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8 A Creative Philosophy of Mathematical Truth

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Platonic philosophy visualizes an ideal realm of absolute truth and beauty of which the physical world is a dim reflection. This ideal reality is perfect, complete and thus static. In stark contrast, the universe we inhabit is spectacularly creative. An almost amorphous cosmic big bang has evolved into an immense universe of galaxies each of which is of a size and complexity that takes ones breath away. On at least one minuscule part of one of the these galaxies, reproducing molecules have evolved into the depth and richness of human conscious experience. There is no reason to think that we at a limit of this creative process. There may be no finite limit to the evolution of physical structure and the evolution of consciousness. This is what the history of the universe, this planet and the facts of mathematics suggest to me. We need a new philosophy of mathematics grounded in our scientific understanding and the creativity that mathematics itself suggests is central to both developing mathematics and the content created in doing do.
Mathematics is both objective and creative. If a TM runs forever, this is logically determined by its program. Yet it takes creativity to develop a mathematical system to prove this. Gödel proved that no formal system that is sufficiently strong can be complete, but there is nothing (except resources) to prevent an exploration over time of every possible formalization of mathematics. As mentioned earlier, it is just such a process that created the mathematically capable human mind. The immense diversity of biological evolution was probably a necessary prerequisite for evolving that mind.
Our species has a capacity for mathematics as a genetic heritage. We will eventually exhaust what we can understand from exploiting that biological legacy through cultural evolution. This exhaustion will not occur as an event but a process that keeps making progress. However there must be a Gödel limit to the entire process even if it continues forever. Following a single path of mathematical development will lead to an infinite sequence of results all of which are encompassed in a single axiom that will never be explored. This axiom will only be explored if mathematics becomes sufficiently diverse. In the long run, the only way to avoid a Gödel limit to mathematical creativity is through ever expanding diversity.
There is a mathematics of creativity that can guide us in pursuing diversity. Loosely speaking the boundary between the mathematics of convergent processes and that of divergent creative processes is the Church-Kleene ordinal or the ordinal of the recursive ordinals. For every recursive ordinal r0 there is a recursive ordinal r1 (r0 ≤ r1) such that there are halting problems decidable by r1 and not by any smaller ordinal. In turn every halting problem is decidable by some recursive ordinal. The recursive ordinals can decide the objective mathematics of convergent or finite path processes. Larger countable ordinals define a mathematics of divergent processes, like biological evolution, that follow an ever expanding number of paths1
The structure of biological evolution can be connected to a divergent recursive process. To illustrate this consider a TM that has an indefinite sequence of outputs that are either terminal nodes or the Gödel numbers of other recursive processes. In the latter case the TM that corresponds to the output must have its program executed and its outputs similarly interpreted. A path is a sequence of integers that corresponds to the output index at each level in the simulation hierarchy. For example the initial path segment (4,1,3) indexes a path that corresponds to the fourth output of the root TM (r4), the first output of r4 (r4,1) and the third output of r4,1 (r4,1,3). These paths have the structure of the tree of life that shows what species were descended from which other species.
Questions about divergent recursive processes can be of interest to inhabitants of an always finite but potentially infinite universe. For example one might want to know if a given species will evolve an infinite chain of descendant species. In a deterministic universe. this problem can be stated using divergent recursive processes to model species. We evolved through a divergent creative process that might or might not be recursive. Quantum mechanics implies that there are random perturbations, but that may not be the final word.
Even with random perturbations, questions about all the paths a divergent recursive process can follow, may be connected to biological and human creativity. Understanding these processes may become increasingly important in the next few decades as we learn to control and direct biological evolution. Today there is intense research on using genetic engineering to cure horrible diseases. In time these techniques will become safe, reliable and predictable. The range of applications will inevitably expand. At that point it will become extremely important to have as deep an understanding as possible of what we may be doing. To learn more about this see[1].
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References

[1]
Paul Budnik. What is and what will be: Integrating spirituality and science. Mountain Math Software, Los Gatos, CA, 2006.

Footnotes:

1In a finite universe there are no truly divergent processes. Biological evolution can be truly divergent only if our universe is potentially infinite and life on earth migrates to other planets, solar systems and eventually galaxies.


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