Brains, Machines and Mathematics
Michael A. Arbib, 1965

This book forms an introduction to the common ground of brains, machines and mathematics. It is for a reader who has heard of cybernetics, information theory, and Gödel's theorem and wants to gain more of an understanding of them. Full use of the book does require a moderate mathematic background, say, a year of college calculus. However, much of the book should be intelligible to the reader who chooses to skip the mathematical proofs.

Biological systems are so much more complicated than the usual systems of physics that we do expect to achieve a fully satisfactory biological mathematics for many years to come. This book strives to introduce the reader to its early stages.

Chapter 1. Neural Nets, Finite Automata, and Turing Machines

1.1 Introductory Neurophysiology

1.2 The McCulloch-Pitts Model

1.3 Finite Automata and Modular Nets

1.4 Finite Automata and Digital Computers

1.5 Turing Machines

1.6 Turing's Hypothesis and Recursive Sets

1.7 Regular and Realizable Events

Bibliography: Chapter 1

Chapter 2. Structure and Randomness

2.1 The Visual System of the Frog

2.1.1 Comparisons

2.2 The Perceptron

2.3 Structure versus Randomness

Chapter 3. The Correction of Errors in Communication and Computation

3.1 Reliable Brains from Unreliable Neurons

3.2 Von Neumann's Multiplexing Scheme

3.3 Shannon's Communication Theory

3.31 A Measure of Information

3.32 Modeling the Source and Channel

3.33 Equivocation and Channel Capacity

3.34 Shannon's Fundamental Theorem for a Discrete Noisy Channel

3.35 Coding

3.4 Communication Theory and Automata

3.5 The Cowan-Winograd Theory of Reliable Automata

Chapter 4. Cybernetics

4.1 Feedback and Oscillation

4.2 Resonant Frequencies in Neural Networks

4.3 Prosthesis and Homeostatis

4.4 Gestalt and Universals

4.5 Some Further Topics

Chapter 5. Gödel's Incompleteness Theorem

5.1 The Foundations of Mathematics

5.2 Revision on Recursion

5.3 Recursive Logics

5.4 Arithmetic Logics

5.5 The Proof of Gödel's Incompleteness Theorem

5.6 The Brain-Machine Controversy


Appendix: Basic Notions of Set Theory

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