Axiomatic Thinking Ii

Author : Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga
Publisher : Springer Nature
Page : 293 pages
File Size : 49,8 Mb
Release : 2022-09-17
Category : Mathematics
ISBN : 9783030777999

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Axiomatic Thinking II by Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga Pdf

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.

Author : Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga
Publisher : Unknown
Page : 0 pages
File Size : 54,6 Mb
Release : 2022
Category : Axioms
ISBN : 8303077791

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Axiomatic Thinking by Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga Pdf

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.

Author : Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga
Publisher : Springer Nature
Page : 209 pages
File Size : 55,7 Mb
Release : 2022-10-13
Category : Mathematics
ISBN : 9783030776572

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Axiomatic Thinking I by Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga Pdf

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.

Author : Andrei Rodin
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 44,6 Mb
Release : 2013-10-14
Category : Philosophy
ISBN : 9783319004044

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Axiomatic Method and Category Theory by Andrei Rodin Pdf

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.

Author : Emily Carson,Renate Huber
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 46,8 Mb
Release : 2006-07-02
Category : Philosophy
ISBN : 9781402040405

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Intuition and the Axiomatic Method by Emily Carson,Renate Huber Pdf

Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant’s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant’s own approach. By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

Author : Alma Steingart
Publisher : University of Chicago Press
Page : 300 pages
File Size : 42,8 Mb
Release : 2023-01-17
Category : Mathematics
ISBN : 9780226824192

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Axiomatics by Alma Steingart Pdf

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.

Author : Jan von Plato
Publisher : Princeton University Press
Page : 392 pages
File Size : 48,5 Mb
Release : 2017-08-02
Category : Science
ISBN : 9780691174174

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The Great Formal Machinery Works by Jan von Plato Pdf

The information age owes its existence to a little-known but crucial development, the theoretical study of logic and the foundations of mathematics. The Great Formal Machinery Works draws on original sources and rare archival materials to trace the history of the theories of deduction and computation that laid the logical foundations for the digital revolution. Jan von Plato examines the contributions of figures such as Aristotle; the nineteenth-century German polymath Hermann Grassmann; George Boole, whose Boolean logic would prove essential to programming languages and computing; Ernst Schröder, best known for his work on algebraic logic; and Giuseppe Peano, cofounder of mathematical logic. Von Plato shows how the idea of a formal proof in mathematics emerged gradually in the second half of the nineteenth century, hand in hand with the notion of a formal process of computation. A turning point was reached by 1930, when Kurt Gödel conceived his celebrated incompleteness theorems. They were an enormous boost to the study of formal languages and computability, which were brought to perfection by the end of the 1930s with precise theories of formal languages and formal deduction and parallel theories of algorithmic computability. Von Plato describes how the first theoretical ideas of a computer soon emerged in the work of Alan Turing in 1936 and John von Neumann some years later. Shedding new light on this crucial chapter in the history of science, The Great Formal Machinery Works is essential reading for students and researchers in logic, mathematics, and computer science.

Author : Anonim
Publisher : Mary Kathryn Thompson
Page : 221 pages
File Size : 42,6 Mb
Release : 2024-06-01
Category : Electronic
ISBN : 9788989693307

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Proceedings of the 6th International Conference on Axiomatic Design by Anonim Pdf

Author : Reinhard Kahle,Michael Rathjen
Publisher : Springer
Page : 561 pages
File Size : 44,7 Mb
Release : 2015-11-02
Category : Mathematics
ISBN : 9783319101033

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Gentzen's Centenary by Reinhard Kahle,Michael Rathjen Pdf

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.

Author : Fabio Minazzi
Publisher : Springer Nature
Page : 415 pages
File Size : 42,8 Mb
Release : 2022-04-01
Category : Science
ISBN : 9783030963323

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Historical Epistemology and European Philosophy of Science by Fabio Minazzi Pdf

This book offers a comprehensive analysis on the evolution of philosophy of science, with a special emphasis on the European tradition of the twentieth century. At first, it shows how the epistemological problem of the objectivity of knowledge and axiomatic knowledge have been previously tackled by transcendentalism, critical rationalism and hermeneutics. In turn, it analyses the axiological dimension of scientific research, moving from traditional model of science and of scientific methods, to the construction of a new image of knowledge that leverages the philosophical tradition of the Milan School. Using this historical-epistemological approach, the author rethinks the Kantian Transcendental, showing how it could be better integrated in the current philosophy of science, to answer important questions such as the relationship between science and history, scientific and social perspectives and philosophy and technology, among others. Not only this book provides a comprehensive study of the evolution of European Philosophy of Science in the twentieth century, yet it offers a new, historical and epistemological-based approach, that could be used to answers many urgent questions of contemporary societies.

