This book offers a comprehensive treatment of the classical decision problem of mathematical logic and of the role of the classical decision problem in modern computer science. The text presents a revealing analysis of the natural order of decidable and undecidable cases and includes a number of simple proofs and exercises.

The theme of this book is formed by a pair of concepts: the concept of formal language as carrier of the precise expression of meaning, facts and problems, and the concept of algorithm or calculus, i.e. a formally operating procedure for the solution of precisely described questions and problems. The book is a unified introduction to the modern theory of these concepts, to the way in which they developed first in mathematical logic and computability theory and later in automata theory, (...) and to the theory of formal languages and complexity theory. Apart from considering the fundamental themes and classical aspects of these areas, the subject matter has been selected to give priority throughout to the new aspects of traditional questions, results and methods which have developed from the needs or knowledge of computer science and particularly of complexity theory. It is both a textbook for introductory courses in the above-mentioned disciplines as well as a monograph in which further results of new research are systematically presented and where an attempt is made to make explicit the connections and analogies between a variety of concepts and constructions. (shrink)

Popper holds to the unity of scientific method: any differences between natural and social science are a product of theory, not a pretheoretical premise. Distin guishing instead pure and applied generalizing sciences, Popper focuses on the different role of laws in each. In generalizing social science, our tools are the logic of the situation, including the rationality principle, and unintended conse quences. Situations contain individuals, but also social entities not reducible to individuals: conspiracy theory is the extreme form of individualism. (...) Action in situations has unintended consequences. Both social and natural laws may be required to explain outcomes. The fate of Popper's ideas is a case study in the logic of the situation. Professional philosophers of social science lean toward individualism and a priorism (either intuitionist or rational choice). There are social and political explanations of this outcome, but little critical engagement with Popper's ideas. (shrink)

This article raises the question of whether standard economics with the general equilibrium model at its core applies situational analysis in a Popperian sense. Contrary to Popper's own view, the authors come to the conclusion that this is not the case. Standard economics fails to represent elements essential to any social situation in an adequate manner. It comprises uncertainty, time and space, social interaction, unintended effects, as well as culture and institutions. The authors suggest, therefore, the socioeconomic context as an (...) alternative approach to analyzing social situations. It consists of four basic elements: (1) dominant worldviews, (2) institutions and technologies, (3) relative prices, and (4) political instruments. The alternative approach was applied with some success to analyzing inter alia problems of unemployment as well as of transformation. (shrink)

European prehistoric decorative art abounds in motifs that are not humble decorative elements but seem to be significant signs. Circles, concentric circles with or without a dot in the centre, circles divided into four, six or eight equal segments (sun/star-crosses) and often round decoration complexes filled with different spiral motifs are generally considered sun symbols by archaeologists. It is predominantly accepted that sun cults dominated the belief system of the European Bronze Age. These symbols can be found as a single (...) sign on many artifacts, especially luxury ones like weapons, jewels and fine pottery. Although they were widespread in Europe in the Bronze Age the Carpathian Basin has an outstanding collection of such archaeological material. A detailed comparative study of atmospheric optic phenomena and Bronze Age symbols revealed similarities between many abstract symbols and manifestations of atmospheric phenomena in the sky. The investigation resulted in the assumption of the significance of such symbols in Bronze Age visual culture. The study attempts to identify, present and reveal the role of simple as well as the more complex celestial symbols in the relationship between prehistoric people and their environment based on this argument. (shrink)

There is sufficient evidence from more recent experiments in psychology that equal retinal stimulus-elements do not lead to equal experiences and reactions except under certain rather specific conditions. An unsophisticated observer will find himself surprised to be able to cover with his own finger a person entering the door of his living room. When the finger is moved to the right or left, thus doing away with the precise retinal coincidence with the person, the observer will soon become unable to (...) recognize intuitively the actual retinal stimulus equality of the two distant things. This usually holds even for the case when he is making every inner effort towards an antagonistic, analytic perceptual attitude of the type which is used by painters or draughtsmen in order to represent the environmental situation in a similar way as it would project itself on a photographic plate or on a retina. (shrink)

The purpose of this paper is to contextualize the study of metaphors within constructivist-informed research, in the hope that this process will orient cognitive scientists to the usefulness of implementing qualitative research methodologies, especially to using the person of the researcher as the primary research instrument. First, I explore some of the differences between Johnson and Lakoff’s Contemporary Metaphor Theory (CMT) and approaches evolving from it on one hand, and the clinical approach to metaphor based on a constructivist therapy model, (...) on the other. CMT has been one of the most significant forces that helped shift cognitive science toward an embodied approach to cognition. While it has succeeded to place physical experience back where it belongs in reason and meaning, CMT has, however, also fallen into some positivist traps which lead to problems such as a dualism, a split between the knower and the known, and with that, to a distrust of introspective, first-person accounts. In the process of finding conceptual metaphors — generalizations that govern metaphorical expressions — CMT often deletes the idiosyncratic characteristics and presuppositions implicit in linguistic metaphors; it divorces them from people’s sensory experiences, the “here and now” and the intent of their communications. The constructivist approach to metaphor that I present here accepts as a priori assumptions much of what workers of CMT are out to prove. In particular, it takes the correlation of conceptual metaphors and physical experience, as well as the unity of language and thought as pragmatic givens. Emulating the constructivist therapist’s approach to metaphors, I show how it is possible to deconstruct conceptual metaphors into minute sensory distinctions, using one’s own person as the main tool, for the purpose of helping people change their experiences in desired ways, at will. I illustrate this process by numerous examples from a wide field of applications, including mathematics education and psychotherapy. (shrink)

Motivation: Paralleling my own transformation from a Platonist to a radical constructivist, mathematics education has been experiencing for more than a decade a movement that started in theoretical foundations mostly originating in von Glasersfeld's work, and then reached professional organizations, which have been leading extensive efforts to reform school mathematics according to constructivist principles. However, the theories espoused by the researchers are, as yet, too abstract to lend themselves readily to implementation in the classroom. N2 - Purpose: I define a (...) shared experiential language (SEL) for the constructivist teacher to embody in order to transform her practice congruently according to constructivist principles. While SEL is comprised of Neuro-Linguistic Programming (NLP) subjective experience distinctions, what "makes it tick" is the constructivist epistemology with its insight that for consistent understanding to happen, new knowledge has to attach to prior experiences in a process of co-construction. Throughout the paper, I elaborate and validate this insight by numerous examples. Practical implications: Utilizing SEL allows understanding of mathematics to be rooted in each student's individual sensory experiences, thus shifting the responsibility for success in mathematics from the students back to those who guide them in co-constructing knowledge. This, in turn, should allow everybody access to understanding and so it should no longer be socially acceptable to fail in mathematics. (shrink)

We capture the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study.1.

The editor of this Journal has asked me to comment on the preceding paper by Professor Hammond in which certain parallels are drawn between relativity physics and my own suggestions concerning the establishment of “representative design” in psychological experimentation. Since I find little basis for criticism of Dr. Hammond's major line of reasoning, I shall concentrate on anticipating possible objections to Hammond's comparison as well as on elucidating my point of view by means of examples, both old and new, that (...) seem pertinent in the present context. (shrink)