The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal topology presents (...) a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

We associate with any game G another game, which is a variant of it, and which we call . Winning strategies for have a lower recursive degree than winning strategies for G: if a player has a winning strategy of recursive degree 1 over G, then it has a recursive winning strategy over , and vice versa. Through we can express in algorithmic form, as a recursive winning strategy, many common proofs of non-constructive Mathematics, namely exactly the theorems of the (...) sub-classical logic Limit Computable Mathematics [6], Hayashi and Nakata [7]). (shrink)

This work concerns constructive aspects of measure theory. By considering metric completions of Boolean algebras – an approach first suggested by Kolmogorov – one can give a very simple construction of e.g. the Lebesgue measure on the unit interval. The integration spaces of Bishop and Cheng turn out to give examples of such Boolean algebras. We analyse next the notion of Borel subsets. We show that the algebra of such subsets can be characterised in a pointfree and constructive way by (...) an initiality condition. We then use our work to define in a purely inductive way the measure of Borel subsets. (shrink)

Originally published in 1963. Can one discern certain regularities in the manoeuvrings and techniques employed by scientists and can these be formulated into the methodological principles of science? What is the origin and basis of such principles? Are they imposed by objective realities, do they derive from conceptual necessities or are they rooted in our own deep seated predilections? This volume investigates these questions and sheds light on the growth mechanism of the evolving structure of science itself.

This volume is a critical edition of the eight-year correspondence between Gottfried Wilhelm Leibniz and Burcher de Volder, professor of philosophy and mathematics at Leiden University. Containing the surviving correspondence between Leibniz and De Volder, the volume also presents a generous selection from the letters between Leibniz and his friend Johann Bernoulli, through whose intercession the correspondence began. Bernoulli acted as intermediary throughout, and the often candid discussions between Leibniz and Bernoulli provide illuminating background to the correspondence proper. Each of (...) the selections appears both in the original Latin and in English translation. (shrink)

The social aspects of causality in medicine and healthcare have been emphasized in recent debates in the philosophy of science as crucial factors that need to be considered to enable, among others, appropriate interventions in public health. Therefore, it seems central to recognize the bearing of social causes (broadly understood, e.g., social inequalities and socio-economic status) in bringing about certain concrete pathologies. Being aware of the relevance of social causes in medicine and healthcare is particularly important in the face of (...) the role that artificial intelligence-based systems (such as machine learning algorithms) are increasingly playing in these high-stakes fields. In fact, these systems bear the dangerous potential of concealing relevant social causes. This is highly problematic not only because it reinforces issues of distributive injustice but also because it can pave the way for issues of epistemic injustice. The central aim of this chapter is to make a first effort to point out possible connections between the importance of recognizing social causes in medicine and healthcare and forms of epistemic injustice. (shrink)