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.
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. Chapter 8 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; and more. Includes 27 figures. Index. 1959 edition.
Author: Maria Antonaccio Assistant Professor of Religion Bucknell University
Publisher: Oxford University Press, USA
Category: Literary Criticism
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.
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.
Thinking about Mathematics covers the range of philosophical issues and positions concerning mathematics. The text describes the questions about mathematics that motivated philosophers throughout history and covers historical figures such as Plato, Aristotle, Kant, and Mill. It also presents the major positions and arguments concerning mathematics throughout the twentieth century, bringing the reader up to the present positions and battle lines.
This fascinating book argues for a new way of looking at the world and at human systems, companies or (Western) society as a whole. Walter R.J. Baets argues that we should let go of our drive to control, manage and organize, in order to be able to create an ideal environment for continuous learning, both for ourselves and for our collaborators. Arguing in favour of a holistic management approach, and very much in opposition to the short-term shareholder value driven approaches that are popular today, Baets’ book develops a logic founded in real life observations, examples and cases that every reader will recognize in their daily practice. It guides the reader to understand an alternative paradigm and allows them finally to be able to work with the dynamics of business on a daily basis. A must-read for students of complexity, strategy and organizational behaviour, this well-researched, well-argued book skilfully guides the reader through this interesting subject.
This book presents a selection of papers from the International Conference Geometrias’17, which was hosted by the Department of Architecture at the University of Coimbra from 16 to 18 June 2017. The Geometrias conferences, organized by Aproged (the Portuguese Geometry and Drawing Teachers’ Association), foster debate and exchange on practical and theoretical research in mathematics, architecture, the arts, engineering, and related fields. Geometrias’17, with the leitmotif “Thinking, Drawing, Modelling”, brought together a group of recognized experts to discuss the importance of geometric literacy and the science of representation for the development of scientific and technological research and professional practices. The 12 peer-reviewed papers gathered here show how geometry, drawing, stereotomy, and the science of representation are still at the core of every act leading to the conception and materialization of form, and highlight their continuing relevance for scholars and professionals in the fields of architecture, engineering, and applied mathematics.
The first collection of critical essays on the work of this most original thinker. Francois Laruelle is one of the most important French philosophers of the last 20 years, and as his texts have become available in English there has been a rising tide of interest in his work, particularly on the concept of 'Non-Philosophy'. Non-philosophy radically rethinks many of the most cutting-edge concepts such as immanence, pluralism, resistance, science, democracy, decisionism, Marxism, theology and materialism. It also expands our view of what counts as philosophical thought, through art, science and politics, and beyond to fields as varied as film, animality and material objects.
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.
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.
This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book’s title, an alogon in the logos of mathematics. The outcome of this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields.