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

NOTICE: This website is being gradually phased out. My new website is hosted by Academia.
However the LISP software for my three Springer books will not be moved to Academia.
We intend to freeze this website, not remove it. — GJC, 22 January 2014

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

Latest Book Covers

# Music by Michael Winter

Michael Winter is a composer, music theorist, and software designer. He co-founded and directs the wulf., a non-profit arts organization that presents music free to the public in Los Angeles. Michael is a firm believer in music making as an exploratory process and in free information; e.g. open source code, free music, etc.

Michael is spending four months in South America, and just performed two evenings at Audio Rebel in Rio de Janeiro. Mike has two pieces inspired by the Omega number. He performed one of these pieces in Rio, "approximating omega." The other is "for gregory chaitin."

I've just added recordings and the open scores for both of these pieces. Enjoy!

• approximating omega: piece by composer Michael Winter sonifying a LISP program for approximating Omega in the limit from below

• for gregory chaitin: piece by composer Michael Winter sonifying the first few bits of the numerical value of an Omega number

# Bit Bang: The Birth of Digital Philosophy

Bit Bang. La nascita della filosofia digitale: An important book just published in Italy by the theologian Andrea Vaccaro (with G. O. Longo). Highly recommended!

The Search for the Perfect Language argues that God cannot be a mathematician, because there is no perfect language for expressing mathematical reasoning (Gödel, 1931), but He could be a programmer, because there are perfect languages for expressing mathematical algorithms (Turing, 1936). Going beyond Turing, algorithmic information theory identifies the most perfect, the most compact, most expressive, such algorithmic languages.

# Books

• Algorithmic Information Theory, Cambridge University Press, 1987.

• Information, Randomness and Incompleteness: Papers on Algorithmic Information Theory, World Scientific, 1987, 2nd edition, 1990.

• Information-Theoretic Incompleteness, World Scientific, 1992.

• The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning, Springer, 1998. Also in Japanese.

• The Unknowable, Springer, 1999. Also in Japanese.

• Exploring Randomness, Springer, 2001.

• Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer, 2002. Also in Portuguese and Japanese.

• From Philosophy to Program Size: Key Ideas and Methods. Lecture Notes on Algorithmic Information Theory from the 8th Estonian Winter School in Computer Science, EWSCS '03, Tallinn Institute of Cybernetics, 2003.

• Meta Math! The Quest for Omega, Pantheon, 2005. Also UK, French, Italian, Portuguese, Japanese and Greek editions.

• With Ugo Pagallo, Teoria algoritmica della complessità, Giappichelli, 2006.

• Thinking about Gödel and Turing: Essays on Complexity, 1970-2007, World Scientific, 2007.

• Matemáticas, Complejidad y Filosofía / Mathematics, Complexity and Philosophy (bilingual Spanish/English edition), Midas, 2011.

• With Newton da Costa and Francisco Antonio Doria, Gödel's Way: Exploits into an Undecidable World, CRC Press, 2012.

• Proving Darwin: Making Biology Mathematical, Pantheon, 2012. Also in Spanish, Italian, Japanese and soon in Chinese.

# 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

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