Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Predicativity and parametric polymorphism of Brouwerian implication

A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten

Design: One, but in different forms

This overview paper defends an augmented cognitively oriented generic-design hypothesis: there are both significant similarities between the design activities implemented in different situations and crucial differences between these and other cognitive activities; yet, characteristics of a design situation (related to the design process, the designers, and the artefact) introduce specificities in the corresponding cognitive activities and structures that are used, and in the resulting designs. We thus augment the classical generic-design hypothesis with that of different forms of designing. We review the data available in the cognitive design research literature and propose a series of candidates underlying such forms of design, outlining a number of directions requiring further elaboration

What's Right With a Syntactic Approach to Theories and Models?

Syntactic approaches in the philosophy of science, which are based on formalizations in predicate logic, are often considered in principle inferior to semantic approaches, which are based on formalizations with the help of structures. To compare the two kinds of approach, I identify some ambiguities in common semantic accounts and explicate the concept of a structure in a way that avoids hidden references to a specific vocabulary. From there, I argue that contrary to common opinion (i) unintended models do not pose a significant problem for syntactic approaches to scientific theories, (ii) syntactic approaches can be at least as language independent as semantic ones, and (iii) in syntactic approaches, scientific theories can be as well connected to the world as in semantic ones. Based on these results, I argue that syntactic and semantic approaches fare equally well when it comes to (iv) ease of application, (iv) accommodating the use of models in the sciences, and (vi) capturing the theory-observation relation

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets