CHAPTER 7

Hilbert and his famous

problem

jack copeland

I

n 1936 mathematics was changing profoundly, thanks to Turing and his fellow revo-

lutionaries Gödel and Church. Older views about the nature of mathematics, such as

those powerfully advocated by the great mathematician David Hilbert, were being

swept away, and simultaneously the foundations for the modern computer era were being

laid. These three revolutionaries were also catching the first glimpses of an exciting new

world—the hitherto unknown and unimagined mathematical territory that lies beyond the

computable.^1

The 1930s revolution

In the 1930s a group of iconoclastic mathematicians and logicians launched the field that we

now call theoretical computer science.^2 These pioneers embarked on an investigation to spell

out the meaning and limits of computation. Pre-eminent among them were Alan Turing, Kurt

Gödel, and Alonzo Church.

These three men are pivotal figures in the story of modern science, and it is probably true

to say that, even today, their role in the history of science is underappreciated. The theoretical

work that they carried out in the 1930s laid the foundations for the computer revolution, and

the computer revolution in turn fuelled the rocketing expansion of scientific knowledge that

characterizes modern times. Previously undreamed of number-crunching power was soon

boosting all fields of scientific enquiry, thanks in large part to these seminal investigations. Yet,

at the time, Turing, Gödel, and Church would have thought of themselves as working in a most

abstract field, far flung from practical computing. Their concern was with the very foundations

of mathematics.