Problems are defined as abstract procedures. An explication of procedures as used in Transparent Intensional Logic and called constructions is presented and the subclass of constructions called concepts is defined. Concepts as closed constructions modulo α- and η-conversion can be associated with meaningful expressions of a natural or professional language in harmony with Church’s conception. Thus every meaningful expression expresses a concept. Since every problem can be unambiguously determined by a concept we can state that every problem is a concept (...) and every concept can be viewed as a problem.Kolmogorov’s idea of a connection between problems and Heyting’s calculus is examined and the non-classical features of the latter are shown to be compatible with realistic logic using partial functions. (shrink)

The core doctrine of ethical intuitionism is that some of our ethical knowledge is non-inferential. Against this, Sturgeon has recently objected that if ethical intuitionists accept a certain plausible rationale for the autonomy of ethics, then their foundationalism commits them to an implausible epistemology outside ethics. I show that irrespective of whether ethical intuitionists take non-inferential ethical knowledge to be a priori or a posteriori, their commitment to the autonomy of ethics and foundationalism does not entail any implausible non-inferential (...) knowledge in areas outside ethics (such as the past, the future, or the unobservable). However, each form of intuitionism does require a controversial stand on certain unresolved issues outside ethics. (shrink)

Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation of (...) mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict. It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century. (shrink)

The shape of contemporary ethics owes a great deal to Henry Sidgwick, through his influence on Rawls, Parfit, and others. No one who reads David Phillips’s outstanding book can be left in the slightest doubt about Sidgwick’s continuing significance for both metaethics and normative ethics. Phillips’s scholarship and his substantive arguments are powerful and insightful, and I find them largely persuasive. So in these remarks I intend merely to raise a few questions about each of his four main..

The paper pores over the recent conceptions of normative judgement developed against the background of advances in psychology and neuroscience. It begins by analyzing what normative claim of morality and law consists of before presenting and criticizing the Social Intuitionist Model of normative judgement developed by Jonathan Haidt. The model poses serious challenges for well-established normative concepts, and the concept of normativity as objective reason for action in particular. A question is asked of what the relationship between philosophical conceptions and (...) the findings of neuroscience should be. The conclusion is that, in the face of new scientific facts, philosophers have no choice but to carefully revise their conceptual schemes. (shrink)

This paper aims to extend the dialogue between social intuitionism and the genetic perspectives of moral psychology, pointing out the contributions and limitations of each one to advance in the understanding of the formulation and transformation of moral judgments. An examination of how the relations between the subject and the object of knowledge have been approached in the light of the contributions of constructivist psychological tradition has been proposed. The relations between emotions, reasoning, and the specific social situation in (...) which such cognitive processes take place it will be examined, taking up particularly the contributions of cultural psychology. By critically examining the three theoretical perspectives, it was possible to point out their contributions and limitations, as well as to appreciate the undoubted contribution of social intuitionism to moral psychology, its limitations and to elucidate the theses that shape this theory. It is precisely by examining the relationships, convergences and profound differences between social intuitionism, constructivism, and cultural psychology in their approaches to moral judgment that it is possible to contribute to promoting their revision and elucidation. (shrink)

The aim of this paper is to defend moral intuitionism, in its new formulations, against the criticism that there is something objectionably non-natural about its conception of moral properties. The force of this complaint depends crucially on what it means to be a non-natural property. I consider a number of ways of drawing the natural/non-natural distinction and argue that, once the notion of 'non-natural property' is sufficiently clarified, it fails to figure in a compelling argument against moral intuitionism.

Fred Seddon answers Michael Huemer's reply, focusing on two central issues in ethics: foundationalism and relativism. On the latter, he argues that Huemer neglects Rand's metaethics and her relational notion of the good.

When Heidegger insists that each of us is distinctive because “the most radical individuation” is both possible and necessary for us, he might mean: it is possible and necessary to be an individual in the most radical way; or it is possible and necessary to engage in the project of becoming a distinct individual in the most radical way; or it is possible and necessary to see the distinct individual that I am, and to do so in the most radical (...) way. Although all three readings are possible and defensible, I’d like to explore in this paper a -type reading of Heidegger’s claim. What would it mean to see the distinct individual that I am in the most radical way? (shrink)

This book publishes for the first time notes from two notebooks of Gödel which formed the basis of a course on intuitionism Gödel delivered at Princeton in the.

