get the best result for the least effort.