First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

The solution of the outer multiplicity problem in the tensor product of U(3) irreducible representations (irreps) developed by Biedenharn et al.(1–7) and realized through the well-known Draayer-Akiyama (DA) computer code(8) is extended to the quantum algebra Uq(3). An analytic formula for special stretched Uq(3) Wigner coefficients, $$\left\langle {(\lambda _1 \mu _1 ) H_1 , (\lambda _2 \mu _2 ) \varepsilon _2 \Lambda _2 m_2 \left| { (\lambda _3 \mu _3 ) H_3 } \right.} \right\rangle _{\max }^q $$ is derived using (...) a projection operator method.(9–10) In this expression Hi denotes the highest weight vector of the (λiμi) irrep; the subscript “max” means coefficients corresponding to a unit tensor operator with a maximal characteristic null space; and q is the usual quantum label so the standard U(3) Wigner coefficient, which is required in the DA code, can be obtained in the q→1 limit of the theory. To illustrate the theory, some Uq(3) Wigner coefficients for the tensor product (22)×(22) are calculated. The procedure for evaluating nonhighest weight Wigner Uq(3) coefficients follow the DA prescription. (shrink)

We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of (...) ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined. (shrink)

The ‘decapitation gambit’ holds that, if physical decapitation normally entails the death of the human being, then physiological decapitation, evident in cases of total brain failure, entails the death of the human being. This argument has been challenged by Franklin Miller and Robert Truog, who argue that physical decapitation does not necessarily entail the death of human beings and that therefore, by analogy, artificially sustained human bodies with total brain failure are living human beings. They thus challenge the current neurological (...) criterion for determining death and argue for a return to the traditional criterion of the irreversible loss of circulation and respiration. In this paper, I defend the decapitation gambit and total brain failure as a criterion for determining death against Miller and Truog's criticism. (shrink)

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$. Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ (...) games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$, but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation. (shrink)

The history behind the ‘Madagascar’ example of Gareth Evans is traced, suggesting that the decisive reference-shift occurred in the 16th, not the 17, century. The difference between this example and the ‘Gödel’ example of Saul Kripke is explained in terms of the distinction between de re and de dicto beliefs and intentions.

After describing Heidegger's critique of metaphysics as ontotheology, I unpack the metaphysical assumptions of several transhumanist philosophers. I claim that they deploy an ontology of power and that they also deploy a kind of theology, as Heidegger meant it. I also describe the way in which this metaphysics begets its own politics and ethics. In order to transcend the human condition, they must transgress the human.

Humanity does not gradually progress from combat to combat until it arrives at universal reciprocity, where the rule of law finally replaces warfare; humanity installs each of its violences in a system of rules and thus proceeds from domination to domination. (Foucault, 1984, 85)In this essay, I take a note from Michel Foucault regarding the notion of biopolitics. For Foucault, biopolitics has both repressive and constitutive properties. Foucault's claim is that with the rise of modern government, the state became exceedingly (...) concerned about the body politic, the bodies that make up the polis, including the health of those bodies. However, Giorgio Agamben claims that Foucault and all western political philosophy misses the relationship between power and Sovereignty, with disastrous results and totalizing tendencies. I explore the case of Terri Schiavo claiming that the social conservatives have attempted to politicize bare life in its legal maneuverings, but I also show how the social liberals open an uncontrollable space between life and death. Both the left and the right miss the aporia at the heart of western political philosophy, and bioethics is complicit in the totalizing effects of contemporary medicine. (shrink)

Despite Vancil’s (1979) proclamation over twenty years ago that topoi have been abandoned in argument theory, this essay contends that topoi should have a vital role in contemporary argumentation theory. Four key areas are identified where topoi are (or can be) essential tools for argumentation: Locating argument, building argument, development of critical thinking, and argument pedagogy. As a result, teachers and students of argument can both benefit from a (re)discovery of topoi.

The philosophical problems of liberty may be classified as those of definition, of justification and of distribution. They are so complex that there is a danger of being unable to see the wood for the trees. It may be helpful, therefore, to provide an aerial photograph of a large part of the wood, namely, the liberty ofindividual persons. But it is, of course, a photograph taken from an individual point of view, as Leibniz would have put it.