Methods developed in a previous paper are employed to define an exact correspondence between the states of a deterministic cellular automaton in 1+1 dimensions and those of a bosonic quantum field theory. The result may be used to argue that quantum field theories may be much closer related to deterministic automata than what is usually thought possible.

Zuse proposed that the universe is being computed by some sort of cellular automaton or other discrete computing machinery, challenging the long-held view that some physical laws are continuous by nature. Calculating Space is the title of MIT's English translation of Konrad Zuse's 1969 Rechnender Raum, the first work on digital physics. This is the LaTeX edition by A. German and H. Zenil based on the MIT's English translation with permission from the MIT and Konrad Zuse's son Horst Zuse. Followed (...) by an afterword by A. German and H. Zenil. (shrink)

The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in ...

Will we ever discover a single scientific theory that explains everything that has ever happened and everything that will happen - a key that unlocks the ...

In books such as The World Within the World and The Anthropic Cosmological Principle, astronomer John Barrow has emerged as a leading writer on our efforts to understand the universe. Timothy Ferris, writing in The Times Literary Supplement of London, described him as "a temperate and accomplished humanist, scientist, and philosopher of science--a man out to make a contribution, not a show." Now Barrow offers the general reader another fascinating look at modern physics, as he explores the quest for a (...) single, unifying theory that will unlock nature's secrets. Theories of Everything is more than a history of science, more than a popular report on recent research and discoveries. Barrow provides a reflective, intelligent commentary on what a true Theory of Everything would be--its ingredients, its limitations, and what it could tell us about the universe. Never before, he writes, have physicists been so confident and so eager in the hunt for this "cosmic Rosetta Stone," as he calls it: "a single all-embracing picture of all the laws of nature from which the inevitability of all things seen must follow with unimpeachable logic." He lays out eight essential ingredients for a Theory of Everything and then explores each in turn, tracing how our knowledge has developed and how scientific discovery relates to our changing philosophy and religious thought in each area. Some of these ingredients are obvious--the laws of nature must be explained, for example, as well as its organizing principles--but others may be surprising, such as broken symmetries and selection biases. A Theory of Everything must account for the fact that the universe is "messy and complicated," he tells us, and for the limitations imposed by the questions we ask and the information we can obtain. The key lies in the remarkable capacity of mathematics to express the fundamental workings of the physical world--a language that the human mind is uniquely equipped to understand and manipulate. Barrow examines what mathematics actually is and describes how it makes the universe intelligible and provides a path to the underlying coherence in nature--which has led, in fact, to arguments that the universe itself is a vast computer. Yet even the most complete theory, even the most comprehensive mathematical explanation, cannot account for the uncomputable varieties of human experience and thought. "No non-poetic account of reality," he writes, "can be complete." In a field where the authorities converse in equations and mathematical notations, John Barrow speaks with the voice of thoughtful and knowledgeable humanist. Written with eloquence and expertise, Theories of Everything establishes a new perspective on humanity's efforts to explain the universe. (shrink)

The Concept of Physical Law is an original and creative defense of the Regularity theory of physical law, the concept that physical laws are nothing more than descriptions of whatever universal truths happen to be instanced in nature. Professor Swartz clearly identifies and analyzes the arguments and intuitions of the opposing Necessitarian theory, and argues that the standard objection to the Regularity theory turns on a mistaken view of what Regularists mean by 'physical impossibility'; that it is impossible to construct (...) an empirical test that can distinguish between events Necessitarians call 'mere accidents' and those they call 'nornologically necessary', and that the Necessitarian theory cannot account fot human beings' free wills. Other topics in this important work include: the distinction between instrumental scientific laws and true physical laws; the distinction between failure and doom; potentialities; miracles and marvels; predictability and uniformity; statistical and numerical laws; and necessity-in-praxis. (shrink)

