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Metabiology Life as Evolving Software by G J Chaitin

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- 1 METABIOLOGY: LIFE AS EVOLVING SOFTWARE METABIOLOGY: a eld parallel to biology, dealing with the random evolution of arti- cial software (computer programs) rather than natural software (DNA), and simple enough that it is possible to prove rigorous theorems or formulate heuristic arguments at the same high level of precision that is common in the- oretical physics.
- 2 The chance that higher life forms might have emerged in this way [by Darwinian evolution] is comparable to the chance that a tornado sweeping through a junkyard might assemble a Boeing 747 from the materials therein. Fred Hoyle. In my opinion, if Darwins theory is as simple, fundamental and basic as its adherents believe, then there ought to be an equally fundamental mathemati- cal theory about this, that expresses these ideas with the generality, precision and degree of abstractness that we are accustomed to demand in pure math- ematics. Gregory Chaitin, Speculations on Biology, Information and Complexity. Mathematics is able to deal successfully only with the simplest of situations, more precisely, with a complex situation only to the extent that rare good fortune makes this complex situation hinge upon a few dominant simple fac- tors. Beyond the well-traversed path, mathematics loses its bearings in a jungle of unnamed special functions and impenetrable combinatorial partic- ularities. Thus, the mathematical technique can only reach far if it starts from a point close to the simple essentials of a problem which has simple essentials. That form of wisdom which is the opposite of single-mindedness, the ability to keep many threads in hand, to draw for an argument from many disparate sources, is quite foreign to mathematics. Jacob Schwartz, The Pernicious Inuence of Mathematics on Science. It may seem natural to think that, to understand a complex system, one must construct a model incorporating everything that one knows about the system. However sensible this procedure may seem, in biology it has repeat- edly turned out to be a sterile exercise. There are two snags with it. The rst is that one nishes up with a model so complicated that one cannot understand it: the point of a model is to simplify, not to confuse. The sec- ond is that if one constructs a suciently complex model one can make it do anything one likes by ddling with the parameters: a model that can predict anything predicts nothing. John Maynard Smith & Eors Szathmary, The Origins of Life.
- 3 Course Notes METABIOLOGY: LIFE AS EVOLVING SOFTWARE G. J. Chaitin Draft October 1, 2010
- 4 To my wife Virginia who played an essential role in this research
- Contents Preface 7 1 Introduction: Building a theory 9 2 The search for the perfect language 19 3 Is the world built out of information? Is everything soft- ware? 39 4 The information economy 45 5 How real are real numbers? 55 6 Speculations on biology, information and complexity 77 7 Metaphysics, metamathematics and metabiology 87 8 Algorithmic information as a fundamental concept in physics, mathematics and biology 101 9 To a mathematical theory of evolution and biological creativ- ity 113 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 History of Metabiology . . . . . . . . . . . . . . . . . . . . . . 114 9.3 Modeling Evolution . . . . . . . . . . . . . . . . . . . . . . . . 116 9.3.1 Software Organisms . . . . . . . . . . . . . . . . . . . . 116 9.3.2 The Hill-Climbing Algorithm . . . . . . . . . . . . . . 116 9.3.3 Fitness . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.3.4 What is a Mutation? . . . . . . . . . . . . . . . . . . . 117 5
- 6 9.3.5 Mutation Distance . . . . . . . . . . . . . . . . . . . . 117 9.3.6 Hidden Use of Oracles . . . . . . . . . . . . . . . . . . 118 9.4 Model A (Naming Integers) Exhaustive Search . . . . . . . . . 119 9.4.1 The Busy Beaver Function . . . . . . . . . . . . . . . . 119 9.4.2 Proof of Theorem 1 (Exhaustive Search) . . . . . . . . 119 9.5 Model A (Naming Integers) Intelligent Design . . . . . . . . . . 120 9.5.1 Another Busy Beaver Function . . . . . . . . . . . . . 120 9.5.2 Improving Lower Bounds on . . . . . . . . . . . . . . 121 9.5.3 Proof of Theorem 2 (Intelligent Design) . . . . . . . . . 123 9.6 Model A (Naming Integers) Cumulative Evolution at Random . 124 9.7 Model B (Naming Functions) . . . . . . . . . . . . . . . . . . . 128 9.8 Remarks on Model C (Naming Ordinals) . . . . . . . . . . . . . 132 9.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10 Parsing the Turing test 143 11 Should mathematics be done dierently because of Godels incompleteness theorem? 149 Bibliography 159 Books by Chaitin 161
