... limit.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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... encounter.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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... random.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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... like.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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... permit,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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... themselves.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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