Download or read book Mathesis Universalis Computability and Proof written by Stefania Centrone and published by Springer Nature. This book was released on 2019-10-25 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a fragment entitled Elementa Nova Matheseos Universalis (1683?) Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects” (“objets quelconques”). It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” (“Gründe”) of others, and the latter are “consequences” (“Folgen”) of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.
Download or read book Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification written by Klaus Mainzer and published by World Scientific. This book was released on 2021-07-27 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.
Download or read book Is Law Computable written by Simon Deakin and published by Bloomsbury Publishing. This book was released on 2020-11-26 with total page 578 pages. Available in PDF, EPUB and Kindle. Book excerpt: What does computable law mean for the autonomy, authority, and legitimacy of the legal system? Are we witnessing a shift from Rule of Law to a new Rule of Technology? Should we even build these things in the first place? This unique volume collects original papers by a group of leading international scholars to address some of the fascinating questions raised by the encroachment of Artificial Intelligence (AI) into more aspects of legal process, administration, and culture. Weighing near-term benefits against the longer-term, and potentially path-dependent, implications of replacing human legal authority with computational systems, this volume pushes back against the more uncritical accounts of AI in law and the eagerness of scholars, governments, and LegalTech developers, to overlook the more fundamental - and perhaps 'bigger picture' - ramifications of computable law. With contributions by Simon Deakin, Christopher Markou, Mireille Hildebrandt, Roger Brownsword, Sylvie Delacroix, Lyria Bennet Moses, Ryan Abbott, Jennifer Cobbe, Lily Hands, John Morison, Alex Sarch, and Dilan Thampapillai, as well as a foreword from Frank Pasquale.
Download or read book The Architecture and Archaeology of Modern Logic written by Ansten Klev and published by Springer Nature. This book was released on with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Handbook of Constructive Mathematics written by Douglas Bridges and published by Cambridge University Press. This book was released on 2023-03-31 with total page 864 pages. Available in PDF, EPUB and Kindle. Book excerpt: Constructive mathematics – mathematics in which 'there exists' always means 'we can construct' – is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.
Download or read book Prawitz s Epistemic Grounding written by Antonio Piccolomini d’Aragona and published by Springer Nature. This book was released on 2023-01-01 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents an in-depth and critical reconstruction of Prawitz’s epistemic grounding, and discusses it within the broader field of proof-theoretic semantics. The theory of grounds is also provided with a formal framework, through which several relevant results are proved. Investigating Prawitz’s theory of grounds, this work answers one of the most fundamental questions in logic: why and how do some inferences have the epistemic power to compel us to accept their conclusion, if we have accepted their premises? Prawitz proposes an innovative description of inferential acts, as applications of constructive operations on grounds for the premises, yielding a ground for the conclusion. The book is divided into three parts. In the first, the author discusses the reasons that have led Prawitz to abandon his previous semantics of valid arguments and proofs. The second part presents Prawitz’s grounding as found in his ground-theoretic papers. Finally, in the third part, a formal apparatus is developed, consisting of a class of languages whose terms are equipped with denotation functions associating them to operations and grounds, as well as of a class of systems where important properties of the terms can be proved.
Download or read book Proof And Computation Digitization In Mathematics Computer Science And Philosophy written by Klaus Mainzer and published by World Scientific. This book was released on 2018-05-30 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes Predicative Foundations, Constructive Mathematics and Type Theory, Computation in Higher Types, Extraction of Programs from Proofs, and Algorithmic Aspects in Financial Mathematics. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.
Download or read book Comprehending the Complexity of Countries written by Hans Kuijper and published by Springer Nature. This book was released on 2022-01-18 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book argues for computer-aided collaborative country research based on the science of complex and dynamic systems. It provides an in-depth discussion of systems and computer science, concluding that proper understanding of a country is only possible if a genuinely interdisciplinary and truly international approach is taken; one that is based on complexity science and supported by computer science. Country studies should be carefully designed and collaboratively carried out, and a new generation of country students should pay more attention to the fast growing potential of digitized and electronically connected libraries. In this frenzied age of globalization, foreign policy makers may – to the benefit of a better world – profit from the radically new country studies pleaded for in the book. Its author emphasizes that reductionism and holism are not antagonistic but complementary, arguing that parts are always parts of a whole and a whole has always parts.
Download or read book Reflections on the Foundations of Mathematics written by Stefania Centrone and published by Springer Nature. This book was released on 2019-11-11 with total page 511 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.
Download or read book Gentzen s Centenary written by Reinhard Kahle and published by Springer. This book was released on 2015-11-02 with total page 563 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The Universe as Automaton written by Klaus Mainzer and published by Springer Science & Business Media. This book was released on 2011-10-01 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Brief is an essay at the interface of philosophy and complexity research, trying to inspire the reader with new ideas and new conceptual developments of cellular automata. Going beyond the numerical experiments of Steven Wolfram, it is argued that cellular automata must be considered complex dynamical systems in their own right, requiring appropriate analytical models in order to find precise answers and predictions in the universe of cellular automata. Indeed, eventually we have to ask whether cellular automata can be considered models of the real world and, conversely, whether there are limits to our modern approach of attributing the world, and the universe for that matter, essentially a digital reality.
