For decades, proponents of artificial intelligence have argued that computers will soon be doing everything that a human mind can do. Admittedly, computers now play chess at the grandmaster level, but do they understand the game as we do? Can a computer eventually do everything a human mind can do? In this absorbing and frequently contentious book, Roger Penrose--eminent physicist and winner, with Stephen Hawking, of the prestigious Wolf prize--puts forward his view that there are some facets of human thinking that can never be emulated by a machine. Penrose examines what physics and mathematics can tell us about how the mind works, what they can't, and what we need to know to understand the physical processes of consciousness. He is among a growing number of physicists who think Einstein wasn't being stubborn when he said his "little finger" told him that quantum mechanics is incomplete, and he concludes that laws even deeper than quantum mechanics are essential for the operation of a mind. To support this contention, Penrose takes the reader on a dazzling tour that covers such topics as complex numbers, Turing machines, complexity theory, quantum mechanics, formal systems, Godel undecidability, phase spaces, Hilbert spaces, black holes, white holes, Hawking radiation, entropy, quasicrystals, the structure of the brain, and scores of other subjects. The Emperor's New Mind will appeal to anyone with a serious interest in modern physics and its relation to philosophical issues, as well as to physicists, mathematicians, philosophers and those on either side of the AI debate.

Following his popular books "The Armchair Economist" and "More Sex Is Safer Sex", Slate columnist Landsburg uses concepts from mathematics, economics, and physics to address the big questions in philosophy.

Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these things possible and in doing so finds a correspondence between real numbers and the real world is the subject of this dazzling book by a writer of extraordinary clarity and stylistic brio. Even as he initiates us into the mysteries of real numbers, functions, and limits, Berlinski explores the furthest implications of his subject, revealing how the calculus reconciles the precision of numbers with the fluidity of the changing universe. "An odd and tantalizing book by a writer who takes immense pleasure in this great mathematical tool, and tries to create it in others."--New York Times Book Review

'In many ways it s the ultimate question in natural science: How does our universe work? Is there a fundamental theory? An incredible amount has been figured out about physics over the past few hundred years. But even with everything that s been done and it's very impressive - we still, after all this time, don t have a truly fundamental theory of physics.' The Wolfram Physics Project is a bold effort to find the fundamental theory of physics. It combines new ideas with the latest research in physics, mathematics and computation in the push to achieve this ultimate goal of science. Written with Stephen Wolfram's characteristic expository flair, this book provides a unique opportunity to learn about a historic initiative in science right as it is happening. A Project to Find the Fundamental Theory of Physics includes: The announcement of the project; A technical introduction; A history section that outlines the backstory which led Wolfram to this point; Background information from computational essays published by the author in 2015 and 2017; Foundational information, including historical notes and an excerpt from Wolfram's; 2002 New York Times Best Seller A New Kind of Science; Hundreds of beautiful full-colour visualisations. Stephen Wolfram is a distinguished scientist and best-selling author, and the creator of some of the world's most respected software systems, including Mathematica, Wolfram|Alpha and Wolfram Language. For more than 30 years he has been the CEO of the global technology company Wolfram Research.

This fascinating exploration of the great discoveries of history's most important mathematicians seeks an answer to the eternal question: Does mathematics hold the key to understanding the mysteries of the physical world?

Doing Physics makes concepts of physics easier to grasp by relating them to everyday knowledge. Addressing some of the models and metaphors that physicists use to explain the physical world, Martin H. Krieger describes the conceptual world of physics by means of analogies to economics, anthropology, theater, carpentry, mechanisms such as clockworks, and machine tool design. The interaction of elementary particles or chemical species, for example, can be related to the theory of kinship—who can marry whom is like what can interact with what. Likewise, the description of physical situations in terms of interdependent particles and fields is analogous to the design of a factory with its division of labor among specialists. For the new edition, Krieger has revised the text and added a chapter on the role of mathematics and formal models in physics. Doing Physics will be of special interest to economists, political theorists, anthropologists, and sociologists as well as philosophers of science.

