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Axiomatic Thinking I

Author: Fernando Ferreira

Publisher: Springer Nature

Published: 2022-10-13

Total Pages: 209

ISBN-13: 3030776573

In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations. Chapter 8 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Axiomatic Method and Category Theory

Author: Andrei Rodin

Publisher: Springer Science & Business Media

Published: 2013-10-14

Total Pages: 285

ISBN-13: 3319004042

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This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

Axiomatic Thinking II

Author: Fernando Ferreira

Publisher: Springer Nature

Published: 2022-09-17

Total Pages: 293

ISBN-13: 3030777995

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Axiomatic Thinking

Author: Fernando Ferreira

Publisher:

Published: 2022

Total Pages: 0

ISBN-13: 9788303077790

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In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Gottingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.

Non-axiomatic Logic

Author: Pei Wang

Publisher: World Scientific

Published: 2013

Total Pages: 275

ISBN-13: 9814440280

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This book provides a systematic and comprehensive description of Non-Axiomatic Logic, which is the result of the author''s research for about three decades.Non-Axiomatic Logic is designed to provide a uniform logical foundation for Artificial Intelligence, as well as an abstract description of the OC laws of thoughtOCO followed by the human mind. Different from OC mathematicalOCO logic, where the focus is the regularity required when demonstrating mathematical conclusions, Non-Axiomatic Logic is an attempt to return to the original aim of logic, that is, to formulate the regularity in actual human thinking. To achieve this goal, the logic is designed under the assumption that the system has insufficient knowledge and resources with respect to the problems to be solved, so that the OC logical conclusionsOCO are only valid with respect to the available knowledge and resources. Reasoning processes according to this logic covers cognitive functions like learning, planning, decision making, problem solving, This book is written for researchers and students in Artificial Intelligence and Cognitive Science, and can be used as a textbook for courses at graduate level, or upper-level undergraduate, on Non-Axiomatic Logic."

Axiomatic

Author: Maria Tumarkin

Publisher: Random House Australia

Published: 2021-05-04

Total Pages: 226

ISBN-13: 1761043587

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Stories are not enough, even though they are essential. And books about history, books of psychology--the best of them take us closer, but still not close enough. Maria Tumarkin's Axiomatic is a boundary-shifting fusion of thinking, storytelling, reportage and meditation. It takes as its starting point five axioms: 'Time Heals All Wounds'; 'History Repeats Itself'; 'Those Who Forget the Past are Condemned to Repeat It'; 'Give Me a Child Before the Age of Seven and I Will Show You the Woman'; and 'You Can't Enter The Same River Twice.' These beliefs--or intuitions--about the role the past plays in our present are often evoked as if they are timeless and self-evident truths. It is precisely because they are neither, yet still we are persuaded by them, that they tell us a great deal about the forces that shape our culture and the way we live.

Axiom's End

Author: Lindsay Ellis

Publisher: St. Martin's Press

Published: 2020-07-21

Total Pages: 397

ISBN-13: 1250256747

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THE INSTANT NEW YORK TIMES BESTSELLER The alternate history first contact adventure Axiom's End is an extraordinary debut from Hugo finalist and video essayist Lindsay Ellis. Truth is a human right. It’s fall 2007. A well-timed leak has revealed that the US government might have engaged in first contact. Cora Sabino is doing everything she can to avoid the whole mess, since the force driving the controversy is her whistleblower father. Even though Cora hasn’t spoken to him in years, his celebrity has caught the attention of the press, the Internet, the paparazzi, and the government—and with him in hiding, that attention is on her. She neither knows nor cares whether her father’s leaks are a hoax, and wants nothing to do with him—until she learns just how deeply entrenched her family is in the cover-up, and that an extraterrestrial presence has been on Earth for decades. Realizing the extent to which both she and the public have been lied to, she sets out to gather as much information as she can, and finds that the best way for her to uncover the truth is not as a whistleblower, but as an intermediary. The alien presence has been completely uncommunicative until she convinces one of them that she can act as their interpreter, becoming the first and only human vessel of communication. Their otherworldly connection will change everything she thought she knew about being human—and could unleash a force more sinister than she ever imagined.

Axiomatics

Author: Alma Steingart

Publisher: University of Chicago Press

Published: 2023-01-17

Total Pages: 300

ISBN-13: 0226824195

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The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart’s history ably demonstrates, mathematics is the social activity in which styles of truth—here, abstraction—become synonymous with ways of knowing.

The Legacy of Kurt Schütte

Author: Reinhard Kahle

Publisher: Springer Nature

Published: 2020-08-10

Total Pages: 502

ISBN-13: 3030494241

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This book on proof theory centers around the legacy of Kurt Schütte and its current impact on the subject. Schütte was the last doctoral student of David Hilbert who was the first to see that proofs can be viewed as structured mathematical objects amenable to investigation by mathematical methods (metamathematics). Schütte inaugurated the important paradigm shift from finite proofs to infinite proofs and developed the mathematical tools for their analysis. Infinitary proof theory flourished in his hands in the 1960s, culminating in the famous bound Γ0 for the limit of predicative mathematics (a fame shared with Feferman). Later his interests shifted to developing infinite proof calculi for impredicative theories. Schütte had a keen interest in advancing ordinal analysis to ever stronger theories and was still working on some of the strongest systems in his eighties. The articles in this volume from leading experts close to his research, show the enduring influence of his work in modern proof theory. They range from eye witness accounts of his scientific life to developments at the current research frontier, including papers by Schütte himself that have never been published before.

Axiomatic Theories of Truth

Author: Volker Halbach

Publisher: Cambridge University Press

Published: 2014-02-27

Total Pages:

ISBN-13: 1316584232

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At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties and shows how the logical results impinge on the philosophical topics related to truth. In particular, he shows that the discussion on topics such as deflationism about truth depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate or professional philosopher in theories of truth.