Download or read book Essays on Husserl s Logic and Philosophy of Mathematics written by Stefania Centrone and published by Springer. This book was released on 2017-09-07 with total page 541 pages. Available in PDF, EPUB and Kindle. Book excerpt: Essays on Husserl’s Logic and Philosophy of Mathematics sets out to fill up a lacuna in the present research on Husserl by presenting a precise account of Husserl’s work in the field of logic, of the philosophy of logic and of the philosophy of mathematics. The aim is to provide an in-depth reconstruction and analysis of the discussion between Husserl and his most important interlocutors, and to clarify pivotal ideas of Husserl’s by considering their reception and elaboration by some of his disciples and followers, such as Oskar Becker and Jacob Klein, as well as their influence on some of the most significant logicians and mathematicians of the past century, such as Luitzen E. J. Brouwer, Rudolf Carnap, Kurt Gödel and Hermann Weyl. Most of the papers consider Husserl and another scholar – e.g. Leibniz, Kant, Bolzano, Brentano, Cantor, Frege – and trace out and contextualize lines of influence, points of contact, and points of disagreement. Each essay is written by an expert of the field, and the volume includes contributions both from the analytical tradition and from the phenomenological one.
Download or read book Digital And The Real World The Computational Foundations Of Mathematics Science Technology And Philosophy written by Klaus Mainzer and published by World Scientific. This book was released on 2017-11-17 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the 21st century, digitalization is a global challenge of mankind. Even for the public, it is obvious that our world is increasingly dominated by powerful algorithms and big data. But, how computable is our world? Some people believe that successful problem solving in science, technology, and economies only depends on fast algorithms and data mining. Chances and risks are often not understood, because the foundations of algorithms and information systems are not studied rigorously. Actually, they are deeply rooted in logics, mathematics, computer science and philosophy.Therefore, this book studies the foundations of mathematics, computer science, and philosophy, in order to guarantee security and reliability of the knowledge by constructive proofs, proof mining and program extraction. We start with the basics of computability theory, proof theory, and information theory. In a second step, we introduce new concepts of information and computing systems, in order to overcome the gap between the digital world of logical programming and the analog world of real computing in mathematics and science. The book also considers consequences for digital and analog physics, computational neuroscience, financial mathematics, and the Internet of Things (IoT).
Download or read book Oskar Becker On the Logic of Modalities 1930 Translation Commentary and Analysis written by Stefania Centrone and published by Springer Nature. This book was released on 2022-04-11 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers the first-ever English translation of Oskar Becker’s Zur Logik der Modalitäten. This essay, published in 1930, is a pioneering yet often neglected contribution in the context of prewar modal logic research in Europe. Becker’s text is complemented by an extended commentary that explains, analyzes and highlights Becker’s accomplishments and the philosophical background of his investigations. The commentary provides an in-depth analysis of all of Becker's important contributions, both from a philosophical and logical perspective, making it a very useful book for scholars in both philosophy and logic.
Download or read book Husserl and Mathematics written by Mirja Hartimo and published by Cambridge University Press. This book was released on 2021-08-05 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: Husserl and Mathematics explains the development of Husserl's phenomenological method in the context of his engagement in modern mathematics and its foundations. Drawing on his correspondence and other written sources, Mirja Hartimo details Husserl's knowledge of a wide range of perspectives on the foundations of mathematics, including those of Hilbert, Brouwer and Weyl, as well as his awareness of the new developments in the subject during the 1930s. Hartimo examines how Husserl's philosophical views responded to these changes, and offers a pluralistic and open-ended picture of Husserl's phenomenology of mathematics. Her study shows Husserl's phenomenology to be a method capable of both shedding light on and internally criticizing scientific practices and concepts.
Download or read book What Algorithms Want written by Ed Finn and published by MIT Press. This book was released on 2017-03-10 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: The gap between theoretical ideas and messy reality, as seen in Neal Stephenson, Adam Smith, and Star Trek. We depend on—we believe in—algorithms to help us get a ride, choose which book to buy, execute a mathematical proof. It's as if we think of code as a magic spell, an incantation to reveal what we need to know and even what we want. Humans have always believed that certain invocations—the marriage vow, the shaman's curse—do not merely describe the world but make it. Computation casts a cultural shadow that is shaped by this long tradition of magical thinking. In this book, Ed Finn considers how the algorithm—in practical terms, “a method for solving a problem”—has its roots not only in mathematical logic but also in cybernetics, philosophy, and magical thinking. Finn argues that the algorithm deploys concepts from the idealized space of computation in a messy reality, with unpredictable and sometimes fascinating results. Drawing on sources that range from Neal Stephenson's Snow Crash to Diderot's Encyclopédie, from Adam Smith to the Star Trek computer, Finn explores the gap between theoretical ideas and pragmatic instructions. He examines the development of intelligent assistants like Siri, the rise of algorithmic aesthetics at Netflix, Ian Bogost's satiric Facebook game Cow Clicker, and the revolutionary economics of Bitcoin. He describes Google's goal of anticipating our questions, Uber's cartoon maps and black box accounting, and what Facebook tells us about programmable value, among other things. If we want to understand the gap between abstraction and messy reality, Finn argues, we need to build a model of “algorithmic reading” and scholarship that attends to process, spearheading a new experimental humanities.
Download or read book Philosophy and the Many Faces of Science written by Dionysios Anapolitanos and published by Rowman & Littlefield. This book was released on 1998 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of original papers by an international group of distinguished philosophers of science impressively demonstrates the links among the philosophic points of view, areas of focus, and methods of treatment used in examining the many facets of scientific inquiry. It will be an indispensable collection for philosophers of science and scientists of various disciplines, including physicists, neuroscientists, and psychologists.
