Download or read book The Incompleteness Phenomenon written by Martin Goldstern and published by CRC Press. This book was released on 2018-10-08 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to mathematical logic takes Gödel's incompleteness theorem as a starting point. It goes beyond a standard text book and should interest everyone from mathematicians to philosophers and general readers who wish to understand the foundations and limitations of modern mathematics.
Download or read book Incompleteness written by Rebecca Goldstein and published by W. W. Norton & Company. This book was released on 2006-01-31 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: "An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
Download or read book Lectures on the Philosophy of Mathematics written by Joel David Hamkins and published by MIT Press. This book was released on 2021-03-09 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
Download or read book Can Mathematics Be Proved Consistent written by Jan von Plato and published by Springer. This book was released on 2021-07-26 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kurt Gödel (1906–1978) shook the mathematical world in 1931 by a result that has become an icon of 20th century science: The search for rigour in proving mathematical theorems had led to the formalization of mathematical proofs, to the extent that such proving could be reduced to the application of a few mechanical rules. Gödel showed that whenever the part of mathematics under formalization contains elementary arithmetic, there will be arithmetical statements that should be formally provable but aren’t. The result is known as Gödel’s first incompleteness theorem, so called because there is a second incompleteness result, embodied in his answer to the question "Can mathematics be proved consistent?" This book offers the first examination of Gödel’s preserved notebooks from 1930, written in a long-forgotten German shorthand, that show his way to the results: his first ideas, how they evolved, and how the jewel-like final presentation in his famous publication On formally undecidable propositions was composed.The book also contains the original version of Gödel’s incompleteness article, as handed in for publication with no mentioning of the second incompleteness theorem, as well as six contemporary lectures and seminars Gödel gave between 1931 and 1934 in Austria, Germany, and the United States. The lectures are masterpieces of accessible presentations of deep scientific results, readable even for those without special mathematical training, and published here for the first time.
Download or read book Goedel s Way written by Gregory Chaitin and published by CRC Press. This book was released on 2011-10-14 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein’s general relativity, as he proved that Einstein’s theory allows for time machines. The Gödel incompleteness theorem - the usual formal mathematical systems cannot prove nor disprove all true mathematical sentences - is frequently presented in textbooks as something that happens in the rarefied realms of mathematical logic, and that has nothing to do with the real world. Practice shows the contrary though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics and even ecology. In this lively treatise, based on Chaitin’s groundbreaking work and on the da Costa-Doria results in physics, ecology, economics and computer science, the authors show that the Gödel incompleteness phenomenon can directly bear on the practice of science and perhaps on our everyday life. This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities. Besides theory, the historical report and personal stories about the main character and on this book’s writing process, make it appealing leisure reading for those interested in mathematics, logic, physics, philosophy and computer sciences. See also: http://www.youtube.com/watch?v=REy9noY5Sg8
Download or read book Paideia written by Anna-Teresa Tymieniecka and published by Springer Science & Business Media. This book was released on 2000-08-31 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt: The education of humanity is the key to the next century's culture, its social and practical life. The main concerns of education are perennial, but the continuous flood of inventions, the technological innovations that re-shape life, calls for a radically new appraisal of the situation, such as only philosophy can provide. Answering the call of humanity for the measure, sense of proportion and direction that could re-orient present and future education, the phenomenology of life - integral and scientific, in a dialogue with the arts, the sciences, and the humanities - proposes an ontopoietic model of life's unfolding as the universal paradigm for this re-orientation. Taking the Human Creative Condition as its Archimedean point, it offers a unique context for a fresh investigation of the concerns of education, both perennial and immediate.
Download or read book When Einstein Walked with G del written by Jim Holt and published by Farrar, Straus and Giroux. This book was released on 2018-05-15 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Jim Holt, the New York Times bestselling author of Why Does the World Exist?, comes an entertaining and accessible guide to the most profound scientific and mathematical ideas of recent centuries in When Einstein Walked with Gödel: Excursions to the Edge of Thought. Does time exist? What is infinity? Why do mirrors reverse left and right but not up and down? In this scintillating collection, Holt explores the human mind, the cosmos, and the thinkers who’ve tried to encompass the latter with the former. With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth. Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot. Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U.S. Constitution contained a terrible contradiction—and whether the universe truly has a future.
Download or read book Incomplete Nature How Mind Emerged from Matter written by Terrence W. Deacon and published by W. W. Norton & Company. This book was released on 2012 with total page 625 pages. Available in PDF, EPUB and Kindle. Book excerpt: Examines the emergent processes that bridge the gap between organisms that think and have consciousness and those that do not and discusses the origins of life, information, and free will.
Download or read book Intelligent Design and Religion as a Natural Phenomenon written by John S. Wilkins and published by Routledge. This book was released on 2017-03-02 with total page 559 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the past decade a strident public debate has arisen about the nature and origin of religions. Controversies include how exactly religion evolved, whether by individual or group selection, if it is adaptive, and if not, whether and how it is a side effect of evolution. This volume focuses on the issue of naturalizing religion: on the ways in which cognitive science and social sciences have treated religion as a natural phenomenon. It questions whether religious behaviour, institutions, and experiences can be explained in natural terms. The editor brings together some of the best published work on the definition of 'religion', intelligent design and the evolution of religion.
Download or read book Mathematical Undecidability Quantum Nonlocality and the Question of the Existence of God written by A. Driessen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: On January 22, 1990, the late John Bell held at CERN (European Laboratory for Particle Physics), Geneva a seminar organized by the Center of Quantum Philosophy, that at this time was an association of scientists interested in the interpretation of quantum mechanics. In this seminar Bell presented once again his famous theorem. Thereafter a discussion took place in which not only physical but also highly speculative epistemological and philosophical questions were vividly debated. The list of topics included: assumption of free will in Bell's theorem, the understanding of mind, the relationship between the mathematical and the physical world, the existence of unobservable causes and the limits of human knowledge in mathematics and physics. Encouraged by this stimulating discussion some of the participants decided to found an Institute for Interdisciplinary Studies (lIS) to promote philosoph ical and interdisciplinary reflection on the advances of science. Meanwhile the lIS has associated its activities with the Swiss foundation, Fondation du Leman, and the Dutch foundation, Stichting Instudo, registered in Geneva and Amsterdam, respectively. With its activities the lIS intends to strengthen the unity between the professional activities in science and the reflection on fun damental philosophical questions. In addition the interdisciplinary approach is expected to give a contribution to the progress of science and the socio economic development. At present three working groups are active within the lIS, i. e. : - the Center for Quantum Philosophy, - the Wealth Creation and Sustainable Development Group, - the Neural Science Group.
Download or read book Handbook of Practical Logic and Automated Reasoning written by John Harrison and published by Cambridge University Press. This book was released on 2009-03-12 with total page 683 pages. Available in PDF, EPUB and Kindle. Book excerpt: The sheer complexity of computer systems has meant that automated reasoning, i.e. the ability of computers to perform logical inference, has become a vital component of program construction and of programming language design. This book meets the demand for a self-contained and broad-based account of the concepts, the machinery and the use of automated reasoning. The mathematical logic foundations are described in conjunction with practical application, all with the minimum of prerequisites. The approach is constructive, concrete and algorithmic: a key feature is that methods are described with reference to actual implementations (for which code is supplied) that readers can use, modify and experiment with. This book is ideally suited for those seeking a one-stop source for the general area of automated reasoning. It can be used as a reference, or as a place to learn the fundamentals, either in conjunction with advanced courses or for self study.
Download or read book The Philosopher s Handbook written by Stanley Rosen and published by Random House Reference. This book was released on 2009-02-19 with total page 656 pages. Available in PDF, EPUB and Kindle. Book excerpt: An ideal introduction for the casual reader and a beneficial reference for the student, The Philosopher's Handbook features the writings of some of the world's most influential philosophers. Based on the premise that all human beings are curious about their existence, Rosen's collection brings together primary excerpts from the works of prominent thinkers such as Plato, Nietzsche, Descartes, Machiavelli, and Kant. Experts in each field have carefully selected the sources and provided brief introductions to help readers gain insight into the readings. Newly revised in order to emphasize its broad appeal, The Philosopher's Handbook is a solid introduction to Western philosophy for all inquiring minds.
Download or read book G del s Theorem written by Torkel Franzén and published by CRC Press. This book was released on 2005-06-06 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature." --- John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel
Download or read book G del s Incompleteness Theorems written by Juliette Kennedy and published by Cambridge University Press. This book was released on 2022-04-14 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
Download or read book Proven Impossible written by Dan Gusfield and published by Cambridge University Press. This book was released on 2023-12-31 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: A highly readable presentation of elementary yet rigorous proofs of profound impossibility theorems for a broad, lay audience.
Download or read book Semantics and Truth written by Jan Woleński and published by Springer Nature. This book was released on 2020-01-01 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a historical (with an outline of the history of the concept of truth from antiquity to our time) and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas (pro and contra) as well as points out various applications of the semantic theory of truth to philosophical problems (truth-criteria, realism and anti-realism, future contingents or the concept of correspondence between language and reality).
Download or read book On Hilbert s Sixth Problem written by Newton C. A. da Costa and published by Springer Nature. This book was released on 2022-01-25 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the premise that a physical theory is an interpretation of the analytico–canonical formalism. Throughout the text, the investigation stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics. The authors start from a presentation of the analytico–canonical formalism for classical mechanics, and its applications in electromagnetism, Schrödinger's quantum mechanics, and field theories such as general relativity and gauge field theories, up to the Higgs mechanism. The analysis uses the main criterion used by physicists for a theory: to formulate a physical theory we write down a Lagrangian for it. A physical theory is a particular instance of the Lagrangian functional. So, there is already an unified physical theory. One only has to specify the corresponding Lagrangian (or Lagrangian density); the dynamical equations are the associated Euler–Lagrange equations. The theory of Suppes predicates as the main tool in the axiomatization and examples from the usual theories in physics. For applications, a whole plethora of results from logic that lead to interesting, and sometimes unexpected, consequences. This volume looks at where our physics happen and which mathematical universe we require for the description of our concrete physical events. It also explores if we use the constructive universe or if we need set–theoretically generic spacetimes.
