Axioms is a work that explores the true nature of human knowledge, in particular the fundamental nature of deductive and inductive reasoning. 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 understood in the first place, and that the rules that govern it are structured and understandable. The rules of thought, logic, and mathematics that we use to structure this understanding are themselves only methodologies for deriving contingent truth based on unprovable axioms, and changing the axioms underlying any mathematical or logical argument often changes the equally valid conclusions.
This book is dedicated to the giants of mathematical and scientific philosophy upon whose backs it stands: Plato, Hume, Descartes, Gödel, 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'' philosophy 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 Mechanics 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.
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''.