Author : Alain Badiou
Publisher : re.press
Page : 182 pages
File Size : 42,9 Mb
Release : 2007
Category : Critical theory
ISBN : 9780980666571

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The Concept of Model by Alain Badiou Pdf

In "The Concept of Model" Alain Badiou establishes a new logical 'concept of model'. Translated for the first time into English, the work is accompanied by an exclusive interview with Badiou in which he elaborates on the connections between his early and most recent work--for which the concept of model remains seminal.

Author : Tom Leinster
Publisher : Cambridge University Press
Page : 457 pages
File Size : 55,6 Mb
Release : 2021-04-22
Category : Language Arts & Disciplines
ISBN : 9781108832700

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Entropy and Diversity by Tom Leinster Pdf

Discover the mathematical riches of 'what is diversity?' in a book that adds mathematical rigour to a vital ecological debate.

Author : National Science Foundation (U.S.)
Publisher : Unknown
Page : 56 pages
File Size : 42,9 Mb
Release : 1970
Category : Science
ISBN : STANFORD:36105122679678

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Course and Curriculum Improvement Projects: Mathematics, Science, Social Sciences by National Science Foundation (U.S.) Pdf

Author : Maria Antonaccio Assistant Professor of Religion Bucknell University
Publisher : Oxford University Press, USA
Page : 258 pages
File Size : 47,8 Mb
Release : 2000-05-31
Category : Literary Criticism
ISBN : 9780198030195

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Picturing the Human : The Moral Thought of Iris Murdoch by Maria Antonaccio Assistant Professor of Religion Bucknell University Pdf

Iris Murdoch has long been known as one of the most deeply insightful and morally passionate novelists of our time. This attention has often eclipsed Murdoch's sophisticated and influential work as a philosopher, which has had a wide-ranging impact on thinkers in moral philosophy as well as religious ethics and political theory. Yet it has never been the subject of a book-length study in its own right. Picturing the Human seeks to fill this gap. In this groundbreaking book, author Maria Antonaccio presents the first systematic and comprehensive treatment of Murdoch's moral philosophy. Unlike literary critical studies of her novels, it offers a general philosophical framework for assessing Murdoch's thought as a whole. Antonaccio also suggests a new interpretive method for reading Murdoch's philosophy and outlines the significance of her thought in the context of current debates in ethics. This vital study will appeal to those interested in moral philosophy, religious ethics, and literary criticism, and grants those who have long loved Murdoch's novels a closer look at her remarkable philosophy.

Author : Ernst Cassirer
Publisher : Routledge
Page : 606 pages
File Size : 46,7 Mb
Release : 2020-09-24
Category : Philosophy
ISBN : 9781000001112

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The Philosophy of Symbolic Forms, Volume 3 by Ernst Cassirer Pdf

“In his Phenomenology of Cognition, Cassirer provides a comprehensive and systematic account of the dynamic process involved in the whole of human culture as it progresses from the world of myth and its feeling of social belonging to the highest abstractions of mathematics, logic and theoretical physics. Cassirer engages with the most sophisticated and cutting-edge work in fields ranging from ethnology to classics, egyptology and assyriology to ethology, brain science and psychology to logic, mathematics and theoretical physics. His command of philosophy, literature, and the arts is superb. Echoing his work on Kant, Cassirer begins The Philosophy of Symbolic Forms with the problem posed by the meaning of being for philosophy since Plato. But Cassirer also shows that this problem gains new significance with Kant and with the development of modern culture. Cassirer weaves his conception of the development of knowledge into a broadly Kantian and German idealist dynamic-historical conception of significance and of experience that refuses to accept a fundamental opposition between literary, philosophical and scientific culture. In consequence of his great vision grounded in careful reflection and argument, Cassirer’s systematic conception of the Copernican cosmopolitan-cosmological revolution is still philosophically and scientifically unmatched in contemporary philosophy on both sides of the Atlantic and of the Pacific.” Pierre Keller, Professor of Philosophy, University of California, Riverside, USA. This new translation makes Cassirer’s seminal work available to a new generation of scholars. Each volume includes a translator’s introduction by Steve G. Lofts, a foreword by Peter E. Gordon, a glossary of key terms, and an index.