Calculus Of Ideas, A: A Mathematical Study Of Human Thought

A Mathematical Study of Human Thought

Ulf Grenander

• Publisher: WSPC
• Year: 2012
• Print ISBN: 9789814383189
• eBook ISBN: 9789814439114
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This monograph reports a thought experiment with a mathematical structure intended to illustrate the workings of a mind. It presents a mathematical theory of human thought based on pattern theory with a graph-based approach to thinking. The method illustrated and produced by extensive computer simulations is related to neural networks. Based mainly on introspection, it is speculative rather than empirical such that it differs radically in attitude from the conventional wisdom of current cognitive science.

Contents:

• An Architecture for the Mind:
  • Introduction
  • Creating Thoughts: Algebra of Thinking
• Personality of a Mind:
  • A Mind Equation
• Two Personalities:
  • Building a GOLEM, a Thinking Machine
  • As Thinking Develops
• Realizing a Mind:
  • MIND and Brain
  • Reflections
  • Doubts and Certainties
• Appendics:
  • Some Famous Mind Theories
  • Consistency of Probability Measure
  • A Modality Lattice
  • Dynamics of Random Process of Thinking
  • Software for GOLEM

Readership: Graduate students in applied mathematics, statistics and cognitive science as well as postgraduates and researchers in these fields.

Part I

An Architecture for the Mind

Chapter 1

Introduction

1.1 A Mathematical Theory of Mind?

The human mind is a mystery. Although it is so close to us — we live in and with it — we do not really understand how it works. Philosophers and thinkers in general have struggled with this question for millennia and much has been learned, most in a vague and unspecific form. Some attempts have also been tried to describe it through logical schemata and in mathematical form. But human thought is (normally) not completely rigid; it is only partly predictable.

We instinctively avoid believing that our thoughts are generated by a more or less mechanical device. We do not want to be seen as machines. Hence we tend to reject statements like the one by Karl Vogt, a 19th century German philosopher, who stated that “the brain produces thoughts as the liver produces bile, or the kidneys produce urine”. But few would deny that the material substrate of thought, the neural system of the brain, obeys the laws of physics/chemistry, so that it is not impossible that there may exist mathematical laws of thought in principle derivable from physic/chemistry. Such laws would have to be probabilistic. The following consists of speculations with no firm support in empirics, just ideas that seem plausible (to the author).

We shall consider thought processes that include logical thinking, but this is only one mode among many. We follow Damasio (1999) who discusses the dominating role of emotions for human thought in an elegant and convincing way. We shall include fear, love, emotions . . . But recall Pascal’s dictum: “The heart has its reasons, of which reason knows nothing.” Indeed, we know only little about the functioning of emotional thought processes. But wait! We are not after a general theory of human thought, indeed we do not believe in such an endeavor. Instead we will try to present only a shell, a scheme only, of human thought that will have to be filled with content different for each individual, setting different values to the (many) mind parameters. This content can have its origin in the genetic and cultural background in which the individual lives, as well as being formed by experiences leading to a dynamically changing mind. Thus we will concentrate on the general architecture of the building rather than on its detailed specification of mortar and bricks.

We shall deal with the mind without reference to the brain. A completely reductionist mind theory would be based on neuro-physiological knowledge, deriving mental processes from what is known about their cerebral substrate. We are certainly in favor of such an approach, but in the absence of a complete brain theory, it is not feasible at present. Instead we shall base the construction on introspection and on what has been learned over the centuries in a less formal setting about the working of the mind by clinicians and what can be found in novels, poetry and plays. This non-positivist attitude is open to the criticism that it leads to no testable hypothesis. We admit that this is true, at least in the immediate future, and accept the criticism.

The last several decades have witnessed remarkable process in the neurophysiology of the brain — many elegant experiments have thrown light on the functioning of neurons, at first for single neurons and more recently for cell assemblies. This has led to an impressive body of empirical knowledge about the brain. Some researchers have tried to increase our understanding of the human mind through mathematical studies of the firing rates of neurons. It seems doubtful to this author whether mathematical work of this type alone will lead to more insight in the human mind than what the purely experimental results have shown. This author is all in favor of such a reductionist approach: it is necessary — but not sufficient! Perhaps such studies can help in understanding how Ratus ratus runs in mazes or how we turn our right hand at some command, but for the understanding of the mind of Homo sapiens they are flagrantly insufficient. We are aware of the many talented and knowledgeable researchers applying mathematical analysis to neural rates, concentrating on neural behavior while neglecting high level activities of the human mind. They seem suspicious of a theory of higher mental faculties. Alas, they include even such perso...

Table of contents

• Front Cover
• Half Title
• Dedication
• Title
• Copyright
• Foreword
• Preface
• Abstract
• Contents
• Part I An Architecture for the Mind
• Part II Personality of a Mind
• Part III Two Personalities
• Part IV Realizing a Mind
• References
• Appendix A Some Famous Mind Theories
• Appendix B Consistency of Probability Measure
• Appendix C A Modality Lattice
• Appendix D Dynamics of Random Process of Thinking
• Appendix E Software for GOLEM