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)

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)

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)

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)

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)

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)

The nature of representation is a central topic in philosophy. This is the first book to connect problems with understanding representational artifacts, like pictures, diagrams, and inscriptions, to the philosophies of science, mind, and art. Can images be a source of knowledge? Are images merely conventional signs, like words? What is the relationship between the observer and the observed? In this clear and stimulating introduction to the problem John V. Kulvicki explores these questions and more. He discusses: the nature (...) of pictorial experience and "seeing in" recognition, resemblance, pretense, and structural theories of depiction images as aids to scientific discovery and understanding mental imagery and the nature of perceptual content photographs as visual prostheses. In so doing he assesses central problems in the philosophy of images, such as how objects we make come to represent other things, and how we distinguish kinds of representation - pictures, diagrams, graphs - from one another. Essential reading for students and professional philosophers alike, the book also contains chapter summaries, annotated further reading, and a glossary. (shrink)

John Kulvicki explores the many ways in which pictures can be meaningful, taking inspiration from the philosophy of language. Pictures are important parts of communicative acts. They express a variety of thoughts, and they are also representations. Kulvicki shows how the meanings of pictures let us put them to a wide range of communicative uses.

This paper seeks to highlight the importance of spatial cognition in Bergson’s Données immédiates by engaging with Robert Watt’s reconstruction of Bergson’s argument that every idea of number involves the idea of space. We focus on the second stage of Watt’s reconstruction, where Bergson argues that only space can provide the distinction required for our counting of otherwise identical items. Watt bases his reconstruction on a premise regarding the possibility that identical objects, in the absence of spatial distinction, might remain (...) identical across different “temporal locations”. Our paper raises the prospect that Bergson is committed to a stronger thesis, namely one implying that identical objects would necessarily remain indistinguishable without the intervention of space. The paper thus concludes by emphasizing the indispensability of space for knowledge according to Bergson. (shrink)

I can’t help but like a book that calls Wittgenstein the greatest philosopher since Kant and then proceeds to show how On Certainty, a manifestly brilliant but understudied book, sheds light on matters under current debate. It is pleasant to see a highly skilled contemporary put texts from the later philosophy under close scrutiny and mine them for insight, and that outside the bounds of familiar Wittgenstein scholarship.

ABSTRACT Historians of science, inasmuch as they are concerned with knowledges and practices rather than institutions, have tended of late to focus on case studies of common processes such as experiment and publication. In so doing, they tend to treat science as a single category, with various local instantiations. Or, alternatively, they relate cases to their specific local contexts. In neither approach do the cases or their contexts build easily into broader histories, reconstructing changing knowledge practices across time and space. (...) This essay argues that by systematically deconstructing the practices of science and technology and medicine (STM) into common, recurrent elements, we can gain usefully “configurational” views, not just of particular cases and contexts but of synchronic variety and diachronic changes, both short term and long. To this end, we can begin with the customary actors’ disciplines of early modern knowledge (natural philosophy, natural history, mixed mathematics, and experimental philosophy), which can be understood as elemental “ways of knowing and working,” variously combined and disputed. I argue that these same working knowledges, together with a later mode—synthetic experimentation and systematic invention—may also serve for the analysis of STM from the late eighteenth century to the present. The old divisions continued explicitly and importantly after circa 1800, but they were also “built into” an array of new sciences. This historiographic analysis can help clarify a number of common problems: about the multiplicity of the sciences, the importance of various styles in science, and the relations between science and technology and medicine. It suggests new readings of major changes in STM, including the first and second scientific revolutions and the transformations of biomedicine from the later twentieth century. It offers ways of recasting both microhistories and macrohistories, so reducing the apparent distance between them. And it may thus facilitate both more constructive uses of case studies and more innovative and acceptable longer histories. (shrink)

Saul Kripke, among others, reads Wittgenstein’s private-language argument as an inference from the idea of rule following: The concept of a private language is inconsistent, because using language entails following rules, and following rules entails being a member of a community. Kripke expresses the key exegetical claim underlying that reading as follows.

