One might imagine that with only a single active cell being updated at each step different parts of the system would inevitably be perceived to progress through time one after another. … But with rules (a) and (b) only a limited number of points in space can ever be reached.

With up to 6 Nand s and 2 variables none of the 16,896 possible axiom systems of this kind work even up to 3-value operators. … If one looks at axiom systems of the form {…  a, a ∘ b  b ∘ a} the first one that one finds that allows only Nand and Nor with 2-value operators is {(a ∘ a) ∘ (a ∘ a)  a, a ∘ b  b ∘ a} . … My guess is that unless one asks about very specific details there is really not—and that standard logic is in a sense distinguished in the end only by its historical context.

And the only way we will normally become convinced of its validity is by understanding how some whole chain of consequences can lead to purposes that happen to fit into our own specific personal context. … We can perfectly well instead talk only about mechanism, and about the way in which the underlying rules for the cellular automaton lead to the behavior we see.

It is thus important to read these notes in parallel with the sections of the main text to which they refer, since some necessary points may be made only in the main text.

On pages 363 – 369 of Chapter 8 , however, I discuss some general issues of modelling, and in Chapter 10 I consider at length not only practical but also foundational questions about perception and to some extent general thinking and consciousness.

In the past, essentially the only large-scale notation available for theoretical science has been traditional mathematical notation. But on its own this would do me little good—for I need to represent not only traditional mathematics, but also more general rules and programs, as well as procedures and algorithms. … And the final and very critical advantage of Mathematica notation is that one can not only read it, but also actually execute it on a computer, and interact with it.

And the reason for this is that with s = {1, 2} the function Union[Complement[s, a], b] corresponding to a  b only ever gets to the 3 elements {{1}, {2}, {1, 2}} . … (One might think that one could force there only ever to be two elements by adding an axiom like a  b ∨ b  c ∨ c  a . But all this actually does is to force there to be only two objects analogous to True and False .)

And normally this can be done with any reliability only when the machine code represents a fairly direct translation that has not been extensively rearranged or optimized.

If I am correct that there is a simple underlying program for the universe, then this means that theoretical physics must at some level have only a very small amount of true physical input—and the rest must in a sense all just be mathematics.

Systems of Limited Size and Class 2 Behavior…But since there are only six possible positions in all, it is inevitable that after at most six steps the dot will always get to a position where it has been before.