In the first few layers of the visual cortex about half the cells respond to elongated versions of similar stimuli, while others seem sensitive to various forms of change or motion.

The discovery from the mid-1800s to the mid-1900s of all sorts of elaborate chemical processes in living systems led biologists often to view life as defined by its ability to maintain fixed overall structure while achieving chemical functions such as metabolism.

And while the paper is at a technical level rather clear, it has never been easy for typical mathematicians to read.

For while it emphasizes calculation rather than proof its symbolic expressions and transformation rules provide an extremely general way to represent mathematical objects and operations—as for example the notes to this book illustrate.

But while this is familiar from algebraic mathematics and from the operation of Mathematica it is not the model of proofs that has traditionally been used in mainstream mathematical logic.

And at least for a while the ordinary axioms of set theory can be used to study the sets that arise.

Then, for example, the rule for the mobile automaton shown on page 71 can be given as {{1, 1, 1}  {0, 1}, {1, 1, 0}  {0, 1}, {1, 0, 1}  {1, -1}, {1, 0, 0}  {0, -1}, {0, 1, 1}  {0, -1}, {0, 1, 0}  {0, 1}, {0, 0, 1}  {1, 1}, {0, 0, 0}  {1, -1}} where the left-hand side in each case gives the value of the active cell and its left and right neighbors, while the right-hand side consists of a pair containing the new value of the active cell and the displacement of its position.

Note that in effect, h x gives the information content of spatial sequences in units of bits per unit distance, while h t gives the corresponding quantity for temporal sequences in units of bits per unit time.

Around 330 BC Aristotle mentioned that instead randomness might just be associated with coincidences outside whatever system one is looking at, while around 300 BC Epicurus suggested that there might be randomness continually injected into the motion of all atoms.

And while at first these postulates might seem incompatible, what Einstein showed was that they are not—at least if modifications are made to the basic laws of mechanics.