Sans les mathématiques on ne pénètre point au fond de la philosophie. Sans la philosophie on ne pénètre point au fond des mathématiques. Sans les deux on ne pénètre au fond de rien. — Leibniz [Without mathematics we cannot penetrate deeply into philosophy. Without philosophy we cannot penetrate deeply into mathematics. Without both we cannot penetrate deeply into anything.] Tribute to Leibniz: Essay on Leibniz, Complexity and Incompleteness

METABIOLOGY: Programming without a Programmer

Darwin's theory of evolution has been described as "design without a designer." Instead we study "programming without a programmer," that is, the evolution of randomly mutating software. We propose a toy model of evolution that can be studied mathematically: the new field of metabiology, which deals with randomly evolving artificial software (computer programs) instead of randomly evolving natural software (DNA).

[Note on George Bernard Shaw (1856-1950): The first use of the term metabiology of which I am aware is in Shaw's 1921 play Back to Methuselah: A Metabiological Pentateuch. A better play of Shaw's that is also on evolution is his 1903 Man and Superman. Shaw's idea-filled plays, which feature lengthy prefaces, were originally meant to be read rather than performed.]

Ursula Molter, Gregory Chaitin and Hernán Lombardi opening the Buenos Aires Mathematics Festival (Argentina, May 2009)

Gregory Chaitin is well known for his work on metamathematics and for the celebrated Ω number, which shows that God plays dice in pure mathematics. He has published many books on such topics, including Meta Math! The Quest for Omega. His latest book, Proving Darwin: Making Biology Mathematical, attempts to create a mathematical theory of evolution and biological creativity. His least technical book is Conversations with a Mathematician.

Chaitin is a professor at the Federal University of Rio de Janeiro and an honorary professor at the University of Buenos Aires, and has honorary doctorates from the National University of Córdoba in Argentina and the University of Maine in the United States. He is also a member of the Académie Internationale de Philosophie des Sciences (Brussels) and the Leibniz-Sozietät der Wissenschaften (Berlin).

Carnival in Rio de Janeiro, 1970. Photo by Peter Albrecht

# Contents

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# LISP Software for Springer Books

## Exploring Randomness (2001)

• Preface

### Part I—Introduction

• Historical introduction—A century of controversy over the foundations of mathematics

• What is LISP? Why do I like it?

• How to program my universal Turing machine in LISP
utm2 code, utm2 run

### Part II—Program-Size

• A self-delimiting Turing machine considered as a set of (program, output) pairs
exec code, exec run

• How to construct self-delimiting Turing machines: the Kraft inequality
kraft code, kraft run

• The connection between program-size complexity and algorithmic probability: $H(x) = -\log_2 P(x) + O(1)$.
Occam's razor: there are few minimum-size programs
occam code, occam run

• The basic result on relative complexity: $H(y|x) = H(x, y) - H(x) + O(1)$
decomp code, decomp run, lemma code, lemma run

### Part IV—Future Work

• Extending AIT to the size of programs for computing infinite sets and to computations with oracles

• Postscript—Letter to a daring young reader

$\Omega = \sum_{\text{program p halts}} 2^{-(\text{size in bits of p})}$ $\Omega_U = \sum_{\text{U(p) halts}} 2^{-|p|}$ $\Omega' = \sum_{n \,=\, 1,\, 2,\, 3,\, ...} 2^{-H(n)}$