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.

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.

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.

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 “laws of thought” followed by the human mind. Different from “mathematical” 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 “logical conclusions” 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, etc.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.

Seventeen-year-old Amrey X has one fatal flaw. She can lie in a city forced to speak the truth. For 200 years, citizens of Axiom have been subjected to a truth serum. Forced to stand oath from the age of seven. And for 10 years Amrey has conformed. Until now. Things are changing...

Before we were all connected, before we were The Hive, there was individualism, privacy, ‘personal space’ . . . so quaint, so unnecessary . . . so dangerous . . . TOMMY PIERRE ANTIKAGAMAC, a star quarterback, is the most followed player in the world’s most popular sport: American football. While off-season training in the unpopulated European Fallowlands, he abruptly finds himself detached from the Hive. Agonizingly alone in his head for the first time in his life, he panics, becomes hopelessly lost, and then is captured by a fringe group of anti-Hive saboteurs. The Freemen, as they call themselves, have concocted an audacious plan to "cataclysmically disrupt the brain of the Hive," and Tommy may just be the key they need to make it successful. But Tommy’s arrival among the Freemen is not as serendipitous as it may appear. Neither he nor his captors suspect that it is not the terrorists, but Tommy, who is the threat to the Hive. And the Hive has ways of protecting itself.

Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

The best leaders not only lead well but also reflect on their leadership long enough and thoughtfully enough to articulate the philosophies that cause them to do so. Whether serving in the marketplace or in ministry, as executives or rank-and-file employees, as salaried staff or volunteer servants, good leaders can pinpoint the rationale for their actions and decisions with the ease of reciting their home address. In Axiom: The Language of Leadership, author Bill Hybels divulges the God-given convictions that have dictated his leadership strategy for more than three decades as senior pastor of Willow Creek Community Church. Oriented toward four key leadership categories ... 1. Vision and strategy (“Promote Shameless Profitability,” “Take a Flyer”) 2. Teamwork and communication (“Obi-Wan Kenobi Isn’t for Hire,” “Disagree without Drawing Blood”) 3. Activity and assessment (“Develop a Mole System,” “Sweat the Small Stuff”) 4. Personal integrity (“Admit Mistakes, and Your Stock Goes Up,” “Fight for Your Family”) ...Axiom brokers accessible wisdom from one leader’s journey, as well as emboldens you to nail down the reasons why you lead like you lead.