# 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 r_{0} there is a recursive
ordinal r_{1} (r_{0} ≤ r_{1}) such that
there are halting problems decidable by r_{1} 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 paths^{1}
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 (r_{4}), the first output of
r_{4} (r_{4,1}) and the third output of
r_{4,1} (r_{4,1,3}). 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:

^{1}In 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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