Computing with real numbers, from archimedes to turing and beyond

The article discusses how to test the usefulness of computation for understanding and predicting continuous phenomena. Fractions are not as easy to produce as whole natural numbers, yet the algorithm for them is fairly straightforward. To produce two-third of an apple, one can slice an apple into three equal parts, then take two of them. If one considers positive rational numbers, there is little divergence between the symbolic representation of the number and the algorithm one needs to construct this number out of apples. The number practically shouts a way to construct it. It requires a nuanced understanding of the underlying dynamical system. It is likely this is the case with other natural dynamical systems; the prerequisite to understanding its computational properties would be understanding its other properties. Indeed, understanding the role non-computability and computational universality play in natural dynamical systems probably requires significant advances in both real computation and dynamical systems.