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It is helpful to have some understanding of the foundations of mathematics to fully grasp the structure and essence dichotomy at the core of this book. This chapter and the next develop the foundations of mathematics and connect these to philosophical issues.

Many people are needlessly turned off by mathematics because it is often taught in a manner that is boring and unnecessarily difficult. Like most people, my mind goes blank when I face a page full of equations. Formal mathematics is important because it is connected with every day life, but it is often taught as if this were irrelevant or not true.

By using English language explanations and connecting mathematics to properties of computers, it is hoped that this chapter and the next will be intelligible and of interest to a wide audience. Experience with mathematical expressions and the idea of a function at the level of high school algebra is helpful. These are defined and explained, but may take thinking about and playing with to fully grasp.

Computer programs are used as a surrogate for logically determined or mechanistic processes. The focus on computers is not to suggest there is anything special about them. The universe allows us to predict future events to a limited degree using logic and mathematics. It is that aspect of mathematics that has practical importance and computers are an accurate metaphor for a logically determined sequence of events.

The computer metaphor suggests a deterministic universe where nothing truly novel happens. But this is not the case. Gödel proved in the 1930's that mathematics cannot be captured in any finite system. Gödel's result suggests that mathematics is an inherently creative endeavor albeit one that can create absolute truth. This result is derived and explored in the next two chapters. I will argue that it is central to understanding the creative evolution of consciousness.

- Structure and consciousness
- Logically determined unsolvable problems
- Formal logic
- Formal mathematics
- Axioms of Set Theory
- Axiom of extensionality
- Axiom of the empty set
- Axiom of unordered pairs
- Axiom of union
- Axiom of infinity

- Infinity
- Cardinal numbers
- Gödel's Incompleteness Theorem
- The Halting Problem

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**Next:** Structure and consciousness **Up:** whatr72h
**Previous:** Extensions of consciousness **Contents**

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