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics (...) such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

Emmy Noether’s many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan’s style of mathematics for Hilbert’s, this note shows her combining them in a way she continued throughout her mature abstract algebra.

Over the course of human history there appears to have been a global shift in moral values towards a broadly ‘liberal’ orientation. Huemer argues that this shift better accords with a realist than an antirealist metaethics: it is best explained by the discovery of mind-independent truths through intuition. In this article I argue, contra Huemer, that the historical data are better explained assuming the truth of moral antirealism. Realism does not fit the data as well as Huemer suggests, whereas antirealists (...) have underappreciated resources to explain the relevant historical dynamics. These resources include an appeal to socialization, to technological and economical convergences, to lessons learned from history, to changes induced by consistency reasoning and to the social function of moral norms in overcoming some of the cooperation problems that globalizing societies face. I point out that the realist’s explanans has multiple shortcomings, that the antirealist’s explanans has several explanatory virtues, and conclude that the latter provides a superior account of the historical shift towards liberal values. (shrink)

Any "classical" theory of truth will satisfy tarski's criterion ("p" is true if and only if p), And the principle of bivalence (every proposition is either true or false). A non-Classical theory may be obtained by rejecting these principles: - in fact it is shown that rejection of the second entails rejection of the first. If the resulting non-Classical theory is formalized, A system structurally isomorphic to either s4 or s5 is obtained. An attempt is made to show that the (...) essential insights of intuitionist logic may be preserved if we replace their complex and in many respects "unintuitive" propositional logic by a theory of truth which is non-Classical in the sense described above. (shrink)

This book defends a form of ethical intuitionism, according to which (i) there are objective moral truths; (ii) we know some of these truths through a kind of immediate, intellectual awareness, or "intuition"; and (iii) our knowledge of moral truths gives us reasons for action independent of our desires. The author rebuts all the major objections to this theory and shows that the alternative theories about the nature of ethics all face grave difficulties.

Vers 1918 Hermann Weyl abandonnait le logicisme et donc la tentative de réduire les mathématiques à la logique et la théorie des ensembles. Au niveau philosophique, ses points de référence furent ensuite Husserl et Fichte. Dans les années 1920 il distingua leurs positions, entre une direction intuitionniste-phénoménologique d’un côté, et formaliste-constructiviste de l’autre. Peu après Weyl, Oskar Becker adopta une distinction similaire. Mais à la différence du phénoménologue Becker, Weyl considérait l’approche active du constructivisme de Fichte comme supérieure à la (...) vision d’essences passive de Husserl. Dans cet essai, je montre ce développement dans la pensée de Weyl. Bien que certains points dans sa réception de Husserl et Fichte ne soient pas soutenables, je vais argumenter en faveur de la distinction fondamentale mentionnée ci-dessus. (shrink)

Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert's style of proof and Gentzen's deductive systems. In this context we call it CED . This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations have led to the "calculational style," an impressive array of (...) techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system , we prove Leibniz's principle for intuitionistic logic and also prove that any valid formula of predicate logic can be proved in I-CED. (shrink)

According to moral intuitionism, at least some moral seeming states are justification-conferring. The primary defense of this view currently comes from advocates of the standard account, who take the justification-conferring power of a moral seeming to be determined by its phenomenological credentials alone. However, the standard account is vulnerable to a problem. In brief, the standard account implies that moral knowledge is seriously undermined by those commonplace moral disagreements in which both agents have equally good phenomenological credentials supporting their (...) disputed moral beliefs. However, it is implausible to think that commonplace disagreement seriously undermines moral knowledge, and thus it is implausible to think that the standard account of moral intuitionism is true. (shrink)

I argue that, given evidence of the factors that tend to distort our intuitions, ethical intuitionists should disown a wide range of common moral intuitions, and that they should typically give preference to abstract, formal intuitions over more substantive ethical intuitions. In place of the common sense morality with which intuitionism has traditionally allied, the suggested approach may lead to a highly revisionary normative ethics.