whole numbers that manages to assert that it itself is unprovable (from a given finite set F of axioms using formal logic). (Gödel's paper is included in the well-known anthology [1].) GF : ``GF cannot be proved from the finite set of axioms F.'' This assertion GF is therefore true if and only if it is unprovable, and the formal axiomatic system F in question either proves falsehoods (because it enables us to prove GF) or fails to prove a true (...) assertion (because it does not enable us to prove GF). If we assume that the former situation is impossible, we conclude that F is necessarily incomplete since it does not permit us to establish the true statement GF. (shrink)

This book introduces readers to the narrative structure of mathematical proofs and why mathematicians communicate that way, drawing examples from classic literature and employing metaphors and imagery.

In his book Gabriele Lolli discusses the notion of proof, which is, according to him, the most important and at the same time the least studied aspect of mathematics. According to Lolli, a theorem is a conditional sentence of the form ‘if T then A’ such that A is a logical consequence of T, where A is a sentence and T is a sentence or a conjunction or set of sentences. Verifying that A is a consequence of T (...) generally involves considering infinitely many interpretations; so it is something which is impossible to do in finite terms. Proofs may serve as ‘shortcuts’ in this respect. A proof is defined by Lolli as any finite argument certifying that A is a consequence of T. A proof is a shortcut in the sense that it spares us considering infinitely many interpretations.The reason for such a very general definition of proof is Lolli's strong belief that mathematics is not a rigid system of explicit rules, but rather a set of tools; as a consequence, there is no prescription as to what a proof should or should not be. Actually, mathematics is historically situated and not timeless, and the history of mathematics is the …. (shrink)

This is a welcome reprint of a book that continues to grow in importance.

Kant’s critique/doctrine distinction tracks the difference between a canon for the understanding’s proper use and an organon for its dialectical misuse. The latter reflects the dogmatic use of reason to attain a doctrine of knowledge with no antecedent critique. In the 1790s, Fichte collapses Kant’s distinction and redefines dogmatism. He argues that deriving a canon is essentially dialectical and thus yields an organon: critical idealism is properly a doctrine of science or Wissenschaftslehre. Criticism is furthermore said to refute dogmatism, by (...) which Fichte means Spinozism. Schelling repudiates Fichte’s position in his Philosophical Letters by reviving Kant’s distinction. First, ‘critique’ for Schelling names a canon that is valid for all systems—including Spinozism—and thus favours none—not even the Wissenschaftslehre. Second, Schelling coins ‘dogmaticism’ to refer to what Kant calls ‘dogmatism’: reason’s uncritical claim to doctrine. Hence, he says all systems—even Fichte’s—are susceptible to the ‘enthusiasm’ of mistaking themselves for doctrines or ‘objects of knowledge’, instead of subjectively valid ‘objects of freedom’. I reconstruct Schelling’s argument for reviving Kant’s distinction, while noting two consequences for German idealism. First, if criticism favours no system, a system’s origin cannot be an act of knowledge, e.g., intellectual intuition, but only an act of endorsement. Second, if dogmaticism entails enthusiasm, a system’s goal cannot be absolute knowledge, but only infinite striving. Hence, by retrieving Schelling’s notion of dogmaticism, we can both gain a better understanding of the Letters and register the impact of a core Kantian distinction on the German idealist project. (shrink)

This book presents G. A. Cohen's Gifford Lectures, delivered at the University of Edinburgh in 1996. Focusing on Marxism and Rawlsian liberalism, Cohen draws a connection between these thought systems and the choices that shape a person's life. In the case of Marxism, the relevant life is his own: a communist upbringing in the 1940s in Montreal, which induced a belief in a strongly socialist egalitarian doctrine. The narrative of Cohen's reckoning with that inheritance develops through a series of sophisticated (...) engagements with the central questions of social and political philosophy. In the case of Rawlsian doctrine, Cohen looks to people's lives in general. He argues that egalitarian justice is not only, as Rawlsian liberalism teaches, a matter of rules that define the structure of society, but also a matter of personal attitude and choice. Personal attitude and choice are, moreover, the stuff of which social structure itself is made. Those truths have not informed political philosophy as much as they should, and Cohen's focus on them brings political philosophy closer to moral philosophy, and to the Judeo-Christian ethical tradition, than it has recently been. (shrink)

