We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts (...) are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a (...) consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. Our dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion. (shrink)

Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this kind can be used to understand (a) a Turing machine receiving extra computational power from a physical process, or (b) an experimenter modelled as a Turing machine performing a test of a known physical (...) theory T. Our earlier work was based upon experiments in Newtonian mechanics. Here we extend the scope of the theory of experimental oracles beyond Newtonian mechanics to electrical theory. First, we specify an experiment that measures resistance using a Wheatstone bridge and start to classify the computational power of this experimental oracle using non-uniform complexity classes. Secondly, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on our ability to measure resistance by the Wheatstone bridge. The connection between the algorithm and physical test is mediated by a protocol controlling each query, especially the physical time taken by the experimenter. In our studies we find that physical experiments have an exponential time protocol, this we formulate as a general conjecture. Our theory proposes that measurability in Physics is subject to laws which are co-lateral effects of the limits of computability and computational complexity. (shrink)

We have begun a theory of measurement in which an experimenter and his or her experimental procedure are modeled by algorithms that interact with physical equipment through a simple abstract interface. The theory is based upon using models of physical equipment as oracles to Turing machines. This allows us to investigate the computability and computational complexity of measurement processes. We examine eight different experiments that make measurements and, by introducing the idea of an observable indicator, we identify three distinct forms (...) of measurement process and three types of measurement algorithm. We give axiomatic specifications of three forms of interfaces that enable the three types of experiment to be used as oracles to Turing machines, and lemmas that help certify an experiment satisfies the axiomatic specifications. For experiments that satisfy our axiomatic specifications, we give lower bounds on the computational power of Turing machines in polynomial time using nonuniform complexity classes. These lower bounds break the barrier defined by the Church-Turing Thesis. (shrink)

An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings . To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of (...) a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Original Tao: Inward Training and the Foundations of Taoist MysticismJohn A. TuckerOriginal Tao: Inward Training and the Foundations of Taoist Mysticism. By Harold D. Roth. New York: Columbia University Press, 1999. Pp. v + 268. Hardcover $29.50.Searching for the origins of things remains a perennial favorite of Western scholars. For millennia, this quest has been at the core of innumerable scholarly projects. However, it has had significantly less (...) appeal to Asian thinkers, many of whom are more accustomed to assuming a continual presence or an eternal, cyclical recurrence, rather than an absolute beginning or creation ex nihilo from which all could be traced. Harold Roth's Original Tao: Inward Training and the Foundations of Taoist Mysticism continues this time-honored line of investigation, applying it to Taoist mystical writings, in a search for what Roth calls "the original tao." While such a line of research might appear perfectly reasonable to Western minds, it seems a bit incongruous with much of Taoist thought, especially when the latter is accepted and explained in terms of its own categories. For Taoists, of course, the uncarved block, the great thoroughfare, the tao itself, is the origin of all things, including mystical approaches to it and the human predicament.Given this anti-genealogical outlook, with its radical de-emphasis of texts and historical individuals, it should come as no surprise that the origins of Taoism are as murky, and hints about the same as misleading, as any could possibly be. The legendary figure Lao Tzu, now widely recognized as a fabricated philosopher meant to supply a set of presumably whimsical origins, and thus mock the appeal to them, is a case in point. Another might be the Chuang Tzu's satirically progressive deconstruction of the notion of a "beginning to being" via positing an infinite regress of beginnings that have not yet begun to be.1 Despite this, there remains the determination (and Roth is by no means alone here) to discover what has been hidden, perhaps intentionally. Without meaning to compromise this quest, we nevertheless might well ask whether the search does not somehow impair its own integrity as a [End Page 307] statement about the subject being explored. Put another way, is not the way that can be discussed the true way?Methodologically, Roth's book is "a study in 'textual archaeology' " that seeks to uncover "long-lost or long-overlooked texts and the interpretation of their significance" (p. 3). Its focus is on a collection of philosophical verses titled Nei-yeh (Inward training), which, as Roth notes, A. C. Graham earlier characterized as "possibly the oldest mystical text in China" (p. 2). While basing his analyses in part on textual, historical, and hermeneutic insights that have resulted from the relatively recent discovery of the Ma-wang-tui manuscripts, Roth's purported discovery of "the original tao" emerged not from the literal, archaeological unearthing of a text hitherto unknown, but from a new level of interpretive, historical appreciation of a writing long known to exist, Inward Training.Inward Training had been known, however, as a portion of the Kuan Tzu, a work typically characterized as "legalist" in philosophical orientation. By analyzing Inward Training via a "mystical hermeneutic," Roth suggests that the verses composing it are not just so much ancient metaphysical poetry, but instead derived from a group of devoted practitioners of breathing meditation who conceptualized their discipline through metaphysical notions regarding both the cosmos and human nature. According to Roth, these practitioners subscribed to the "guiding and pulling" methods often associated with "dietary and sexual regimens," practices later criticized in texts such as the Chuang Tzu and Huai-nan Tzu. The cosmology generated by these practitioners, and recorded in Inward Training, was, notably enough, similar to what emerged later in the Lao Tzu, a text mistakenly considered, for millennia, to be the origin of the way. Thus Roth resituates Inward Training, moving it from the more recent peripheries of antiquity to the very headwaters of the ancient tradition, as the original way.In radically rethinking the beginnings of Taoism without invoking the traditionally accepted but apparently ahistorical claims of a Lao-Chuang filiation, Roth proposes a new... (shrink)

