But does one really need a language that has the kind of sequential grammatical structure of ordinary human language? … But my uniform experience has been that if one wants to specify processes of any significant complexity in a fashion that can reasonably be understood then the only realistic way to do this is to use a language—like Mathematica—that has essentially an ordinary sequential grammatical structure. … For while ordinary human language has little trouble describing repetitive and even nested patterns, it seems to be able to do very little with more complex patterns—which is in a sense why this book, for example, depends so heavily on visual presentation.

But the existence of a compressed description does not on its own imply computational reducibility. … So this is why regularities that we recognize by these methods do indeed reflect the presence of computational reducibility. … But if behavior that we see looks complex to us, does this necessarily mean that it can exhibit no computational reducibility?

But the ideas that I have developed are general enough that they do not apply just to nature. And indeed in this section what I will do is to show that they can also be used to provide important new insights on fundamental issues in mathematics. … So where does this similarity come from?

But what does it take to establish that such incompleteness will actually occur in a specific system? The basic way to do it is to show that the system is universal. But what exactly does universality mean for something like an axiom system?

But if one has such a system, how does one decide what questions are interesting to ask about it? Without the guidance of known theorems, the obvious thing to do is just to look explicitly at how the system behaves—perhaps by making some kind of picture. And if one does this, then what I have found is that one is usually immediately led to ask questions that run into phenomena like undecidability.

And most often the stated reason for this would be that they do not seem to fit into any general framework of mathematical results, but instead just seem like isolated random mathematical facts. In doing mathematics, it is common to use terms like difficult, powerful, surprising and deep to describe theorems. But what do these really mean?

But the discoveries in this book have made it clear that in fact such computation is quite common in all sorts of systems that do not show anything that we would normally consider intelligence. And indeed it seems likely that for example an ordinary physical process like fluid turbulence in the gas around a star should rather quickly do more computation than has by most measures ever been done throughout the whole course of human intellectual history. … But here again I do not believe that this is correct.

Doing this certainly required experience in all sorts of different areas of science. But perhaps most crucial for me was that the process was a bit like what I have ended up doing countless times in designing Mathematica: start from elaborate technical ideas, then gradually see how to capture their essential features in something amazingly simple. And the fact that I had managed to make this work so many times in Mathematica was part of what gave me the confidence to try doing something similar in all sorts of areas of science.

In most cases there is no easy way to do this, and in fact there is little choice but just to run the cellular automaton and see what it does. … For while class 4 is above class 3 in terms of apparent complexity, it is in a sense intermediate Rare examples of borderline cellular automata that do not fit squarely into any one of the four basic classes described in the text.

Does one see the same kinds of phenomena as on page 377 ? The pictures on the next page suggest that indeed one does. … So where does this randomness come from?