Au Congrès de philosophie scientifique de 1935 comme lors du lancement de l’Encyclopédie internationale de la science unifiée, Federigo Enriques était reconnu par les néo-positivistes comme un de leurs pères fondateurs, sans qu’il fût tout à fait d’accord. À Paris, Enriques représentait le groupe des philosophes des sciences italiens et son nom était lié au journal Scientia, ouvert aux contributions des positivistes logiques. Ces derniers, désireux de constituer un front commun pour lutter contre les philosophies idéalistes et métaphysiques alors dominantes, (...) surestimaient sans doute la force du groupe italien, comme ils sous-estimaient les critiques d’Enriques sur l’usage de la logique formelle ou sur l’importance du langage. Dès les débuts de son implication en philosophie, il avait souligné la nécessité d’une fondation psychologique des concepts scientifiques, et en 1901, il avait donné un exemple de ce type d’analyse pour les postulats de la géométrie. Il n’appréciait guère les travaux logiques de Giuseppe Peano, affirmant que son formalisme manquait de bases psychologiques. Cependant, contre les mouvements anti-rationalistes de son temps, il défendait la valeur cognitive de la science et, dans ce climat d’appréciation mutuelle, les différences passaient au second plan. (shrink)

A highly original reading of Immanuel Kant that demonstrates his interest in the social realm of human interaction.

Containing selected papers on the fundamentals and applications of Complexity Science, this multi-disciplinary book presents new approaches for resolving complex issues that cannot be resolved using conventional mathematical or software models. Complex Systems problems can occur in a variety of areas such as physical sciences and engineering, the economy, the environment, humanities and social and political sciences. Complexity Science problems, the science of open systems consisting of large numbers of diverse components engaged in rich interaction, can occur in a variety (...) of areas such as physical sciences and engineering, the economy, the environment, humanities and social and political sciences. The global behaviour of these systems emerges from the interaction of constituent components and is unpredictable but not random. The key attribute of Complex Systems is the ability to self-organise and adapt to unpredictable changes in their environment. (shrink)

A leading figure in neuroaesthetics makes the case that aesthetic experience can be meaningfully measured by the tools of neuroscience.

G.E. Moore, more than either Bertrand Russell or Ludwig Wittgenstein, was chiefly responsible for the rise of the analytic method in twentieth-century philosophy. This selection of his writings shows Moore at his very best. The classic essays are crucial to major philosophical debates that still resonate today. Amongst those included are: * A Defense of Common Sense * Certainty * Sense-Data * External and Internal Relations * Hume's Theory Explained * Is Existence a Predicate? * Proof of an External World (...) In addition, this collection also contains the key early papers in which Moore signals his break with idealism, and three important previously unpublished papers from his later work which illustrate his relationship with Wittgenstein. (shrink)

Fichte assigns ‘intellectual intuition’ a new meaning after Kant. But in 1799, his doctrine of intellectual intuition is publicly deemed indefensible by Kant and nihilistic by Jacobi. I propose to defend Fichte’s doctrine against these charges, leaving aside whether it captures what he calls the ‘spirit’ of transcendental idealism. I do so by articulating three problems that motivate Fichte’s redirection of intellectual intuition from being to acting: (1) the regress problem, which states that reflecting on empirical facts of consciousness leads (...) only to further facts and so cannot yield a first principle; (2) the rhapsody problem, which states that the categories form a haphazard set and so lack necessity unless they derive from a first principle; and (3) the nihilism problem, which states that a first principle cannot lie outside our cognition of it, lest it be the cause of our cognition and, being first, the cause of all our actions, reducing us to machines. Crucially, Fichte’s three motivating problems are in fact aspects of a single problem. Leaving any aspect unsolved spoils putative solutions to the other two. Consequently, Fichte requires a single unified solution to all three, which his doctrine of intellectual intuition provides. (shrink)

This chapter contains section titled: I II III IV V VI VII VIII IX X XI XII Coda Appendix Acknowledgements.

A model for the ideal state includes discussions of the nature and application of justice, the role of the philosopher in society, the goals of education, and the effects of art upon character.