John Tucker; IX—Rules Automata and Mathematics, Proceedings of the Aristotelian Society, Volume 70, Issue 1, 1 June 1970, Pages 161–180, https://doi.org/10.1093.

John Tucker; II.—Godel and Epimenides, Proceedings of the Aristotelian Society, Volume 59, Issue 1, 1 June 1959, Pages 25–48, https://doi.org/10.1093/aristoteli.

John Tucker; III—Constructivity and Grammar, Proceedings of the Aristotelian Society, Volume 63, Issue 1, 1 June 1963, Pages 45–66, https://doi.org/10.1093/aris.

John Tucker; X—Constructivity, Consistency, and Natural Languages, Proceedings of the Aristotelian Society, Volume 67, Issue 1, 1 June 1967, Pages 145–168, http.

Ogyû Sorai was one of the greatest philosophers of early modern Japan. This volume, a monumental work of scholarship, offers for the first time in any Western language unabridged and fully annotated translations of Sorai’s masterpieces. The Bendô and Benmei are works of political philosophy that define the theoretical foundation for a leadership exercising total power, the best remedy, in Sorai’s view, for a regime in crisis. The translations are based on the 1740 woodblock edition, the first major edition of (...) these seminal texts published during the Tokugawa period. In his commentary, John Tucker situates the Bendô and Benmei in relation to Neo-Confucianism via what is known as "philosophical lexicography." This genre, which links Sorai’s thinking with Neo-Confucianism, is traced to the early-thirteenth-century Song dynasty text the Xingli ziyi by Chen Beixi. Although Sorai was an unrelenting critic the Neo-Confucian formulations of the great Song synthesizer Zhu Xi, his thinking remained, due to its genre, methodology, and conceptual repertory, essentially a radical revision of Neo-Confucian discourse. Tucker’s introduction also examines the reception of Sorai’s two Ben during the remainder of the Tokugawa, calling attention to radical tendencies in later developments of Sorai’s thought as well as to the increasingly scathing critiques of his "Chinese" approach to philosophy, language, and politics. Finally, it traces the vicissitudes of the two Ben in modern Japanese intellectual history and their role in the formation of the ideas of Meiji intellectuals such as Nishi Amane and Kato Hiroyuki. As before, however, Sorai came under attack—this time for his supposed irreverence toward the throne, the Japanese people, and the imperial nation-state. Though an unpopular philosophy in early twentieth-century Japan, in the postwar years Sorai’s thought was interpreted as an important modernizing force. While it critiques such ideologically grounded attempts to cast Sorai’s Bendô and Benmei as theoretical contributions to political modernization, Tucker’s study nevertheless acknowledges that Sorai’s masterworks, in their concern for language analysis as the way to solve philosophical problems, share significant common ground with the analytic approach to philosophy pioneered by various twentieth-century Anglo-American philosophers. (shrink)