René Descartes (1596–1650) is one of the towering and central figures in Western philosophy and mathematics. His apothegm “ Cogito, ergo sum ” marked the birth of the mind-body problem, while his creation of so-called Cartesian coordinates have made our physical and intellectual conquest of physical space possible. But Descartes had a mysterious and mystical side, as well. Almost certainly a member of the occult brotherhood of the Rosicrucians, he kept a secret notebook, now lost, most of which was written in code. After Descartes’s death, Gottfried Leibniz, inventor of calculus and one of the greatest mathematicians in history, moved to Paris in search of this notebook—and eventually found it in the possession of Claude Clerselier, a friend of Descartes. Leibniz called on Clerselier and was allowed to copy only a couple of pages—which, though written in code, he amazingly deciphered there on the spot. Leibniz’s hastily scribbled notes are all we have today of Descartes’s notebook, which has disappeared. Why did Descartes keep a secret notebook, and what were its contents? The answers to these questions lead Amir Aczel and the reader on an exciting, swashbuckling journey, and offer a fascinating look at one of the great figures of Western culture.

In this remarkable work of fiction, astrophysicist Janna Levin reimagines the lives of two of the most important and influential minds of our time. The narrator is a scientist herself, a physicist obsessed with Kurt Gödel, the greatest logician of many centuries, and with Alan Turing, the extraordinary mathematician, breaker of the Enigma Code during World War II. “They are both brilliantly original and outsiders,” the narrator tells us. “They are both besotted with mathematics. But for all their devotion, mathematics is indifferent, unaltered by any of their dramas . . . Against indifference, I want to tell their stories.” Which she does in a haunting, incantatory voice, the two lives unfolding in parallel narratives that overlap in the magnitude of each man’s achievement and demise: Gödel, delusional and paranoid, would starve himself to death; Turing, arrested for homosexual activities, would be driven to suicide. And they meet as well in the narrator’s mind, where facts are interwoven with her desire and determination to find meaning in the maze of their stories: two men devoted to truth of the highest abstract nature, yet unable to grasp the mundane truths of their own lives. A unique amalgam of luminous imagination and richly evoked historic character and event—A Madman Dreams of Turing Machines is a story about the pursuit of truth and its effect on the lives of two men. A story of genius and madness, incredible yet true.

Explains the history and tenets of mathematics including the relationship of mathematics to the other sciences.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

Carl Gustav Jung (1875-1961) and Wolfgang Ernst Pauli (1900 - 1958) were two of the greatest thinkers of modern times. Jung is the founder of Analytical Psychology and is one of the best known researchers in the field of Dream Interpretation and Symbolization. Pauli was one of the pioneers of Quantum Physics. In 1945, Pauli received the Nobel Prize in Physics after being nominated by Albert Einstein for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle" involving spin theory underpinning the structure of matter and the whole of chemistry. Together they collaborated on this book that was First published in 1952 in German as Naturerklarung und Psyche. C. G. Jung: Synchronizitat als ein Prinzip akausaler Zusammenhange. W. Pauli: Der Einfluss archetypischer Vorstellungen auf die Bildung naturwissenschaftlicher Theorien bei Kepler. However, our reviewer, himself an eminent professor of mathematics, concludes, "After thoroughly studying their writings for many months now, I have come to see clearly that they are both utterly mad.""

When you think about it, it seems obvious: The only mathematical ideas that human beings can have are ideas that the human brain allows. We know a lot about what human ideas are like from research in Cognitive Science. Most ideas are unconscious, and that is no less true of the mathematical ones. Abstract ideas, for the most part, arise via conceptual metaphor-a mechanism for projecting embodied (that is, sensory-motor) reasoning to abstract reasoning. This book argues that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Even the real numbers, the imaginary numbers, trigonometry, and calculus are based on metaphorical ideas coming out of the way we function in the everyday physical world. This book is about mathematical ideas, about what mathematics means-and why. The authors believe that understanding the metaphors implicit in mathematics will make mathematics make more sense. Moreover, understanding mathematical ideas and how they arise from our bodies and brains will make it clear that the brain's mathematics is mathematics, the only mathematics we know or can know. -- Publisher description.

