We will overview the results in an informal approach to constructive reverse mathematics, that is reverse mathematics in Bishop’s constructive mathematics, especially focusing on compactness properties and continuous properties.

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that "every mapping is strongly extensional", "every sequentially nondiscontinuous mapping is sequentially continuous", and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tsĕitin theorem.

We show, within the framework of Bishop's constructive mathematics, that (sequential) completeness of the locally convex space $\mathcal{D} (\mathbb{R})$ of test functions is equivalent to the principle BD-N which holds in classical mathemtatics, Brouwer's intuitionism and Markov's constructive recursive mathematics, but does not hold in Bishop's constructivism.

Classically, weak König's lemma and Brouwer's fan theorem for detachable bars are equivalent. We give a direct constructive proof that the former implies the latter.

Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, (...) completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Löf’s type theory, called \, where mTT can be easily interpreted. Then to show the consistency of \ with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints \ by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment \ we interpret consists of first order intensional Martin-Löf’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors -rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of \ correctly. In particular the universe of \ is modelled by means of \-fixpoints following a technique due first to Aczel and used by Feferman and Beeson. (shrink)

We extend Bishopʼs concept of function spaces to the concept of pre-function spaces. We show that there is an adjunction between the category of neighbourhood spaces and the category of Φ-closed pre-function spaces. We also show that there is an adjunction between the category of uniform spaces and the category of Ψ-closed pre-function spaces.

In this essay I present the philosophy of Nishida Kitarō, the father of the Kyoto School tradition, as an alternative version of transcendental philosophy. While following the steps of Kant and Husserl in important ways, Nishida attempts to take the transcendental approach even further, finally leaving us with a different view of reality that is neither realist nor idealist, but, rather, radically realist. I call this radical transcendental position, “awakened realism”. Due to word limitations, I can only set the stage (...) for further discussion on what kind of implication this may have for transcendental philosophy. But if I succeed in making my presentation clear enough, it should naturally ignite discussion on how far we can radicalize transcendental philosophy without making it into something else. (shrink)

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.

We give a variant of the Gödel-Gentzen-negative translation, and a syntactic characterization which entails conservativity result for formulas.

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{2}^{0}}$$\end{document}-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.

Medical assistance in dying (MAiD) was legalized in Canada following the Carter v. Canada ruling of 2015. In spite of legalization, the ethics of MAiD remain contentious. The bioethical literature has attempted to differentiate MAiD from withdrawing life-sustaining treatment (WLT) in an effort to examine the nature of the moral difference between the two. However, this research has often neglected the firsthand experiences of the clinicians involved in these procedures. By asking physicians if they perceive the major bioethical accounts as (...) clinically useful, we seek to distinguish between aspects of the contemporary bioethical landscape which are useful at the bedside and those which are divorced from the realities faced by clinicians. We applied a qualitative descriptive approach to explore physicians’ experiences and bioethical distinctions in providing MAiD and WLT. Semi-structured interviews were conducted with 21 physicians, and the transcripts were thematically analyzed to identify common patterns and divergences in their perspectives. Three core themes were found: (1) consensus on MAiD’s moral equivalence with WLT despite differences between the practice, (2) discord regarding the use of the term ‘killing’, and (3) disjuncture between bioethical debates and practice. Theme 1 comprised of three sub-themes: (1.1) no moral difference between MAiD and WLT, (1.2) physician versus underlying medical condition as cause of death, and (1.3) relief of suffering. In order to have practical utility for clinical practice, it is essential for bioethicists to engage in dialogue with patients and their medical providers pursuing MAiD or WLT. Theoretical debates that are divorced from the realities of terminal illness do not assist physicians with navigating the ethical terrain of ending a patient’s life. This research captures meaningful accounts regarding MAiD and WLT that is rooted in the lived experience of the providers of these services in order for bioethical debates to have substantive impact in clinical practice and in legislation surrounding future health policies. (shrink)

Several studies have shown that our visual system may construct a “summary statistical representation” over groups of visual objects. Although there is a general understanding that human observers can accurately represent sets of a variety of features, many questions on how summary statistics, such as an average, are computed remain unanswered. This study investigated sampling properties of visual information used by human observers to extract two types of summary statistics of item sets, average and variance. We presented three models of (...) ideal observers to extract the summary statistics: a global sampling model without sampling noise, global sampling model with sampling noise, and limited sampling model. We compared the performance of an ideal observer of each model with that of human observers using statistical efficiency analysis. Results suggest that summary statistics of items in a set may be computed without representing individual items, which makes it possible to discard the limited sampling account. Moreover, the extraction of summary statistics may not necessarily require the representation of individual objects with focused attention when the sets of items are larger than 4. (shrink)

The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the predicative methods available in constructive type theory and constructive set theory.

In this paper, I argue that later Nishida's analysis of self-awareness (jikaku) provides a new perspective on the nature of self-awareness as understood in the philosophical literature today. I argue that the contemporary literature deals with two kinds of self-awareness; the higher-order theory understands self-awareness to be an objectified awareness and the phenomenological tradition generally understands self-awareness to be, at least primarily, a non-objectified awareness. In light of this, I first give an account of Nishida's ?acting-intuition? with reference to the (...) ?historical body.? Then I argue that later Nishida's analysis of self-awareness depicts the inadequacy of the two kinds of self-awareness as both being stuck in the subject-object dichotomy which, according to Nishida, is but one mode of experience, namely the mode of difference. Nishida provides an account that ultimately foregrounds the experiential mode of unity that negates any sense of differentiation between the subject and object. (shrink)

In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T.