This book is a philosophical examination of the main stages in our journey from hominid to human. It deals with the nature and origin of language, the self, self-consciousness, and the religious ideal of a return to Eden. It approaches these topics through a philosophical anthropology derived from the later writings of Wittgenstein. The result is an account of our place in nature consistent with both a hard-headed empiricism and a this-worldy but religiously significant mysticism.

Saul Kripke, among others, reads Wittgenstein’s private-language argument as an inference from the idea of rule following: The concept of a private language is inconsistent, because using language entails following rules, and following rules entails being a member of a community. Kripke expresses the key exegetical claim underlying that reading as follows.

In his commentary on Euclid, Proclus develops what he takes to be an important Platonic critique of the epistemology of abstraction. As I argue, his argument closely reflects terminology and concepts from Plato’s Philebus. Both emphasize the priority—in reality and in our awareness—of the precise over the imprecise. Specifically, Proclus’s famous notion of the psychical “projection” of intermediate mathematical entities, while having no technically exact precedent in Plato, finds a conceptual neighbor in the Philebus’s suggestion that philosophical arithmeticians “posit” pure (...) units for counting. Likewise, for both our self-engagement in mathematical thinking serves to clarify the independence of the precise sciences—both in their content and in their practice—from perception. Thus, as I argue, Plato and Proclus, with their different terms and nuances, develop a shared conception scientific inquiry in which an activity of “creative discovery” plays a central role. (shrink)

Wittgenstein's later philosophy and the doctrines of Mahayana Buddhism integral to Zen coincide in a fundamental aspect: for Wittgenstein language has, one might say, a mystical base; and this base is exactly the Buddhist ideal of acting with a mind empty of thought. My aim is to establish and explore this phenomenon. The result should be both a deeper understanding of Wittgenstein and the removal of a philosophical objection to Zen that has troubled some people.

This study examines Plato's dialogue on the good life and argues, most centrally, that the "pleasures of learning" exemplify, for Socrates, the possibility of good becoming or change.

Health reform debate understandably focuses on large system design. We should not omit attention to the “last mile” problem of physician payment theory. Achieving fundamental goals of integrative, patient-centered primary care depends on thoughtful financial support. This commentary describes the nature and importance of innovative primary care payment programs.

In the 'Hippias Major' Socrates uses a counter-example to oppose Hippias‘s view that parts and wholes always have a "continuous" nature. Socrates argues, for example, that even-numbered groups might be made of parts with the opposite character, i.e. odd. As Gadamer has shown, Socrates often uses such examples as a model for understanding language and definitions: numbers and definitions both draw disparate elements into a sum-whole differing from the parts. In this paper I follow Gadamer‘s suggestion that we should focus (...) on the parallel between numbers and definitions in Platonic thought. However, I offer a different interpretation of the lesson implicit in Socrates‘s opposition to Hippias. I argue that, according to Socrates, parts and sum-wholes may share in essential attributes; yet this unity or continuity is neither necessary, as Hippias suggests, nor is it impossible, as Gadamer implies. In closing, I suggest that this seemingly minor difference in logical interpretation has important implications for how we should understand the structure of human communities in a Platonic context. (shrink)

1. The early philosophy--language as picture -- 2. Logic and ontology -- 3. "My world and its value" -- 4. The later philosophy--views and reviews -- 5. Method and essense -- 6. Meaning -- 7. Criteria -- 8. Knowing, naming, certainty, and idealism -- 9. The private language argument -- 10. Logical necessity and rules -- 11. Philosophy of mathematics -- 12. Persons -- 13. Psychology and conceptual relativity -- 14. Aesthetics, ethics, and religion -- 15. Elective affinities.

While historians of scientific method have recently called attention to the views of many of John Stuart Mill's contemporaries on the relation between probability and inductive inference, little if any note has been taken of Mill's own vigorous attack on the received "Laplacean" interpretation of probability in the first edition of the System of Logic. This paper examines the place of Mill's critique, both in the overall framework of his philosophy, and in the tradition of assessing the so-called "probability (...) of causes". It also offers an account of why, in later editions of the work, Mill appears to adopt a much more sympathetic stance toward the received view. (shrink)

Language Games 2 This chapter provides some background necessary for subsequent discussions by sketching in the idea of a language game, thereby giving a ...