CHANCE is an accessible study of Probability, Randomness, Coincidences, Uncertainty and Games. Mathematician Amir D Aczel leads you through simple explanations of each of these forces in life, and explains how you can use your new-found knowledge of these mathematical paradoxes to succeed (or at least have the best CHANCE of succeeding) in gambling, poker, love, who will win the next election and how to choose a housemate or puppy. Each of the chapters is short, self-contained and simple so readers can drop in and out, look up a problem or a theory and then go ahead and use it. For all the maths nerds, gamblers and curious people in your life.

This book represents a new approach to philosophy. It treats philosophy as not a collection of systems, but as a study of problems. It recognizes in traditional philosophical systems the historical function of having asked questions rather than having given solutions. Professor Reichenbach traces the failures of the systems to psychological causes. Speculative philosophers offered answers at a time when science had not yet provided the means to give true answers. Their search for certainty and for moral directives led them to accept pseudo-solutions. Plato, Descartes, Spinoza, Kant, and many others are cited to illustrate the rationalist fallacy: reason, unaided by observation, was regarded as a source of knowledge, revealing the physical world and "moral truth." The empiricists could not disprove this thesis, for they could not give a valid account of mathematical knowledge. Mathematical discoveries in the early nineteenth century cleared the way for modern scientific philosophy. Its advance was furthered by discoveries in modern physics, chemistry, biology, and psychology. These findings have made possible a new conception of the universe and of the atom. The work of scientists thus altered philosophy completely and brought into being a philosopher with a new attitude and training. Instead of dictating so-called laws of reason to the scientist, this modern philosopher proceeds by analyzing scientific methods and results. He finds answers to the age-old questions of space, time, causality, and life; of the human observer and the external world. He tells us how to find our way through this world without resorting to unjustifiable beliefs or assuming a supernatural origin for moral standards. Philosophy thus is no longer a battleground of contradictory opinions, but a science discovering truth step by step. Professor Reichenbach, known for his many contributions to logic and the philosophy of science, addresses this book to a wider audience. He writes for those who do not have the leisure or preparation to read in the fields of mathematics, symbolic logic, or physics. Besides showing the principal foundations of the new philosophy, he has been careful to provide the necessary factual background. He has written a philosophical study, not a mere popularization. It contains within its chapters all the necessary scientific material in an understandable form—and, therefore, conveys all the information indispensable to a modern world-view. The late Hans Reichenbach was Professor of Philosophy at the University of California, Los Angeles. His previous books include

"Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation." "Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians."--BOOK JACKET

"Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation." "Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians."--BOOK JACKET

"Circularity" is the story of a Janus-face conceptual structure, that on the one hand led to deep scientific discoveries, and on the other hand is used to trick the mind into believing the impossible. Alongside mathematical revolutions that eventually led to the invention of the computer, the book describes ancient paradoxes that arise from circular thinking. Another aspect of Circularity, its ability to entertain, leads to a surprising insight on the time old question " What is humor."

First published originally in German in 1921, "Tractatus Logico-Philosophicus" is the only book-length philosophical work by Austrian philosopher Ludwig Wittgenstein published during his lifetime. Based upon notes made during his time serving in World War I, the treatise is composed of 526 numbered declarative statements that are presented without argument and were seen by Wittgenstein as being self-evident. Wittgenstein's style is succinct and sparse and his ideas are presented in a highly logical, organized, and scientific manner. Wittgenstein proposes that the solution to all major philosophical problems can be solved only with logic and a scientific analysis of language and its meaning. His idea that philosophy is not about metaphysics and mysticism, but about critical and rational thought led to a massive shift in philosophical theories after its publication. The treatise was influential with many of the "logical positivists" of the Vienna Circle, such as Rudolf Carnap and Friedrich Waismann, and to philosophers such as Bertrand Russell. Considered one of the most significant and influential philosophical works of the twentieth century, this treatise is an important read for any student of the history of language and ideas. This edition is printed on premium acid-free paper.