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The mathematics of creativity



The mathematics of creativity

The most remarkable thing about conscious experience is its evolution to every increasing richness and complexity. Through Gödel's Incompleteness Theorem we know that the creative evolution of structure can never be captured in finite form. It is an open ended ever expanding process. There is a hierarchy of mathematical truth that characterizes levels of abstraction or self reflection such as the self reflection that is a defining characteristic of human consciousness. Gödel proved that this hierarchy cannot be finitely described However it can be fully developed by exploring an every increasing number of paths without selecting a best or correct path as biological evolution has done in creating the human mathematical mind.

The diversity of exploring all possible paths need not be a blind exploration of everything. One can prune the search tree without limiting the richness of structure that can be explored. For example one does not need to explore extensions of a system that contradict that system. Exploring more complex systems requires more resources and there is an inevitable tradeoff between allocating resources to the paths that seem most promising and widening the search of alternatives. This is not just an abstraction but is central to creative development in science, mathematics, technology and the economy.

It is possible to develop the mathematics that sets boundary conditions for creativity. These cannot optimize creativity but they can establish regions outside of which creativity is certain to be suboptimal. A trivial example is the necessity for continually increasing both the number of paths explored and the resources devoted to the most promising paths. This suggests that concentration of resources at the expense of diversity in an expanding economy will be suboptimal in the long run.

In spite of Gödel's Incompleteness Theorem most mathematicians do not see the development of mathematics as in inherently divergent creative process. They most commonly try to extend mathematics through axioms of infinity. These are simple assumptions about infinite sets that have powerful combinatorial implications. Mathematicians strive for results that can be directly apprehended by the human mind and this limits their ability to develop direct intuition about combinatorially complex structures that can only be effectively dealt with using computers.

Mathematics plays an important role in the study of evolution including evolutionary psychology. But that analysis focuses on steady state solutions. For example mathematics is often used to compute the expected frequency of different mutations. The much more difficult problem of evolutionary creativity is poorly understood. There is a lot of interest in the emergence of complexity but little work that I know of tries to connect this with the hierarchies of mathematical truth and complexity.

Number is the mediator between the reality of here and now and our deeper existence as part of the unbounded creative process of evolving consciousness as Carl Jung believed[6, par 778][9]. If we survive as a species, the future not just of our species but of the entire evolutionary process will fall into our hands.


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