Online research in philosophy

Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especially that of Friedman et al. and Feferman et al. Some recent results (of Pour-El and Richards and of the author) are then presented bearing on the first question on the need for non-constructive analysis, especially for quantum physics. Despite the impressive work of Bishop et al. in constructive analysis, Hilbert's objection to intuitionism still carries significant force, and may be decisive depending in part on one's conception of "physics".

In a series of recent papers, Walter Sinnott-Armstrong has developed a novel argument against moral intuitionism. I suggest a defense on behalf of the intuitionist against Sinnott-Armstrong’s objections. Rather than focus on the main premises of his argument, I instead examine the way in which Sinnott-Armstrong construes the intuitionistic position. I claim that Sinnott-Armstrong’s understanding of intuitionism is mistaken. In particular, I argue that Sinnott-Armstrong mischaracterizes non-inferentiality as it figures in intuitionism. To the extent that Sinnott-Armstrong’s account of intuitionism has been adopted by others uncritically, intuitionists have cause for concern. I develop an alternative, and more accurate, reading of what is non-inferential about intuitionistic moral knowledge. In light of this alternative reading, certain elements of Sinnott-Armstrong’s case against intuitionism are significantly weakened. But perhaps more importantly, this paper helps clarify what circumspect intuitionists mean when they claim that some moral knowledge is non-inferential.

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.

This paper outlines and defends a moderate intuitionism. The point of departure is the intuitionism of W. D. Ross (1930) in The Right and the Good, conceived as ethically pluralist and epistemologically rationalist. The paper articulates a conception of self-evidence – including mediate as well as immediate kinds – appropriate to a moderate intuitionism, explores some of the resources and varieties of that position, and considers some problems and prospects for a rationalist version of intuitionism. The final section addresses the issue of how best to conceive the nature and grounds of prima facie duty, the problem of whether intuitionism can adequately deal with conflicts of prima facie duties, and the question of how satisfactorily a moderate intuitionism can account for the epistemic status of moral judgments of overall duty and their connection with rational action.

According to moral intuitionism, moral properties are objective, but our cognitions of them are not always based on premises. In this paper, I develop a novel version of moral intuitionism and argue that this new intuitionism is worthy of closer attention. The intuitionistic theory I propose, while inspired by the early twentieth-century intuitionism of W. D. Ross, avoids the alleged errors of his view. Furthermore, unlike Robert Audi's contemporary formulation of intuitionism, my theory has the resources to account for the noninferential character of particular, as opposed to merely general, moral beliefs. I achieve this result by avoiding the appeal to self-evidence to explain the possibility of noninferential moral knowledge.

We show, within the framework of Bishop's constructive mathematics, that (sequential) completeness of the locally convex space $\mathcal{D} (\mathbb{R})$ of test functions is equivalent to the principle BD-N which holds in classical mathemtatics, Brouwer's intuitionism and Markov's constructive recursive mathematics, but does not hold in Bishop's constructivism.

The main ideas behind Brouwer’s philosophy of Intuitionism are presented. Then some critical remarks against Intuitionism made by William Tait in “Against Intuitionism” [Journal of Philosophical Logic, 12, 173–195] are answered.

This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.