Mark van Atten

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§1. The continuity principle. There are two principles that lend Brouwer’s mathematics the extra power beyond arithmetic. Both are presented in Brouwer’s writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is(More)
Contents 1. Introduction 426 2. G ¨ odel's position in the 1950's—a stalemate 428 2.1. Inconclusive arguments 428 2.2. Realism and rationalism 430 2.3. Epistemological parity 434 2.4. A way out? 437 3. G ¨ odel's turn to Husserl's transcendental idealism 439 3.1. Varieties of idealism 439 3.2. G ¨ odel and German Idealism 440 3.3. The turn to Husserl's(More)
Peter Aczel: Predicate logic over a type setup There are a variety of closely related notions aimed at capturing the abstract structure of type dependency in the syntax and semantics of dependent type theories. Examples of such notions are category with attributes, category with families, category with display maps, contextual category, comprehension(More)
Kripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C◦ of the classically correct part of I, with an “end of time” axiom ET(More)
David Hilbert opened ‘Axiomatic Thought’ [15] with the observation that ‘the most important bearers of mathematical thought,’ for ‘the benefit of mathematics itself have always [. . . ] cultivated the relations to the domains of physics and the [philosophical] theory of knowledge.’ We have in L.E.J. Brouwer and Kurt Gödel two of those ‘most important(More)
This article shows that an apparent puzzle in finance and accounting is resolved by changing from classical to constructive (more specifically, intuitionistic) mathematics. Our position is that it is unproblematic if real-world actors behave inconsistently with nonconstructive mathematical results. Thus the solution to our puzzle lies in a deeper challenge(More)