: Most discussions of Yamaga Soko's philosophical development as a Confucian scholar in Tokugawa Japan suggest that in his later years he moved away from Confucianism and toward a religio-philosophical celebration of Japan's supposed uniqueness. It is shown here, however, that Soko's nativism, set forth in his Chucho jijitsu, was later eclipsed by his final philosophical work, the Gengen hakki, wherein he articulated a kind of naturalistic numerology, based vaguely on the Yijing. This shift in Soko's thought can be viewed (...) as a return to Neo-Confucianism, the earliest philosophical paradigm that he had embraced. Moreover, the successive shifts in his thinking can be understood in terms of the vicissitudes of his life, first in his exile to the Kansai area, near the ancient imperial capital of Kyoto, and then later in his pardon and return to Edo, the shogun's capital. Perhaps most importantly, this final shift in Soko's thought reveals that this prominent early modern thinker did reach his philosophical climax not in defiant opposition to Neo-Confucianism, nor in a sustained celebration of Japan's political traditions and their superlative nature, but instead in a return to modes of metaphysics akin to those typically deployed by Neo- Confucians themselves in their attempts to understand the changing nature of the cosmos and their political place within its flux. (shrink)

This study examines Itō Jinsai’s 伊藤仁斎 (1627–1705) criticisms of the Great Learning (C: Daxue 大學 J: Daigaku). Three primary sources are considered: Jinsai’s Shigi sakumon 私擬策問 (Personal Essays, 1668); the Daigaku teihon 大學定本 (The Definitive Text of the Great Learning, manuscript 1685); and his essay, “Daigaku wa Kōshi no isho ni arazaru no ben” 大學非孔氏之遺書辨 (The Great Learning is not a Writing Confucius Transmitted, 1705), appended to his Gomō jigi 語孟字義. The study suggests that Jinsai’s critical inclinations grew from his (...) acceptance of Zhu Xi’s views about the value of doubt for progress in learning. The study also suggests that Jinsai’s thinking on the Great Learning had political implications derived in many respects from Jinsai’s overall approach to philosophizing via analysis of words and their meanings. (shrink)

Written as companion texts, the Bendō 弁道 and the Benmei 弁名 present Ogyū Sorai’s most mature and comprehensive expression of his philosophical thought. Sorai modestly spoke of the texts in a letter to a student, Uno Shirō 宇野士朗, calling them “my humble achievements”. In another letter to a student, Yamagata Shūnan 山県周南, Sorai related that after a prolonged bout with ill-health, he feared passing like the morning dew. Therefore, he took up his writing brush and completed the two works. Sorai (...) added that while more than a millennium had passed since Confucius’ death, the Way had only been clarified in recent times. Yet rather than boast of this, Sorai suggested that his hand in the process had been by heaven’s decree. With the two works, he added that even if he passed away soon, his life would not have been wasted. (shrink)

Shibusawa Eiichi, widely known as the father of Japanese capitalism, was also one of the more outspoken advocates of Confucius’ learning in modern Japan. This paper examines Shibusawa's The Analects and the Abacus in relation to Max Weber's assessment of Confucian cultures and their inability to develop, early on, capitalism. Without making grand claims about Confucianism and capitalism, the paper suggests that Weber's life and thought constitute considerable counterevidence vis‐à‐vis Weber's thesis. The paper also examines Shibusawa's thoughts about China in (...) an effort to offer a historical contextualization of Shibusawa's promotion of Confucius’ thought. (shrink)

Volume one. History -- volume two. Philosophy -- volume three. Religion -- volume four. Translations.

