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.

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.

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)

We introduce Kripke models for propositional substructural logics with strong negation, and show the completeness theorems for these logics using an extended Ishihara's canonical model construction method. The framework presented can deal with a broad range of substructural logics with strong negation, including a modified version of Nelson's logic N$^-$, Wansing's logic COSPL, and extended versions of Visser's basic propositional logic, positive relevant logics, Corsi's logics and M\'endez's logics.

Ishiharaʼs boundedness principleBD-N was introduced in Ishihara [5] and has turned out to be most useful for constructive analysis, see e.g. Ishihara [6]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from NN to N is sequentially continuous and in which the axiom of choice from NN (...) to N holds. Since the latter is known to be inconsistent with the statement that all functions from NN to N are continuous these models refute BD-N. (shrink)

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.

The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem (...) for series holds but BD-|$\mathbb{N}$|does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD-|$\mathbb {N}$|. We show that this is the case when the property is weak-permutable convergence. (shrink)

The article analyzes literature addressing the problem of reality in modern anti-realist theories. The purpose of the review is to expand the circle of researchers, including not only representatives of analytical philosophy, but also those of the phenomenological tradition, since the principle of phenomenological reduction corresponds to the general conceptual attitude of anti-realists, and in methodological terms, phenomenology more consistently implements the program of anti-realism. The principle of anti-realist philosophy is shown as exemplified in solutions of the “difficult problem of (...) consciousness” within analytical philosophy and also in the development of a phenomenological attitude. Such a combination of two, by far the most respectable, philosophical schools makes it possible to more accurately determine the contribution of theorists of both directions to the theory of reality, to identify problematic nodes, internal disagreements and points of intersection of the ideas of phenomenologists and representatives of analytical philosophy in understanding the nature of reality. The works of modern researchers of the problems of reality R. Berghofer, V. Vasyukov, Ya. Ishihara, V. Ladov, L. Makeeva, R. Pils, A. Fursov, D. Chalmers and many others, as well as the works of E. Husserl, W. Quine, H. Putnam’s works, which have become philosophical classics, allow us to clarify the ideas about reality that have developed in the anti-realist discourse from different sides. New studies of the principle of phenomenological reduction by Yu. Himanka and the origins of the formation of phenomenological philosophy by N.V. Motroshilova make it possible to clarify the contribution of phenomenologists to the anti-realist tradition. (shrink)

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 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.

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.

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.

It has been suggested that preparing to teach and teaching are conditionally effective in enhancing one’s own learning. This paper focuses on interactivity—the level of teacher-student interaction in expected or actual teaching—as the potential key to understanding and controlling the variability in the effectiveness of learning by preparing to teach and teaching. By summarizing and reanalyzing the results of previous studies, I suggest that the learning benefits of studying with the expectation of direct teaching (i.e., teaching a student face-to-face) are (...) greater than those of studying with the expectation of indirect teaching (i.e., teaching a student indirectly by creating a lecture video, providing written explanations, or using other means) and that learning by direct teaching surpasses learning by explaining to oneself or indirect teaching at least after preparing to do so. Next, three candidate explanations for the impact of interactivity are discussed: the advantages of asking and answering questions, obtaining additional information about and from one’s student, and enhancing one’s motivation to process learning material deeply while preparing to teach and teaching. Finally, I conclude with the remaining questions and directions for future research. (shrink)

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)

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

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)

Study of Vaiśeṣika philosophy with special reference to Vaiśeṣikadaśapadārthaśāstra.

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 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.

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.

The question of the existence of cultural universals immediately leads us to the problem of intercultural communication and of so-called incommensurability. Over the last few decades, these topics have been the subject of controversy in the philosophy of science, and the stock of universalism has been falling as a result of the rise of Kuhn’s paradigm theory and Quine’s thesis of the indeterminacy of translation. Nowadays cultural pluralism or relativism is rather dominant among philosophers and has begun to appear plausible, (...) too, from an anthropological or sociological point of view. But philosophically, as is well known, relativism harbors many difficulties and paradoxes. As philosophers, therefore, we cannot accept imprudent relativism just as it is. (shrink)

To explain why students learn effectively by teaching, explaining to others in particular, Koh and colleagues advanced the retrieval practice hypothesis, which attributes the learning benefits entirely to the effect of practicing retrieval, that is, effortfully recalling to-be-taught information for the provision of instructional explanations. After delineating the rationale behind the retrieval practice hypothesis, the current situation of research, and the limitations of the existing approach, this paper proposes three tests for the evaluation of the hypothesis that address whether explaining (...) to others after initial studying surpasses restudying in learning performance, whether the amount of effort to retrieve to-be-taught information from memory moderates the learning effects of explaining to others, and whether the degree of elaboration during retrieval practice positively predicts the outcomes of learning by merely recalling to-be-taught information. Evidence is examined regarding whether each test is passed, and future directions for research on the retrieval practice hypothesis are discussed. (shrink)

はたして科学は万能なのか。19世紀以来の「科学主義」は、現在なお「自然主義」の衣を纏って生き続けている。本書は「科学主義」と「反科学主義」をともに退け、科学を自然というテクストを解読する解釈学的営みと して捉え直す。ハンソンとクーンの「新科学哲学」、クワインの「知識の全体論」、ウィトゲンシュタインの「アスペクト論」を手がかりに、科学哲学に「科学的理性批判」という本来の哲学的課題を担わせることを目指す 。ここに新たに論文3篇を加え、サイエンス・ウォーズや実在論/反実在論など、現代の哲学状況と切り結ぶスリリングな論考。.

古代・中世のアリストテレス的自然観を克服し、信仰や迷信から独立することで17世紀に近代「科学」は誕生した。しかしパラダイム転換はくり返され、20世紀には科学技術に伴うリスクも叫ばれるようになる。科学哲 学の第一人者がこうした決定的な転換点に光をあてながら、知の歴史のダイナミズムへと誘う。科学神話が揺らぐ今だからこそもう一度深く掘り下げる、入門書の決定版。.

はたして科学は万能なのか。19世紀以来の「科学主義」は、現在なお「自然主義」の衣を纏って生き続けている。本書は「科学主義」と「反科学主義」をともに退け、科学を自然というテクストを解読する解釈学的営みと して捉え直す。ハンソンとクーンの「新科学哲学」、クワインの「知識の全体論」、ウィトゲンシュタインの「アスペクト論」を手がかりに、科学哲学に「科学的理性批判」という本来の哲学的課題を担わせることを目指す 。ここに新たに論文3篇を加え、サイエンス・ウォーズや実在論/反実在論など、現代の哲学状況と切り結ぶスリリングな論考。.

The severe accident at the Fukushima-Daiichi nuclear power plant caused by the March 11 Great East Japan Earthquake in 2011 was a typical disaster in the age of “trans-science,” which means the situation that science and politics are closely connected and inseparable. The stage of trans-science requires a philosophy of trans-science instead of a philosophy of science such as logical positivism. I would like to characterize norms for techno-scientists in the risk society as RISK, which includes Regulatory deliberation, Intergenerational ethics, (...) Social accountability and Knowledge-product liability. (shrink)

The March 11, 2011, Great East Japan Earthquake caused extensive damage to the Tōhoku district of Japan and gave rise to many arguments concerning the meaning of “disaster” as well as the road to recovery. In particular, the severe accident of the Fukushima-Daiichi nuclear power plant reminded us of the overconfidence of science and technology. In this article, I will discuss concepts such as “disaster of civilization,” “impermanence,” “betweenness,” and the double structure of the Japanese view of nature.