We deal with a restricted form WC-N' of the weak continuity principle, a version BT' of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N'. BT' + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.

精神障害の分類と診断が難しいなか、哲学や精神医学は精神障害とどのように向きあうのか。思考吹入や妄想についての哲学的考察に加え、現象学的精神病理学や精神分析の現状、そして現代の精神医学のさまざまなアプロ ーチの到達点を総観する。.

In this note, we prove that certain theories of (many‐sorted) intuitionistic predicate logic are closed under the independence of premiss rule (IPR). As corollaries, we show that and extended by some non‐classical axioms and non‐constructive axioms are closed under IPR.

Consider the problem which set V of propositional variables suffices for whenever, where, and ⊢c and ⊢i denote derivability in classical and intuitionistic implicational logic, respectively. We give a direct proof that stability for the final propositional variable of the (implicational) formula A is sufficient; as a corollary one obtains Glivenko's theorem. Conversely, using Glivenko's theorem one can give an alternative proof of our result. As an alternative to stability we then consider the Peirce formula. It is an easy consequence (...) of the result above that adding a single instance of the Peirce formula suffices to move from classical to intuitionistic derivability. Finally we consider the question whether one could do the same for minimal logic. Given a classical derivation of a propositional formula not involving ⊥, which instances of the Peirce formula suffice as additional premises to ensure derivability in minimal logic? We define a set of such Peirce formulas, and show that in general an unbounded number of them is necessary. (shrink)

How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by-product, (...) the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine-Borel property for coverings by countably many possibly empty open balls. (shrink)

We systematically study several principles and give a principle which is weaker than disjunctive Markov’s principle. We also show that the principle is underivable and strictly weaker than MP∨ in certain extensions of the system EL of elementary analysis.

We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.

Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic.

Intercultural Phenomenology explores the nature of reality by engaging in a cross-cultural dialogue between two of the most influential philosophical traditions of the 20th century. Drawing on ideas from phenomenology, Japanese philosophy and Zen Buddhism, it follows the philosophers who changed their perception of the world by choosing to suspend judgement. Guided by this philosophical method known as the "epoché", or suspension of judgment in ancient Greek, it is an introduction to the philosophy and practice of letting objects in the (...) world speak for themselves. (shrink)

The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by Aczel and (...) Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class; in particular, every complete separable metric space automatically is a set. (shrink)

In this paper, we deal with compact operators on a Hilbert space, within the framework of Bishop's constructive mathematics. We characterize the compactness of a bounded linear mapping of a Hilbert space into C n , and prove the theorems: Let A and B be compact operators on a Hilbert space H , let C be an operator on H and let α ϵ C . Then α A is compact, A + B is compact, A ∗ is compact, CA (...) is compact and if C ∗ exists, then AC is compact; An operator on a Hilbert space has an adjoint if and only if it is weakly compact. (shrink)

In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule ) for intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of \ and \ which are closed under \ but do not derive the independence of premiss axiom.

This special issue explores the significance of collectivism in modern and contemporary Japanese art. Japanese artists banded together throughout the twentieth century to work in collectives, reflecting and influencing each evolution of their culture. Illuminating the interplay between individual and community throughout Japan’s tumultuous century, the contributors to this issue examine both the practical internal operations of the collectives and the art that they produced. One contributor studies the art societies of prewar imperial Japan, whose juried art salons defined a (...) new _nihonga_ painting tradition. While recent scholarly work on art produced during World War II has tended to neglect the collectivist tradition, this issue covers wartime groups like the Art Unit for Promoting the Munitions Industry and the important questions they pose about the relationship between artists and the state. Art collectives in post-occupation Japan gained prominence working in the experimental vanguard of the global art scene in painting, sculpture, design, and intermedia projects. Adding a crucial dimension to the study of Japanese art and modernism, this issue explores how these groups attempted to accommodate the creative paradox of individualism within collectivism. _Contributors: Maki Kaneko, Kuroda Raiji, John Szostak, Miwako Tezuka, Ming Tiampo, Reiko Tomii, Alicia Volk, Midori Yoshimoto Reiko Tomii_ is an independent art historian and curator in New York. She is coauthor of _Xu Bing_. _Midori Yoshimoto_ is Associate Professor of Art History and curator of two galleries at New Jersey City University. She is the author of _Into Performance: Japanese Women Artists in New York_. (shrink)

This book provides linguists with a clear, critical, and comprehensive overview of theoretical and experimental work on information structure. Leading researchers survey the main theories of information structure in syntax, phonology, and semantics as well as perspectives from psycholinguistics and other relevant fields. Following the editors' introduction the book is divided into four parts. The first, on theories of and theoretical perspectives on information structure, includes chapters on topic, prosody, and implicature. Part 2 covers a range of current issues in (...) the field, including focus, quantification, and sign languages, while Part 3 is concerned with experimental approaches to information structure, including processes involved in its acquisition and comprehension. The final part contains a series of linguistic case studies drawn from a wide variety of the world's language families. This volume will be the standard guide to current work in information structure and a major point of departure for future research. (shrink)

Constructive mathematics – mathematics in which ‘there exists’ always means ‘we can construct’ – is enjoying a renaissance. Fifty years on from Bishop’s groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject’s myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, (...) constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches. (shrink)