We can gain insight into the archetypes through the finite and
infinite in mathematics. The for TM s is
the most elementary example of the relationship between the finite
and infinite. A TM will either halt or not halt. This is an
absolute fact. There is no general method to *decide* the
question for each TM yet for each TM there is a mathematical
principle that will decide the question for that TM . As TM s
become more complex new undiscovered mathematics must be
understood. There is no finite way to encompass mathematical
truth even when that truth is constrained
to statements about the future states of a computer following the
precise deterministic steps of a program. This is important not
only for questions that refer to an indefinite future. It is rare
that one can model a system in complete detail. Most useful
mathematics involves that ability to decide things about a system
with less than perfect knowledge of its state. Often such
mathematics falls in the same category of the halting problem with
respect to decidability . It is based on
general mathematical properties of a system and the implications of
those properties from mathematics.

We must create meaning to broaden our understanding of absolutely true mathematics. The idea of well foundedness

for all paths encoded by a real number (as discussed in Chapter on page ) is a powerful mathematical concept. It provides strong extensions of elementary induction . Is this concept created or discovered? There are not and cannot be real numbers or well founded structures in our finite particular experience. Yet the concepts are relevant to a potentially infinite universe . Evolution creates the brain structures that in turn create these concepts. They evolve because of the advantage of building models of reality that can be extrapolated into the future with logic.

The unknowable creative aspect of the properties of numbers and the unknowable creative aspect of matter are the same thing. It is this creativity that has expressed itself in our world as it is today and that continually unfolds in ways that we can never predict or control. The archetypal images of the human psyche are formed from this creative process and point towards it.

The mathematics of creativity as described in Chapter allow us to know with mathematical precision some of the properties and constraints of creativity. It allows us to make connections between some human instincts and general mathematical properties. It opens the Jungian notion of archetype to mathematical analysis. This does not lessen the divine mystical nature of archetype. On the contrary it shows the divine mystical nature of mathematics.

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