- Preface Biology and mathematics are like oil and water, they do not mix. Never- theless this course will describe my attempt to express some basic biological principles mathematically. Ill try to explain the raison detre of what I call my metabiological approach, which studies randomly evolving computer programs rather than biological organisms. I want to thank a number of people and organizations for inviting me to lecture on metabiology; the interaction with audiences was extremely stimu- lating and helped these ideas to evolve. Firstly, I thank the IBM Watson Research Center, Yorktown Heights, where I gave two talks on this, including the world premiere talk on metabiol- ogy. Another talk on metabiology in the United States was at the University of Maine. In Argentina I thank Veronica Becher of the University of Buenos Aires and Victor Rodriguez of the University of Cordoba for their kind invitations. And I am most grateful to the University of Cordoba, currently celebrating its 400th anniversary, for the honorary doctorate that they were kind enough to bestow on me. In Chile I spoke on metabiology several times at the Valparaiso Complex Systems Institute, and in Brazil I included metabiology in courses I gave at the Federal University of Rio de Janeiro and in a talk at the Federal University in Niteroi. Furthermore I thank Bernd-Olaf Kuppers for inviting me to a very stim- ulating meeting at his Frege Centre for Structural Sciences at the University of Jena. And I thank Ilias Kotsireas for organizing a Chaitin-in-Ontario lecture se- ries in 2009 in the course of which I spoke on metabiology at the University of Western Ontario in London, at the Institute for Quantum Computing in Waterloo, and at the Fields Institute at the University of Toronto. The chap- 7
- 8 Chaitin: Metabiology ter of this book on is based on a talk I gave at Wilfrid Laurier University in Waterloo. Finally, I should mention that the chapter on The Search for the Perfect Language was rst given as a talk at the Hebrew University in Jerusalem in 2008, then at the University of Campinas in Brazil, and nally at the Perimeter Institute in Waterloo, Canada. The chapter on Is Everything Software? was originally a talk at the Technion in Haifa, where I also spoke on metabiology at the University of Haifa, one of a series of talks I gave there as the Rothschild Distinguished Lecturer for 2010. These were great audiences, and their questions and suggestions were extremely valuable. Gregory Chaitin, August 2010
- Chapter 1 Introduction: Building a theory This is a course on biology that will spend a lot of time discussing Kurt Godels famous 1931 incompleteness theorem on the limits of formal mathematical reasoning. Why? Because in my opinion the ultimate historical perspective on the signicance of incompleteness may be that Godel opens the door from mathematics to biology. We will also spend a lot of time discussing computer programs and software for doing mathematical calculations. How come? Because DNA is presumably a universal programming language, which is a language that is rich enough that it can express any algorithm. The fact that DNA is such a powerful programming language is a more fundamental characteristic of life than mere self-reproduction, which anyway is never exactfor if it were, there would be no evolution. Now a few words on the kind of mathematics that we shall use in this course. Starting with Newton mathematical physics is full of what are called ordinary dierential equations, and starting with Maxwell partial dierential equations become more and more important. Mathematical physics is full of dierential equations, that is, continuous mathematics. But that is not the kind of mathematics that we shall use here. The secret of life is not a dierential equation. There is no dierential equation for your spouse, for an organism, or for biological evolution. Instead we shall concentrate on the fact that DNA is the software, its the programming language for life. It is true that there are (ordinary) dierential equations in a highly suc- 9
- 10 Chaitin: Metabiology cessful mathematical theory of evolution, Wright-Fisher-Haldane pop- ulation genetics. But population genetics does not say where new genes come from, it assumes a xed gene pool and discusses the change of gene frequencies in response to selective pressure, not bio- logical creativity and the major transitions in evolution, such as the transition from unicellular to multicellular organisms, which is what interests us. If we arent going to use anymore the dierential equations that popu- late mathematical physics, what kind of math are we going t

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