Implications for Mathematics and Its Foundations

Much of what I have done in this book has been motivated by trying to understand phenomena in nature. 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.

At some rather abstract level one can immediately recognize a basic similarity between nature and mathematics: for in nature one knows that fairly simple underlying laws somehow lead to the rich and complex behavior we see, while in mathematics the whole field is in a sense based on the notion that fairly simple axioms like those on the facing page can lead to all sorts of rich and complex results.

So where does this similarity come from? At first one might think that it must be a consequence of nature somehow intrinsically following mathematics. For certainly early in its history mathematics was specifically set up to capture certain simple aspects of nature.

But one of the starting points for the science in this book is that when it comes to more complex behavior mathematics has never in fact done well at explaining most of what we see every day in nature.

Yet at some level there is still all sorts of complexity in mathematics. And indeed if one looks at a presentation of almost any piece of modern mathematics it will tend to seem quite complex. But the point is that this complexity typically has no obvious relationship to anything we see in nature. And in fact over the past century what has been done in mathematics has mostly taken increasing pains to distance itself from any particular correspondence with nature.

So this suggests that the overall similarity between mathematics and nature must have a deeper origin. And what I believe is that in the end it is just another consequence of the very general Principle of Computational Equivalence that I discuss in this chapter.

For both mathematics and nature involve processes that can be thought of as computations. And then the point is that all these computations follow the Principle of Computational Equivalence, so