The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines

Ontology-Driven Semantic Data Integration in Open Environment

Collaborative intelligence in the context of information management can be defined as A shared intelligence that results from the collaboration between various information systems . In open environments, these collaborating information systems can be heterogeneous, dynamic and loosely-coupled. Information systems in open environment can also possess a certain degree of autonomy. The integration of data residing in various heterogeneous information systems is essential in order to drive the intelligence efficiently and accurately. Because of the heterogeneous, loosely-coupled, and dynamic nature of open environment, the integration between these information systems in the data level is not efficient. Several approaches and models have been proposed in order to perform the task of data integration. Many of the existing approaches for data integration are designed for closed environment, tightly-coupled systems and enterprise data integration. They make explicit, or implicit, assumptions about the semantic structure of the data. Because of the heterogeneous and loosely-coupled nature of open environment, such assumptions are deemed unintuitive. Data integration approaches based on model that are extensional in nature are also inadequate for open environment. This is because they do not account for the dynamic nature of open environment. The need for an adequate model for describing data integration systems in open environment is quite evident. Intensional based modeling is found to be an adequate and natural choice for modeling in open environment. This is because it addresses the dynamic and loosely-coupled nature of open environment. In this work, an intensional model for the conceptualization is presented. This model is based on the theory of Properties Relations and Propositions (PRP). The proposed description takes the concepts, relations, and properties as primitive and as such, irreducible entities. The formal intensional account of both Ontology and Ontological Commitment are also proposed in light of the intensional model for conceptualization. An intensional model for ontology-driven mediated data integration in open environment is also proposed. The proposed model accounts for the dynamic nature of open environment and also intensionally describes the information of data sources. The interface between global and local ontologies and the formal intensional semantics of the query answering are then described

The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines

Minimal Theologies

Originally published in in 2004. What, at this historical moment "after Auschwitz," still remains of the questions traditionally asked by theology? What now is theology's minimal degree? This magisterial study, the first extended comparison of the writings of Theodor W. Adorno and Emmanuel Levinas, explores remnants and echoes of religious forms in these thinkers' critiques of secular reason, finding in the work of both a "theology in pianissimo" constituted by the trace of a transcendent other. The author analyzes, systematizes, and formalizes this idea of an other of reason. In addition, he frames these thinkers' innovative projects within the arguments of such intellectual heirs as Jürgen Habermas and Jacques Derrida, defending their work against later accusations of "performative contradiction" (by Habermas) or "empiricism" (by Derrida) and in the process casting important new light on those later writers as well. Attentive to rhetorical and rational features of Adorno's and Levinas's texts, his investigations of the concepts of history, subjectivity, and language in their writings provide a radical interpretation of their paradoxical modes of thought and reveal remarkable and hitherto unsuspected parallels between their philosophical methods, parallels that amount to a plausible way of overcoming certain impasses in contemporary philosophical thinking. In Adorno, this takes the form of a dialectical critique of dialectics; in Levinas, that of a phenomenological critique of phenomenology, each of which sheds new light on ancient and modern questions of metaphysics, ethics, and aesthetics. For the English-language publication, the author has extensively revised and updated the prize-winning German version