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If everything that exists is finite, what is the metaphysical status of mathematical questions about a potentially infinite process? Such questions tells us something about the constraints and possibilities in a potentially infinite universe . They can be thought of as the mathematics of creativity for they can help us to determine principles that foster creativity and avoid stagnation. In a sense such questions have an absolute meaning. All the mathematics that I think is absolutely meaningful refers to a recursively enumerable collection of finite events. The way in which the truth of those events is related to the truth of the statement does not have any well defined canonical form. Any statements of this nature that seem to be meaningful in a potentially infinite universe are candidates for the mathematics of creativity. Questions like the continuum hypothesis cannot be cast in this form.
We can apply the laws of logic to tell us about processes that continue for ever. That is important if we are concerned about will happen in an unbounded future. Insofar as we extend our identity beyond the necessary illusion of our individuality to the deeper truth of are unity and identity with the creative thrust of all that is, we will have a spiritual interest in an indefinite future. As we shall discuss in Chapter we can begin to understand from mathematics why freedom and diversity are so important to the human species. We may even be able to improve the creativity of the economy by applying the mathematics of creativity.
Understanding higher levels of definability is a creative process that cannot be characterized. However we can characterize decidability relative to some level of definability. In this way we can reason about developing a hierarchy of increasingly more powerful mathematical systems with respect to decidability.
Suppose we wish to set up a mechanistic process that will nondeterministically (following an increasing number of paths) evolve mathematical systems that are increasingly stronger with respect to decidability. We want there to be at least one path that correctly decides (for example) the halting problem for every TM. It is trivial to do this since we can set up a nondeterministic process that will enumerate every initial segment of every real number along some path such that the union of those segments along the path is that real. How can we optimize this process? There are many possibilities. Instead of directly enumerating the statements we are interested in we can assign truth values to a formal system such as first order arithmetic . We can then enumerate the consequences of those assumptions and eliminate any path that is inconsistent. There are tradeoffs between the effort but in enumerating these consequences and the effort put into following more divergent paths. We can ask how do we best allocate resources between these alternatives to maximize the rate at which we precede along at least one correct path.
This is similar to the tradeoffs between more complex organisms and more descendants that have evolved biologically. It is possible that arguments like the above can help us understand evolution. Biological systems are subject to the tradeoff between diversity and complexity of individuals for similar reasons. The complexity can help an individual better deal with and understand his situation but one can also better respond to ones environment through more diversity. In each case part of the issue is how well one can internally model `mathematically' aspects of ones environment and situation.
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