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The finite and the infinite



 

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 gif on page gif ) 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 gif 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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