Axioms is a work that explores the true nature of human knowledge, in particular the fundamental nature of deductive and inductive rea- soning. It begins by embracing Hume’s Skepticism and Descartes’ one “certain” thing, and then looking for a way out of the solipsistic hell this leaves one in in terms of “certain” knowledge. Indeed, to the extent that philosophy in the past has sought to provide certain answers to virtually any question at all, philosophy itself proves to be bullshit – all philosophical arguments ultimately come back to at least one unprovable premise, usually unstated, and can be refuted by simply asserting “I don’t agree with your premises.” The way out is to give up the idea of certain knowledge. All non- immediate knowledge of the world is based on these premises, which are unprovable assumptions, or axioms. To understand the world around us, we have to begin by making all sorts of assumptions – such as the assumption that there is a real world out there to be un- derstood in the first place, and that the rules that govern it are struc- tured and understandable. The rules of thought, logic, and mathe- matics that we use to structure this understanding are themselves only methodologies for deriving contingent truth based on unprov- able axioms, and changing the axioms underlying any mathematical or logical argument often changes the equally valid conclusions. Axioms by Robert G. Brown Dedication This book is dedicated to the giants of mathematical and scientific philosophy upon whose backs it stands: Plato, Hume, Descartes, G¨odel, Bayes, Shannon, Cantor, Cox, Jaynes, and many more, too many to count, actually. I do wish to explicitly acknowledge Cox’s The Algebra of Inference, Jaynes’ Probability Theory: The Logic of Science, and MacKay’s Information Theory, Inference, and Learning Algorithms, which collectively establish what is very likely “the” rigorous basis for knowledge expressed as a contingent degree of belief and many of its connections to worlds both concrete and abstract. It is also dedicated to my philosophy professor and guru at Duke, George Roberts, who had an enormous impact on me as I pursued an “invisible” philos- ophy major at Duke to accompany my physics major (invisible because at the time Duke had no way of acknowledging the completion of a Bachelor of Science in one discipline and a Bachelor of Arts in another). Finally, it is dedicated to my good friends and colleagues in the Duke Physics Department, especially Richard Palmer (for teaching me about Jaynes, Bayes, maximum entropy, and complex systems in general way back in Statistical Me- chanics in grad school) and Mikael Ciftan, who has been as a second father to me for nearly thirty years now. No book is written in a vacuum. I have been extraordinarily fortunate to have had the support and encouragement and love of many, many people over a lifetime. My family, my friends, my colleagues (who are also my friends) on the beowulf list, and my many, many students: This book is for you all. Notice Although this book inevitably contains a certain amount of mathematics and science (often expressed as “natural philosophy” or “mathematical philosophy”, it is not intended to be a mathematical or scientific treatise. Indeed, its basic subject is not physics but metaphysics, our basis for knowledge itself rather than any particular thing that we “know” (or rather think that we know) about the world. It is written to be as accessible as possible to as general an audience as possible. So don’t be intimidated – you can read this, and understand it, even if you aren’t terribly good at “math”. Copyright Notice Copyright Robert G. Brown 2007, all rights reserved. Lulu Press www.lulu.com Contents Preface v Introduction 3 I Reason and its Limitations 11 1 Wanted: Answers to some Big Questions 13 1.1 PrimatePhilosophy ......................... 13 1.2 BigQuestions,BigAnswers . 18 1.3 TheAdvantageofReason . 30 2 The Laws of Thought 35 2.1 Making a Living as an Early Philosopher . 35 2.2 ThoughtandtheJoyofSets. 39 2.2.1 Quantum Theory and the Laws of Thought . 43 2.3 AssertionsandExperience. 49 3 Formal Set Theory 53 3.1 Naive Set versus Axiomatic Set Theories . 53 3.2 ThePowerSet ............................ 56 3.3 SetTheoryandtheLawsofThought. 60 3.4 TheNullSet ............................. 67 i ii CONTENTS 3.5 ABitofFormalism.......................... 77 3.6 SetTheoryofThought........................ 79 3.7 Summary ............................... 79 4 Logic 85 4.0.1 The Formal Problem with the Laws of Thought . 90 5 What’s an Axiom? 97 6 G¨odel 107 7 Paradoxes 117 7.1 Fun with Logic: Contradictions and Null Results . 117 8 The Pit of Existential Despair 131 II Philosophy 135 9 Philosophy is Bullshit 137 9.1 PhilosophyisBullshit . 