Arnold Beckmann (original) (raw)
Related papers
The Foundations of Computability Theory
Springer eBooks, 2020
The paradoxes discovered in Cantor's set theory sometime around 1900 began a crisis that shook the foundations of mathematics. In order to reconstruct mathematics, freed from all paradoxes, Hilbert introduced a promising program with formal systems as the central idea. Though the program was unexpectedly brought to a close in 1931 by Gödel's famous theorems, it bequeathed burning questions: "What is computing? What is computable? What is an algorithm? Can every problem be algorithmically solved?" This led to Computability Theory, which was born in the mid-1930s, when these questions were resolved by the seminal works of Church, Gödel, Kleene, Post, and Turing. In addition to contributing to some of the greatest advances of twentieth-century mathematics, their ideas laid the foundations for the practical development of a universal computer in the 1940s as well as the discovery of a number of algorithmically unsolvable problems in different areas of science. New questions, such as "Are unsolvable problems equally difficult? If not, how can we compare their difficulty?" initiated new research topics of Computability Theory, which in turn delivered many important concepts and theorems. The application of these is central to the multidisciplinary research of Computability Theory. Aims Monographs in Theoretical Computer Science usually strive to present as much of the subject as possible. To achieve this, they present the subject in a definitiontheorem-proof style and, when appropriate, merge and intertwine different related themes, such as computability, computational complexity, automata theory, and formal-language theory. This approach, however, often blurs historical circumstances, reasons, and the motivation that led to important goals, concepts, methods, and theorems of the subject. vii viii Preface My aim is to compensate for this. Since the fundamental ideas of theoretical computer science were either motivated by historical circumstances in the field or developed by pure logical reasoning, I describe Computability Theory, a part of Theoretical Computer Science, from this point of view. Specifically, I describe the difficulties that arose in mathematical logic, the attempts to recover from them, and how these attempts led to the birth of Computability Theory and later influenced it. Although some of these attempts fell short of their primary goals, they put forward crucial questions about computation and led to the fundamental concepts of Computability Theory. These in turn logically led to still new questions, and so on. By describing this evolution I want to give the reader a deeper understanding of the foundations of this beautiful theory. The challenge in writing this book was therefore to keep it accessible by describing the historical and logical development while at the same time introducing as many modern topics as needed to start the research. Thus, I will be happy if the book makes good reading before one tackles more advanced literature on Computability Theory.
2012
CiE 2009 was the fifth in the series of conferences associated with the Association for Computability in Europe. The conferences had been organized by an informal network of European scientists from 2005 onwards. At the conference CiE 2008 in Athens, this network decided to give itself a more formal structure by founding the Association for Computability in Europe. The conference in Heidelberg was the first conference after this important step for our community. The main aim of the association is to promote the development, particularly in Europe, of computability-related science, ranging over mathematics, computer science, and applications in various natural and engineering sciences such as physics and biology, including the promotion of the study of philosophy and history of computing as it relates to questions of computability. CiE 2009 aimed at bridging the gap from the theoretical methods of mathematical and meta-mathematical flavour to the applied and industrial questions of computational practice. The conference brought together computer scientists, mathematicians, physicists, biologists, engineers, and explored the historical and philosophical foundations of the field. As is usual in computer science, CiE 2009 had a regular pre-proceedings volume published in the Lecture Notes in Computer Science:
From Gödel to Einstein: Computability between logic and physics at CiE 2006
Theoretical Computer Science, 2008
The seven papers in this special issue arose from the conference CiE 2006: Logical Approaches to Computational Barriers, held at the University of Wales Swansea in July, 2006. CiE 2006 was the second of a new series of conferences associated with the interdisciplinary network Computability in Europe.
Chapman & Hall/CRC applied algorithms and data structures series, 1998
I explore the conceptual foundations of Alan Turing's analysis of computability, which still dominates thinking about computability today. I argue that Turing's account represents a last vestige of a famous but unsuccessful program in pure mathematics, viz., Hilbert's formalist program. It is my contention that the plausibility of Turing's account as an analysis of the computational capacities of physical machines rests upon a number of highly problematic assumptions whose plausibility in turn is grounded in the formalist stance towards mathematics. More speciÿcally, the Turing account conates concepts that are crucial for understanding the computational capacities of physical machines. These concepts include the idea of an " operation " or " action " that is " formal, " " mechanical, " " well-deÿned, " and " precisely described, " and the idea of a " symbol " that is " formal, " " uninterpreted, " and " shaped ". When these concepts are disentangled, the intuitive appeal of Turing's account is signiÿcantly undermined. This opens the way for exploring models of hypercomputability that are fundamentally dierent from those currently entertained in the literature.
Review of B. Jack Copeland, Carl J. Posy, and Oron Shagrir (eds.), Computability: Turing, Gödel, Church, and Beyond, The MIT Press, 2013, 376pp., $20.00 (pbk), ISBN 9780262527484.
Computability. Computable Functions, Logic, and the Foundations of Mathematics
The Bulletin of Symbolic Logic, 2002
One of the first analyses of the notion of computability, and certainly the most influential, is due to Turing. Alan M. Turing, from "On Computable Numbers, with an Application to the Entscheidungsproblem, " 1936 The "computable"numbers may be describedbriefly as the real numbers whose expressionsas a decimalare calculableby [mite means. ... Accordingto my definition, a number is computable, if its decimal can be written down bya machine. p. 116
Directions for Computability Theory Beyond Pure Mathematical
Mathematical Problems from Applied Logic II
This paper begins by briefly indicating the principal, non-standard motivations of the author for his decades of work in Computability Theory (CT), a.k.a. Recursive Function Theory. Then it discusses its proposed, general directions beyond those from pure mathematics for CT. These directions are as follows. 1. Apply CT to basic sciences, for example, biology, psychology, physics, chemistry, and economics. 2. Apply the resultant insights from 1 to philosophy and, more generally, apply CT to areas of philosophy in addition to the philosophy and foundations of mathematics. 3. Apply CT for insights into engineering and other professional fields. Lastly, this paper provides a progress report on the above non-pure mathematical directions for CT, including examples for biology, cognitive science and learning theory, philosophy of science, physics, applied machine learning, and computational complexity. Interweaved with the report are occasional remarks about the future.
Logical Approaches to Computational Barriers: CiE 2006
Journal of Logic and Computation, 2007
is an informal network of European scientists working on computability theory, including its foundations, technical development, and applications. Among the aims of the network is to advance our theoretical understanding of what can and cannot be computed, by any means of computation. Its scientific vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and machines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory or relativity. Computations may be very general, depending upon the foundations of set theory; or very specific, using the combinatorics of finite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, and computational learning. Applications are everywhere, especially, in algebra, analysis and geometry, or data types and programming.