コンピュータの歴史、アルゴリズムの理論の解説を経て、未解決であるミレニアム問題のひとつ、「P≠NP問題」に迫ります!

Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions (...) and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

Recently we proposed "quantum language" (or, the linguistic Copenhagen interpretation of quantum mechanics"), which was not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the linguistic turn of Descartes=Kant epistemology. We believe that quantum language is the language to describe science, which is the final goal of dualistic idealism. Hence there is a reason to want to clarify, from the quantum linguistic point of view, the following problems: "brain in a vat argument", "the Cogito proposition", (...) "five-minute hypothesis", "only the present exists", "Copernican revolution", "McTaggart's paradox", and so on. In this paper, these will be discussed and clarified in quantum language. That is, these are not in quantum language. Also we emphasize that Leibniz's relationalism in Leibniz-Clarke correspondence is regarded as one of the most important parts of the linguistic Copenhagen interpretation of quantum mechanics. This paper is the revised version of the paper: Open Journal of Philosophy, 2018 Vol.8, No.5, 466-480). (shrink)

The studies on the cAMP‐requiring mutants and their suppressors in the yeast, Saccharomyces cerevisiae, revealed that cAMP‐dependent protein phosphorylation is involved in the G1 phase of the cell cycle, in conjugation, and in the post‐meiotic stage of sporulation, and that inhibition of cAMP‐dependent protein phosphorylation is required to induce meiotic division.

メタ倫理学、規範倫理学、応用倫理学を体系的に概説した入門書。大学における倫理学教育のミニマム・エッセンスを体系的に学べるユニークなテキスト。.

Web-based learning resources provide learners with hyperspace where they can navigate in a self-directed way to learn the contents included in the Web pages. The navigation involves making a sequence of the pages visited, which is called navigation path. However, learners often fail in making the navigation path due to a cognitive overload, which is caused by diverse cognitive efforts at comprehending the contents in Web pages, and monitoring the navigation process such as planning and reflection of navigation path. In (...) the self-directed learning, in particular, it is difficult for learners to maintain the navigation monitoring. Focusing on navigation planning, this paper addresses the issue of how to facilitate learners' navigation monitoring to promote their learning in hyperspace provided by Web-based learning resources. Our approach to this issue is to provide learners with a space, in which they can see through the learning resources to plan a navigation path, apart from hyperspace. In this paper, we also demonstrate an assistant system for the navigation planning, which is composed of hyperspace map, page previewer, and path previewer. These facilities give learners an overview of Web pages and navigation path to help them make a navigation path plan without visiting Web pages in hyperspace. This paper also describes a case study with the assistant system. The results indicate that the system facilitates navigation particularly in a more complicated hyperspace. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing.

RésuméDans la partie duŠifāʾqui porte “sur les plantes”, Avicenne aboutit à cette conclusion inattendue que celles-ci ne seraient pas vivantes. Cette thèse surprend étant donné l'opinion d'Aristote voulant que tout ce qui a une âme est vivant. Cet article montre qu'Avicenne a évolué quant à la question de la vie des plantes. Il commence par adopter la conception aristotélicienne selon laquelle les plantes sont vivantes dans la mesure où elles sont dotées d'une âme, comme il ressort de son œuvre précoce (...) qu'est leCompendium sur l'âme.Ensuite cependant, une fois qu'Avicenne a analysé les facultés de l'âme comme il le fait principalement dans sonCanon de Médecine, il est conduit à conclure que les plantes ne pouvaient pas être dites vivantes. Cet article illustre ainsi l'intérêt qu'il y a, lorsqu'on étudie le développement de la pensée d'Avicenne, à prendre en compte l'impact de ses découvertes médicales sur ses réflexions philosophiques. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...) are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (shrink)

We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

Recently we proposed the linguistic Copenhagen interpretation of quantum mechanics, which is called quantum language or measurement theory. This theory is valid for both quantum and classical systems. Thus, we think that quantum language is one of the most powerful scientific theories, like statistics, and thus, it is the scientific completion (i.e., the destination) of dualistic idealism. If so, we can introduce the concept “progress” in the dualistic idealism. For example, we can assert that [Plato → Descartes → Kant → (...) Wittgenstein → quantum language], where [“X” → “Y”] means that “Y” is more like quantum language than “X”. In this paper, we will study the problem of universals from the perspective of quantum language (i.e., from the scientific perspective of ignoring any religious perspective). And we can be confident of the progress of both of the two time series [Plato → Anselmus → Thomas Aquinas → quantum language] and [Descartes → Thomas Aquinas → quantum language] in dualistic idealism. The reader may find it surprising that Scholastic philosophy is more scientific than Cartesian-Kantian philosophy. However, this is because Descartes gave up the pursuit of “universals” and presented dualism as a visible “mind-matter dualism” so that it could be familiar to the general public. This made the Cartesian-Kantian philosophy socially successful, but unscientific. The problem of universals has not always been clear in the long history of philosophy. The reader should be convinced that the reason is that the problem of universals has been discussed in an incomplete non-scientific dualism to this day. (shrink)

Paul Erdős was a mathematicianpar excellencewhose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. Hismodus operandiwas to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressing and ever reaching, and hismodus vivendiwas to be itinerant (...) in the world, stimulating and interacting about mathematics at every port and capital. (shrink)

Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse. This article address the historical emergence of the infinite in set theory, and how we are to take the (...) infinite in and out of mathematics.The first section discusses the emergence of the infinite as a matter of method in set theory. The second section discusses the infinite in and out of mathematics, and how it is to be taken. (shrink)

Azriel Levy did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions (...) to set theory. (shrink)

Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership (...) distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. (shrink)

This study examined how different components of working memory are involved in the acquisition of egocentric and allocentric survey knowledge by people with a good and poor sense of direction (SOD). We employed a dual-task method and asked participants to learn routes from videos with verbal, visual, and spatial interference tasks and without any interference. Results showed that people with a good SOD encoded and integrated knowledge about landmarks and routes into egocentric survey knowledge in verbal and spatial working memory, (...) which is then transformed into allocentric survey knowledge with the support of all three components, distances being processed in verbal and spatial working memory and directions in visual and spatial working memory. In contrast, people with a poor SOD relied on verbal working memory and lacked spatial processing, thus failing to acquire accurate survey knowledge. Based on the results, a possible model for explaining individual differences in spatial knowledge acquisition is proposed. (shrink)

The author reflects on his long career as an educational psychologist and on the role of literature in his vision of psychological science. The author followed with great interest the major developments in psychology around the world, but he felt himself progressively alienated from the revealing power of art and literature. At one moment, he realized that a simple narrative constitutes the most profound and also the most effective means of transmitting genuine insights about teaching from one generation to another. (...) This narrative reveals a career guided by high ideals and compromised by the harsh realities of the discipline of psychology. (shrink)

This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory.