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Author: Martin Goldstern
Publisher: CRC Press
ISBN: 1439863539
Category : Mathematics
Languages : en
Pages : 262
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Book Description
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.
Author: Martin Goldstern
Publisher: CRC Press
ISBN: 1439863539
Category : Mathematics
Languages : en
Pages : 262
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Book Description
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.
Author: Rebecca Goldstein
Publisher: W. W. Norton & Company
ISBN: 0393327604
Category : Biography & Autobiography
Languages : en
Pages : 299
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Book Description
"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
Author: Jan von Plato
Publisher: Springer Nature
ISBN: 3030508765
Category : Mathematics
Languages : en
Pages : 271
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Book Description
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.
Author: Gregory Chaitin
Publisher: CRC Press
ISBN: 1136587640
Category : Mathematics
Languages : en
Pages : 162
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Book Description
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.
Author: Andrei Ionuț Mărăşoiu
Publisher: Taylor & Francis
ISBN: 1040125220
Category : Philosophy
Languages : en
Pages : 266
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Book Description
This volume explores how understanding relates to conscious experience. In doing so, it builds bridges between different philosophical disciplines and provides a metaphysically robust characterization of understanding, both in and beyond science. The past two decades have witnessed growing interest from epistemologists, philosophers of science, philosophers of mind and ethicists in the nature and value of intellectual understanding. This volume features original essays on understanding and the phenomenal experiences that underlie it. The chapters are divided into three thematic sections. Part 1 provides theoretical characterizations of understanding, including Henk de Regt’s defense of a contextual theory of scientific understanding and a debate on whether scientific inference and explanatory power are necessary or central features of understanding. Part 2 explores how conscious experience and understanding are related. The chapters articulate a phenomenal theory of understanding and address themes that are connected to understanding, including awareness, transformative experiences and exemplification. Finally, Part 3 is devoted to domain-specific inquiries about understanding, such as logical proofs, particle physics and moral understanding. Understanding and Conscious Experience will be of interest to scholars and advanced students working in the philosophy of science, epistemology, philosophy of mind, ethics and phenomenology.
Author: Joel David Hamkins
Publisher: MIT Press
ISBN: 0262542234
Category : Mathematics
Languages : en
Pages : 350
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Book Description
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.
Author: Anna-Teresa Tymieniecka
Publisher: Springer Science & Business Media
ISBN: 9780792363194
Category : Education
Languages : en
Pages : 516
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Book Description
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.
Author: Leon Horsten
Publisher: MIT Press
ISBN: 0262297760
Category : Philosophy
Languages : en
Pages : 178
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Book Description
A philosopher proposes a new deflationist view of truth, based on contemporary proof-theoretic approaches. In The Tarskian Turn, Leon Horsten investigates the relationship between formal theories of truth and contemporary philosophical approaches to truth. The work of mathematician and logician Alfred Tarski (1901–1983) marks the transition from substantial to deflationary views about truth. Deflationism—which holds that the notion of truth is light and insubstantial—can be and has been made more precise in multiple ways. Crucial in making the deflationary intuition precise is its relation to formal or logical aspects of the notion of truth. Allowing that semantical theories of truth may have heuristic value, in The Tarskian Turn Horsten focuses on axiomatic theories of truth developed since Tarski and their connection to deflationism. Arguing that the insubstantiality of truth has been misunderstood in the literature, Horsten proposes and defends a new kind of deflationism, inferential deflationism, according to which truth is a concept without a nature or essence. He argues that this way of viewing the concept of truth, inspired by a formalization of Kripke's theory of truth, flows naturally from the best formal theories of truth that are currently available. Alternating between logical and philosophical chapters, the book steadily progresses toward stronger theories of truth. Technicality cannot be altogether avoided in the subject under discussion, but Horsten attempts to strike a balance between the need for logical precision on the one hand and the need to make his argument accessible to philosophers.
Author: Jim Holt
Publisher: Farrar, Straus and Giroux
ISBN: 0374717842
Category : Science
Languages : en
Pages : 385
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Book Description
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.
Author: Raymond M. Smullyan
Publisher: Knopf
ISBN: 0307962466
Category : Mathematics
Languages : en
Pages : 286
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Book Description
Forever Undecided is the most challenging yet of Raymond Smullyan’s puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel’s famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives’ identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel’s two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided!
