^{1}
- Roger Penrose has argued that quantum effects in the brain
allow mathematicians to transcend the limitation of Gödel's
proof[7]. This is not necessary
to explain the mathematically capable human mind. All that requires
is the enormous diversity of biological evolution as discussed
below. There is no significant evidence to support Penrose's idea.
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^{2}
- There are classes of what might be considered practical
problems, like the computer halting problem, that no finite formal
system can solve for all examples. The computer halting problem
asks if an ideal computer with a specific program will run forever
or eventually halt. It is straight forward to construct a computer
program that will halt if and only if the existing formalization of
mathematics is inconsistent.
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^{3}
- Quantum randomness is a closed question for the majority of
physicists, but a minority, including the 1999 Nobel prize winner,
Gerard t Hooft[8], are
unconvinced.
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^{4}
- If the fundamental laws of physics turn out to be discrete and
not continuous, as many prominent physicists suspect, there are no
true chaotic processes, but discrete models can simulate the
effects of chaos to a high enough accuracy that there may be no
practical difference in most circumstances.
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^{5}
- Of course a finite planet will ultimately impose limits.
However we may eventually develop robotic space ships with all our
knowledge and the capacity to develop civilization on planets in
distant solar systems. The universe may be potentially infinite.
Every previous boundary has been greatly expanded. Cosmology is of
necessity a highly speculative science that is continually
evolving.
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^{6}
- The ideas in this paper are more fully developed in
*What is and what will be: Integrating spirituality and
science*[1]. There is a
related video, `Mathematical Infinity and Human Destiny', on
Google Video.
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