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Mathematics

Mathematics can teach us about possible transformations of structure and thus consciousness. Through mathematics we can extrapolate from the experience that is the essence of the human mind to what life and conscious experience may evolve into.

The capacity for subtle self reflection is a defining characteristic of human consciousness. This capacity characterizes a level of conscious experience. In mathematics it is common to measure the power of a mathematical system by the level of self reflection or iteration that is definable within the system. Gödel's Incompleteness Theorem implies that there is no finite limit to the levels of subtle self reflection in finite systems.

At the beginning of the 20th century a famous mathematician, Hilbert, proposed the construction of a formula or mechanistic process for deciding all mathematical questions. Gödel proved this was impossible[3] .

Mathematics then and now is based on formal systems. In effect these are mechanistic processes or computer programs for enumerating theorems. Gödel proved that any consistent formal system powerful enough to define the primitive recursive functions had statements in the system that could not be decided within the system. The primitive recursive functions are a fragment of elementary mathematics powerful enough to define a Universal Turing Machine.

Expanding the level of self reflection seems to expand the nature of conscious experience as has happened in the evolutionary hierarchy culminating in human experience. Evolution creates such structures because they have survival value. They allow better prediction of the consequences of ones actions.

Gödel's theorem implies that unbounded evolution of self reflecting structures is possible only if diversity expands without limit. Evolution operates on itself. Competition and cooperation between branches on the tree of life provide the rich ecosystem from which a structure as sophisticated as the human mind has evolved. Through man the evolutionary process has become conscious of itself and is able to use mathematics to understand aspects of its structure. We can derive the boundary conditions that will allow us to take conscious control of evolution without stifling creativity.


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