## A New Kind of Science

### Cellular Automata

Science depends heavily on mathematics and in particular on continuous mathematics. Scientific laws are often in the form of differential equations, which may not have a solution in closed form. Furthermore, such mathematics makes the assumption that space is infinitely divisible and that a coordinatising it via real numbers with infinitely many digits makes physical sense. Many people have therefore wondered whether it it possible to use discrete mathematics instead. In this book Stephen Wolfram demonstrates a possible way of doing this.

The book is based on cellular automata - that is arrays of boxes where the contents of each box changes according to some rule based on the box's neighbours. The book looks at cellular automata in one, two and more dimensions, as well as similar discrete systems which change over time according to simple rule. Their behaviour is illustrated with plenty of pictures, which helps in the aim of linking the behaviour of the real world to the behaviour of such simple systems. For instance, the pigmentation patterns of many animals bear a striking resemblance to patterns obtained through the evolution of cellular automata. On a smaller scale, Wolfram examines how the fundamental behaviour of space-time can be modelled with such discrete systems, showing the wide applicability of such ideas.

### Computational Equivalence

Wolfram doesn't limit himself to rewriting science in a discrete form. He also aims to demonstrate what he calls 'The principle of computational equivalence'. This claims that many systems - both physical and theoretical - are universal, in the sense that they can be used to perform arbitrary computations. One new way of thinking that he brings to us is that of using a brute-force computational approach to investigate the space of possible behaviours of a system, rather than looking for simple cases which can be examined with pencil and paper. For instance to investigate the theorems which can be proved from a set of axioms, he sets a computer to generate each of them in turn. If half of the possible statements turn out to be theorems then it indicates that the axiom system is likely to be complete (although short gtheorems might have long proofs). Likewise with computers - he looks at the behaviour of Turing machines one by one to see what they compute.