The imperative that Japanese philosophy faces today, I assert, is the imperative of environmental philosophy. It is an imperative that has decidedly global origins and indisputable global significance. In discussing this imperative, I revive some age-old, perhaps idealistic, and even romantic themes from East Asian Confucian thinking in the hopes that they might become more central motifs of Japanese philosophizing, charting a way forward in the wake of Fukushima, toward a more sustainable future. In the process, I critique admixtures of (...) environmentalism and nationalism, seeking to elevate instead an ecologically sound philosophical perspective that is more globally inclined than narrowly nationalistic. (shrink)

This chapter surveys the philosophical vicissitudes of the ancient Confucian classic, the Mencius, in Japanese history, from the earliest references in the eighth century through contemporary times. It highlights the contested, controversial reception of the Mencius which no doubt had virtually everything to do not with its position on human nature but rather its relatively unequivocal readiness to confront the problem of tyrannic government and deal with it in no uncertain terms, remonstrating with those tyrants willing to listen and then, (...) if need be, punishing those pretending to be legitimate rulers. Accompanying its forthright activism, the Mencius also emphasized the role of the people in mediating heaven’s ultimate authority over those in power. Needless to say, the Mencius’s positions on these matters did not make it a popular text with those presuming to have authority and control at the highest levels, though it did have an enduring appeal to politically-minded people who understood fully the Confucian position that rulers should rule virtuously and compassionately as the father and mother of the people. (shrink)

This paper explores Ogyū Sorai’s 荻生徂徠 thinking on the most sensational and controversial incident of eighteenth-century Japan, and perhaps the most well-known in all Japanese history, the forty-seven rōnin incident of 1701–1703. Viewed in relation to his lifework, Sorai’s views on the incident are significant insofar as they reveal the extent to which his philosophical thinking was occasionally shaped decisively by neither ancient Chinese nor later Confucian texts, Neo- or otherwise, but instead by formative life-experiences he had as a youth (...) living in exile. No doubt, Confucian and Neo-Confucian notions, which Sorai knew in-depth, helped him filter, epistemologically, events, issues, and most importantly what he understood to be righteous and just behaviour in a polity. Yet ironically enough, Sorai’s thinking on the rōnin incident shows that however philosophically erudite, cosmopolitan, and urbane he was as an intellectual, his appraisals of things sometimes harked back to rural experiences he had early on with some of the most primitive and foundational expressions of human agency, civilization, and socio-political ethics in early-eighteenth century Japan. In the process, Sorai’s essay reveals, through its later resonance with some Meiji thought, how his views on those same primitive expressions remained relevant even in modern contexts. (shrink)

This study traces the impact of Ch'en Pei-hsi's Hsing-li tzu-i on the rise of philosophical lexicography in Tokugawa Japan . It suggests that the appearance of copies of the 1553 Korean edition of Pei-hsi's Tzu-i, brought to Japan in the wake of Toyotomi Hideyoshi's invasion of Korea , crucially influenced both understandings of and reactions to Neo-Confucianism in Tokugawa Japan. Pei-hsi's Tzu-i, the study relates, served as the literary template for several early Tokugawa works, including Fujiwara Seika's Kana seiri , (...) and Hayashi Razan's Shunkansho and Santokusho . The Tzu-i also went through several Japanese editions, beginning with the 1632 Sino-Japanese edition, based on Razan's kambun rendition of the Tzu-i, a vernacular explication of the text by Razan, the Seiri jigi genkai, published in 1659, an annotated edition of the 1632 Sino-Japanese edition, printed in 1670, and the 1668 publication of a later, significantly variant, Chinese edition of Pei-hsi's Tzu-i. ;Razan's Seiri jigi genkai promoted Neo-Confucian learning as expounded in the Tzu-i, but also called attention to etymologically problematic terms of that learning. The dissertation suggests that Razan's Genkai influenced both the literary form and the philosophical content of Yamaga Soko's Seikyo yoroku , Ito Jinsai's Gomo jigi , and Ogyu Sorai's Benmei , thus linking Razan's critical promotion of Pei-hsi's Tzu-i to a series of conceptually based "School of Ancient Learning" critiques of Neo-Confucianism. (shrink)