Ernest Nagel once remarked that it is fortunately not necessary to be clear about scientific philosophy and methodology in order to practice good science. He went on to say, "Even eminent scientists can make unholy spectacles of themselves when they don the mantle of philosophy and attempt to discuss the broad implications of their specialized labors." Feynman's recent venture into the philosophy of science is, unfortunately, a lucid illustration of the validity of Nagel's observations. The book is a rather literal (...) transcription of seven semi-popular lectures delivered by Feynman at Cornell in November of 1964. Feynman is a wonderful iconoclast, a legendary virtuoso of the bongo drums and—by the way—a Nobel laureate and one of the great theoretical physicists. But this is all the more reason to expect something a little more intellectually solid than is offered in this volume. It was obviously Feynman's intent to discuss some compelling problems which lie on the borderline of physics and philosophy with an intelligent lay audience. But few will be enlightened by Feynman's clever, if uninformative, discourse about such issues as the relation of mathematics to physics, symmetry and conservation laws, and probability and uncertainty. Feynman's finest hour undoubtedly comes in the lecture on quantum mechanics, in which some aspects of the wave-particle duality are presented. But the value of much of the rest of the book is severely qualified by the author's penchant for imprecise and confusing statements and, in some cases, outright mystery-mongering. This is a great pity because theoretical physics, especially in the hands of Richard Feynman, is a source of ideas which are most worth discussing.—H. P. K. (shrink)

NOW IN PAPERBACK"€"Starting from a collection of simple computer experiments"€"illustrated in the book by striking computer graphics"€"Stephen Wolfram shows how their unexpected results force a whole new way of looking at the operation of our universe.

A Computable Universe is a collection of papers discussing computation in nature and the nature of computation, a compilation of the views of the pioneers in the contemporary area of intellectual inquiry focused on computational and informational theories of the world. This volume is the definitive source of informational/computational views of the world, and of cutting-edge models of the universe, both digital and quantum, discussed from a philosophical perspective as well as in the greatest technical detail. The book discusses the (...) foundations of computation in relation to nature. It focuses on two main questions: What is computation? How does nature compute? The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and their philosophical implications. The volume provides a state-of-the-art collection of technical papers and non-technical essays representing a field that takes information and computation to be key to understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse's "Calculating Space", and a panel discussion transcription on the topic, featuring worldwide experts (including a Nobel prize) in quantum mechanics, physics, cognition, computation and algorithmic complexity.A Computable Universe is a collection of papers discussing computation in nature and the nature of computation, a compilation of the views of the pioneers in the contemporary area of intellectual inquiry focused on computational and informational theories of the world. This volume is the definitive source of informational/computational views of the world, and of cutting-edge models of the universe, both digital and quantum, discussed from a philosophical perspective as well as in the greatest technical detail. The book discusses the foundations of computation in relation to nature. It focuses on two main questions: What is computation? How does nature compute? The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and their philosophical implications. The volume provides a state-of-the-art collection of technical papers and non-technical essays representing a field that takes information and computation to be key to understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse's "Calculating Space", and a panel discussion transcription on the topic, featuring worldwide experts (including a Nobel prize) in quantum mechanics, physics, cognition, computation and algorithmic complexity. (shrink)

An introduction to the developments of nonstandard models. Beginning with Godel's incompleteness theorem, it covers the prime models, cofinal extensions, and extensions, Gaifman's construction of a definable type, Tennenbaum's theorem and Friedman's theorem on indicators, ending with a chapter on recursive saturation and resplendency.

Introduction Major scientific revolutions are rarely, if ever, started deliberately. They can be "in the air" for a long time before the first recognizable ...

Standard canonical quantum mechanics makes much use of operators whose spectra cover the set of real numbers, such as the coordinates of space, or the values of the momenta. Discrete quantum mechanics uses only strictly discrete operators. We show how one can transform systems with pairs of integer-valued, commuting operators $P_i$ and $Q_i$ , to systems with real-valued canonical coordinates $q_i$ and their associated momentum operators $p_i$ . The discrete system could be entirely deterministic while the corresponding (p, q) system (...) could still be typically quantum mechanical. (shrink)