Cellular Automata

CA versus mathematical sequence

The cellular automaton (CA) is a concept pioneered by Stanislaw Ulam and John von Neumann. Cells in a grid have states that are calculated from the states of their predecessors. Cellular automata, have applications in modelling physical processes, from urban development to simulations of gas behaviour. Konrad Zuse [[i]] and Edward Fredkin [[ii]] suggested the idea that CAs are the underlying mechanism of physical reality.

Although CAs are used as models to simulate quantum events, the idea that they fundamentally create our reality is far from gaining main-stream acceptance in spite of work by Rudy Rucker[8] and Stephen Wolfram [[iii]].

A cellular automaton can be equivalent to a mathematically defined set, but the concept implies that it requires a machine to run on.

If the Mandelbrot set, or even simpler sets, like the positive integers, do not need a physical world to define them, but exist quite apart from our universe, then other sets must also exist.

Sets are unordered collections of numbers, a sequence is a set with an order. We are interested in multi-dimensional sequences, but let us start with a one-dimensional sequence, let’s take the positive integers.

We define the sequence with a function where each number depends on the value of its predecessors:


and some initial condition:



We would like to define a function that generates our universe in a similar way. There are values at position p, and these values depend on the values of the predecessors. We need 4 things for that:

  • A number system to describe values and coordinates
  • A structure, defining the predecessors of each point
  • A formula that calculates the value at each point depending on the values of its predecessors
  • An initial condition.

As an alternative to “sequence where the value at each point is defined by the value of its predecessors” we could use the equivalent term “cellular automaton” with an initial condition, a definition of neighbours equivalent to our predecessors and a generating formula.

We could also look at the positive integers as a CA. It is not a perfect choice of words because it conjures up concepts of CA clocks, rules, and lattice topologies, as well as hardware running the CA simulation. If the universe is a CA, it invites the question “which computer is running it?”

A mathematical ordered set does not need a computer to exist.

In my view it seems rather more sensible to consider our universe to be a mathematical sequence independent of space and time and to set aside the concept that it is being somehow simulated on a giant computer.

[[i]] Konrad Zuse, Rechnender Raum

[[ii]] Edward Fredkin, An Introduction to Digital Philosophy, International Journal of Theoretical Physics, Vol 42, 2003

[[iii]] Stephen Wolfram, A new kind of science

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