Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes on to characterize this (...) reality as an ‘ideal reality’ that ‘governs’ the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by a what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views (...) on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism. (shrink)

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...) of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)

This thesis explores the significance of Godel's Theorem for an understanding of law as rules, and of legal adjudication as rule-following. It argues that Godel's Theorem, read through Wittgenstein's understanding of rules and language as a contextual activity, and through Derrida's account of 'undecidability,' offers an alternative account of the relationship of judging to justice. Instead of providing support for the 'indeterminacy' claim, Godel's Theorem illuminates the predicament of undecidability that structures any interpretation and every legal decision, and which constitutes (...) the opening to justice. The first argument in this thesis examines Godel's proof, Wittgenstein's views on rules, and Derrida's undecidability, as manifestations of a common concern with the limits of what can be formalized. The meta-argument examines their misinterpretation and misappropriation within legal theory as a case study of just what they mean about meaning, context, and justice as necessarily co-implicated. (shrink)