139 10 The Fundamental Axioms of Religion: 147 10.1 The Common Memes of Religious Scripture . 149 10.2 Memetic analysis of the Superorganism . 156 10.3 Religion versus Science: Axiom Wars . 166 11 The Fundamental Axioms of Science 171 12 The Fundamental Axioms of Human Society 173 13 Critique of Specific Philosophies 175 13.1 Why Descartes Proof of God’s Existence is Bullshit . 177 13.2 WhySolipsismisBullshit . 185 CONTENTS iii 13.3 Why Science (Natural Philosophy) is Bullshit . 188 13.4 Why Logical Positivism is Bullshit . 193 13.5 Why the Ontological Proof of God’s Existence is Bullshit . 198 13.6 Why the Intelligent Design “Theory” is Bullshit . 205 13.7 Why Religious Scripture is (without exception) Bullshit . 213 13.8 Why Life, Liberty and Pursuit of Happiness is Bullshit . 213 13.9 Proof by Silliness: Fairy Philosophy . 213 III Axioms 215 14 Meta-Axioms 217 14.1 TheAxiomofOpen-Mindedness . 218 14.2 Shaving the Barber with the Razor of Ockham . 221 14.3 TheEstheticPrinciple . 222 14.4 TheAxiomofRomance . 223 14.5 TheAxiomofDeity ......................... 225 15 You “Are” Your Axioms 227 15.1 TheAxiomofOpen-Mindedness . 233 15.2 Shaving the Barber with the Razor of Ockham . 235 15.3 TheEstheticPrinciple . 236 16 Conclusions 239 A Galileo and St. Bellarmine 241 iv CONTENTS Preface The work in your hands is, I would like to emphasize, not a scholarly work. I’m sloppy about references, for example – sometimes they are there, sometimes they aren’t. Often they will be in the form of a “wikinote” like this one1 . For example, when I misquote (say) The Hitchhiker’s Guide to the Galaxy, I assume that we share some cultural context there and that you’ll recognize that “42”2 is the answer to it all, the reason for Existence itself (and know where it comes from). And hopefully laugh. Putting a wikinote on things like this is a form of hedging my bets, so to speak. It is intended to be: Fun to read. Maybe even funny in parts (unusual for philosophical works • unless they are written by Terry Pratchett3 ). Educational. Not quite textbook quality educational, maybe, but there’s • a lot of historical stuff and many True Facts inside. Some of which might even be true (but don’t count on it). 1Wikipedia: http://www.wikipedia.org/wiki/Wikipedia. This is a Wikipedia reference. In an online or active version of this book, this footnote becomes a working “hot link” to a useful Wikipedia article. For people reading a paper copy, I can only hope that either you are already amazingly literate, well read, and know offhand all about that of which I speak, or that you read this book somewhere near a web browser. By the way, it should become clear from my frequent use of this as a Universal Resource that in my opinion Wikipedia is well on its way toward becoming the crowning achievement of human civilization – literally an online, free repository for all non-encumbered human knowledge, such as it is. 2Wikipedia: http://www.wikipedia.org/wiki/42. See? All human knowledge. Try looking up “42” on Encyclopedia Brittanica and it will laugh at you and return all sorts of irrelevant facts from World War II. 3Wikipedia: http://www.wikipedia.org/wiki/Discworld. Author of the Discworld novels and a perfect master of all that is in this work and then some. In fact, Terry Pratchett could be the world’s greatest living philosopher. Scary, that. v vi PREFACE Shocking and Scandalous. Well, not really. I hardly ever refer to sex outside • of memetic intercourse and occasional references to Frodo Baggins’ sexual history (or lack thereof). Nevertheless, if you are a True Believer in almost anything you’ll probably be both shocked and scandalized when I argue, hopefully persuasively, that True Belief (as opposed to contingent belief) is in fact rather silly. Mind you, there are occasional pockets of math and logic to wade through in the text below, and it is more than a bit hard on proselytic religions of the highly organized and militant sort that seem to be a source of violent misery in much of the world today. In case you have any personal doubts about whether or not you can manage the math, let me summarize the entire book right here, right now. If you can understand the following, you can cope with the set theory and probability theory or even skip it and it won’t matter. If you agree with the following, you can probably even skip buying the book! But please don’t! If you do, I don’t make any money, and my goal in writing it is to save the world and bring about an age of world peace and understanding, and incidental to this, to make money. Anyway, here it is, the fundamental truth about what any conscious mind knows of the Universe and everything in it: Everything that we “know” of the world in which we live outside of our • immediate, real-time experience of our own sensory inputs is known by inference.