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)

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)

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)

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)

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.

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)

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.

. Because of similarities between some implications of Michael Polanyi's theory of personal knowledge and intelligent design, claims have been made that his theory provides support to the project of intelligent design. This essay contends that, when Polanyi's reflections on a Ideological framework for contextualizing evolutionary biology are properly understood as a heuristic vision, his position contrasts sharply with the empirical claims made on behalf of intelligent design.

. This essay attempts to explore the senses in which religious meanings may be understood to be grounded ontologically and in which they may be validly accepted as true. It begins by outlining Wolfhart Pannenberg’s proposal for conceiving the scientific status of theology and his formulation of the question of theological truth. Then certain epistemological presuppositions are challenged in light of Michael Polanyi’s theory of knowledge. Finally a revised understanding is proposed in Polanyian terms. Here in their primordial sense religious (...) meanings are based in the act of breaking out toward the ground of our tacit foreknowledge. In their primary sense religious symbolizations are accepted as human creations and judged to be valid insofar as they integrate meaningfully all the disparate elements of our experience. (shrink)

This review discusses Weightman's interpretation of Torrance's appropriation of Polanyi's theory of science; Weightman shows how Torrance develops a contemporary “natural”theology, moving beyond Barthian roots, but he argues Torrance misconstrues Polanyi's understanding of “religion” and God. I support Weightman's account, acknowledging much of his argument regarding the nature of religion, but I question whether his constructivist view of God can support the role it must play in Polanyi's thought.

In ‘Wittgenstein and Qualia’ Ned Block argues for the existence of inverted spectra and those ineffable things, qualia. The essence of his discussion is a would-be proof, presented through a series of pictures, of the possible existence of an inverted spectrum. His argument appeals to some remarks by Wittgenstein which, Block holds, commit the former to a certain ‘dangerous scenario’ wherein inverted spectra, and consequently qualia live and breath. I hold that a key premise of this proof is incoherent. Furthermore, (...) Block’s dangerous scenario does not follow from Wittgenstein’s innocent one, as Block believes it does, but rather is in conflict with it. (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.

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.

These remarks are comments on Tihamér Margitay’s criticisms of Polanyi’s so-called “correspondence thesis” in his recent essay “From Epistemology to Ontology.”.

This review essay argues that the emphasis on the personal commitments sustaining all knowledge, while permitting some fruitful insights into structural parallels between Newman's and Polanyi’s epistemological positions, finally is not fully satisfactory for developing a theological program. Moleski’s effort to develop such theological insights may be advanced if it were supplemented by incorporating a more detailed structural analysis of the illative sense and of tacit knowing.

The aim of this essay is to display a congruence between several important features of Augustine’s theory of knowledge, including our knowledge of the world (sapientia) and our knowledge of the standards guiding our thought (sapientia), and Michael Polanyi’s theory of personal knowledge. Its purpose is to commendan interpretation of Polanyi’s thought which situates his major insights within an Augustinian intellectual tradition and which thereby offers fruitful possibilities for theological reflection, particularly on the reality of God.

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

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)

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 deals with some large tracts of Wittgenstein’s writings concerning representation and the mental. Its defining characteristic, and one of its main strengths, is an extensive use of material in the background of Wittgenstein’s Tractatus and Investigations. Stern quotes from and discusses remarks from unpublished manuscripts, including the Big Typescript, little-studied published writings such as the Tractatus notebooks, “Some Remarks on Logical Form,” Philosophical Remarks, Philosophical Grammar, as well as lecture notes by Moore, King and Lee, and others. How (...) much of these writings the book reproduces is suggested by the fact that its appendix, giving the original German of the previously unpublished matter translated in the main text, is eleven small-print pages long; and there are far more quotations from the published part of the corpus. The result is a work in which, for a run of pages, the ratio of quoted material to comment can be as high as one to one ; and in which most pages contain a significant amount of indented quotation. The main justification offered for the wholesale inclusion of this material is that it provides a context that can help determine what problem Wittgenstein was addressing in a given passage from his two main works. Stern’s mastery of that material is impressive, as shown for example in his pellucid outline of Wittgenstein’s philosophical development after 1920. (shrink)