Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a (...) simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual. (shrink)

We offer a novel interpretation of the argumentative role that Meditation IV plays within the whole of the Meditations. This new interpretation clarifies several otherwise head-scratching claims that Descartes makes about Meditation IV, and it fully exonerates the Fourth Meditation from either raising or exacerbating Descartes’ circularity problems.

The notion that our society, its education system and its intellectual life, is characterised by a split between two cultures – the arts or humanities on one hand, and the sciences on the other – has a long history. But it was C. P. Snow's Rede lecture of 1959 that brought it to prominence and began a public debate that is still raging in the media today. This 50th anniversary printing of The Two Cultures and its successor piece, A Second (...) Look features an introduction by Stefan Collini, charting the history and context of the debate, its implications and its afterlife. The importance of science and technology in policy run largely by non-scientists, the future for education and research, and the problem of fragmentation threatening hopes for a common culture are just some of the subjects discussed. (shrink)

The notion that our society, its education system and its intellectual life, is characterised by a split between two cultures – the arts or humanities on one hand and the sciences on the other – has a long history. But it was C. P. Snow's Rede lecture of 1959 that brought it to prominence and began a public debate that is still raging in the media today. This fiftieth anniversary printing of The Two Cultures and its successor piece, A Second (...) Look features an introduction by Stefan Collini, charting the history and context of the debate, its implications and its afterlife. The importance of science and technology in policy run largely by non-scientists, the future for education and research, and the problem of fragmentation threatening hopes for a common culture are just some of the subjects discussed. (shrink)

There have been many claims that quantum mechanics plays a key role in the origin and/or operation of biological organisms, beyond merely providing the basis for the shapes and sizes of biological molecules and their chemical affinities. These range from Schr¨odinger’s suggestion that quantum fluctuations produce mutations, to Hameroff and Penrose’s conjecture that quantum coherence in microtubules is linked to consciousness. I review some of these claims in this paper, and discuss the serious problem of decoherence. I advance some further (...) conjectures about quantum information processing in bio-systems. Some possible experiments are suggested. © 2004 Elsevier Ireland Ltd. All rights reserved. (shrink)

Institute of Theoretical Astronomy, University of Cambridge, Cambridge, UK 3fS. received 28th August 1970, in final revised form 1st July 1971..

Offering an original perspective on the central project of Descartes' Meditations, this book argues that Descartes' free will theodicy is crucial to his refutation of skepticism. A common thread runs through Descartes' radical First Meditation doubts, his Fourth Meditation discussion of error, and his pious reconciliation of providence and freedom: each involves a clash of perspectives-thinking of God seems to force conclusions diametrically opposed to those we reach when thinking only of ourselves. Descartes fears that a skeptic could exploit this (...) clash of perspectives to argue that Reason is not trustworthy because self-contradictory. To refute the skeptic and vindicate the consistency of Reason, it is not enough for Descartes to demonstrate that our Creator is perfect; he must also show that our errors cannot prove God's imperfection. To do this, Descartes invokes the idea that we err freely. However, prospects initially seem dim for this free will theodicy, because Descartes appears to lack any consistent or coherent understanding of human freedom. In an extremely in-depth analysis spanning four chapters, Ragland argues that despite initial appearances, Descartes consistently offered a coherent understanding of human freedom: for Descartes, freedom is most fundamentally the ability to do the right thing. Since we often do wrong, actual humans must therefore be able to do otherwise-our actions cannot be causally determined by God or our psychology. But freedom is in principle compatible with determinism: while leaving us free, God could have determined us to always do the good. Though this conception of freedom is both consistent and suitable to Descartes' purposes, when he attempts to reconcile it with divine providence, Descartes's strategy fails, running afoul of his infamous doctrine that God created the eternal truths. (shrink)

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure $\langle \omega; +, P\rangle$ , where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of $\langle\omega; S, P\rangle$ is decidable, where S is the successor function. The latter result is proved using a general (...) result of A. L. Semenov on decidability of monadic theories, and a proof of Semenov's result is presented. (shrink)