Machines, logic and quantum physics

x1. Mathematics and the physical world. Genuine scientiÞc knowledge cannot be certain, nor can it be justiÞed a priori. Instead, it must be conjec-tured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics. This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, Òthe universe is written in the language of mathematicsÓ [7]. GalileoÕs intro-duction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on sup-posedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science uses mathematics to express quantita-tive descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world Ñ