This essay challenges the widely accepted principle that a person is morally responsible for what he has done only if he could have done otherwise. The author considers situations in which there are sufficient conditions for a certain choice or action to be performed by someone, So that it is impossible for the person to choose or to do otherwise, But in which these conditions do not in any way bring it about that the person chooses or acts as he (...) does. In such situations the person may well be morally responsible for what he chooses or does despite his inability to choose or to do otherwise. Finally the author considers certain suggestions for revising the principle he rejects or for replacing it with a principle of an altogether different kind. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)

There is growing interest in the use of neurointerventions to reduce the risk that criminal offenders will reoffend. Commentators have raised several ethical concerns regarding this practice. One prominent concern is that, when imposed without the offender’s valid consent, neurointerventions might infringe offenders’ right to bodily integrity. While it is commonly held that we possess a moral right to bodily integrity, the extent to which this right would protect against such neurointerventions is as-yet unclear. In this paper, we will assess (...) whether, why, and how severely three forms of neurointervention might infringe the right to bodily integrity. We show that the severity of the infringement of the right to bodily integrity differs across different forms of neurointervention. Moreover, we argue that mental and behavioral effects of neurointerventions could in some cases be relevant to determining the severity of infringements of the right to bodily integrity. (shrink)

In the Vocation of Man, Fichte makes the striking claim that life is eternal, rational, our true being, and the final cause of nature in general and of death in particular. How can we make sense of this claim? I argue that the public lectures that compose the Vocation are a popular expression of Fichte’s pre-existing commitment to what I call immortalism, the view that life is the unconditioned condition of intelligibility. Casting the I as an absolutely self-active or living (...) power enables his philosophical account of both the intelligibility of objects and of what it means to die. This explains his characterization of the Wissenschaftslehre as a system in which ‘all is act, movement, and life’. In Sect. 1, I explain the context of Jacobi’s use of ‘caput mortuum’ as a term for the thing in itself. In Sects. 2 and 3, I reconstruct two immortalist arguments from Fichte’s Jena period. The first is that the I’s self-activity rules out the existence of the thing in itself and thereby vanquishes ‘death’s head’. The second is that, insofar as the I charges us with the moral perfection of nature, it is the final cause of our entire life, including its end. This is to say that the I puts our every moment into question, even our last. In Sect. 4, I use these immortalist arguments to interpret the striking claim from the Vocation. (shrink)

This paper looks at what David Kaplan's work on indexicals can teach us about logic and the philosophy of logic, and also what Kaplan's logic (i.e. the Logic of Demonstratives) can teach us about indexicals. The lessons are i) that logical consequence is not necessary truth-preservation, ii) that that the linguistic doctrine of necessary truth (also called conventionalism about modality) fails, and iii) that there is a kind of barrier to entailment between non-context-sensitive and context-sensitive claims.

Philosophy of Tsong-kha-pa Blo-bzang-grags-pa's Lam rim chen mo.

Conference proceedings on Mongolian contributions to Buddhist knowledge.

Throughout his career, Schelling assigns knowledge of the absolute first principle of philosophy to intellectual intuition. Schelling's doctrine of intellectual intuition raises two important questions for interpreters. First, given that his doctrine undergoes several changes before and after his identity philosophy, to what extent can he be said to “hold onto” the same “sense” of it by the 1830s, as he claims? Second, given that his doctrine of intellectual intuition restricts absolute idealism to what he calls a “science of reason”, (...) which he says cannot prove the absolute’s existence, what other doctrine does he require in order to prove this? I will answer these questions by tracing the shifts in Schelling’s doctrine of intellectual intuition from the 1790s to the 1830s and drawing out its evolving methodological role within his science of intelligibility. (shrink)

Modern science faces a series of problems that undermine confidence in its reliability. To solve these problems, we must reflect on what makes science work and what leads it astray. This book is about Science, its strengths and weaknesses. The papers that scientists write form a vast resource of evidence and theory that is doubling about every ten years, along with the number of scientists. The size of this resource makes it hard for it to be used effectively by scientists, (...) and along with this are growing concerns about its integrity and reliability. This book looks at the scientific method, as currently applied and as understood philosophically by scientists, the social structure of science, and the structure of scientific communication through journals. It includes case studies of controversies, and looks at how fields develop over time. (shrink)