The mathematicization of nature.

This thesis defends the Quine-Putnam indispensability argument for mathematical realism and introduces a new indispensability argument for a substantial conception of truth. Chapters 1 and 2 formulate the main components of the Quine-Putnam argument, namely that virtually all scientific laws quantify over mathematical entities and thus logically presuppose the existence thereof. Chapter 2 contains a detailed discussion of the logical structure of some scientific theories that incorporate or apply mathematics. Chapter 3 then reconstructs the central assumptions of Quine's argument, concluding (provocatively) that "science entails platonism". Chapter 4 contains a brief discussion of some major theories of truth, including deflationary views (redundancy, disquotation). Chapter 5 introduces a new argument against such deflationary views, based on certain logical properties of truth theories. Chapter 6 contains a further discussion of mathematical truth. In particular, non-standard conceptions of mathematical truth such as "if-thenism" and "hermeneuticism". Chapter 7 introduces the programmes of reconstrual and reconstruction proposed by recent nominalism. Chapters 8 discusses modal nominalism, concluding that modalism is implausible as an interpretation of mathematics (if taken seriously, it suffers from exactly those epistemological problems allegedly suffered by realism). Chapter 9 discusses Field's deflationism, whose central motivating idea is that mathematics is (pace Quine and Putnam) dispensable in applications. This turns on a conservativeness claim which, as Shapiro pointed out in 1983, must be incorrect (using Godel's Theorems). I conclude in Chapter 10 that nominalistic views of mathematics and deflationist views of truth are both inadequate to the overall explanatory needs of science

Abstract Consequence and Logics - Essays in Honor of Edelcio G. de Souza

Edelcio G. de Souza is a Brazilian logician and philosopher who has researches in the domains of abstract logic, non-classical systems, philosophy of science and the foundations of mathematics. This book is in his honor with the purpose of celebrating his 60th birthday. It contains some articles connected with the above topics and other subjects in logical investigations

What the heck is Logic? Logics-as-formalizations, a nihilistic approach

Logic is about reasoning, or so the story goes. This thesis looks at the concept of logic, what it is, and what claims of correctness of logics amount to. The concept of logic is not a settled matter, and has not been throughout the history of it as a notion. Tools from conceptual analysis aid in this historical venture. Once the unsettledness of logic is established we see the repercussions in current debates in the philosophy of logic. Much of the battle over the ‘one true logic’ is conceptually talking past each other. The theory of logics-as-formalizations is presented as a conceptually open theory of logic which is Carnapian in flavour and grounding. Rudolf Carnap’s notions surrounding ‘external’ and ‘pseudo-questions’ about linguistic frameworks apply to formalizations, thus logics, as well. An account of what formalizations are, a more structured sub-set of modelling, is given to ground the claim that logics are formalizations. Finally, a novel account of correctness, the COFE framework, is developed which allows the notions of logical monism, pluralism and nihilism to be more precisely formulated than they currently are in the discourse

Logic-Language-Ontology

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem of Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”

Infinite Hermeneutics: Events, Globalization, and the Human Condition

Thesis advisor: Patrick H. ByrneIt has been held in philosophical practice that some matters of reflection have more import than others, and that some are so significant that they may be termed "first philosophy." In contemporary Continental philosophy, the term "event" has become a watchword for a profound change in the orientation of philosophic thought. Indeed, one may say that the discourse surrounding events marks the first decisive development in philosophy since Martin Heidegger penned Being and Time. This is not to say, however, that any consensus has emerged concerning either the character of events, or more importantly what they entail for the meaning of human historical consciousness. To provide such statements, ones that have at least a relative superiority with respect to their rivals, might thus be considered the basic task for first philosophy today. It is to accomplish this double aim that the present work is devoted. These two tasks, articulating the character of events and their significance for human historical consciousness, are here assayed by a movement that is itself double, by a movement of suspicion and affirmation. In the specific case, the present work undertakes a retrieval of Heidegger's understanding of "Ereignis" (or event) after passing through a hermeneutics of suspicion, posed by the criticisms of the contemporary French philosopher Alain Badiou, and returning to an articulation of "Emergence" as a complementary hermeneutics of affirmation. The method by which I undertake this inquiry is what may be called an "infinite hermeneutics," which I intend to be opposed to "finite hermeneutics." By this latter program, "finite hermeneutics," I mean any form of philosophical hermeneutics that is committed to the thesis that human understanding (Verstehen) is finite, or that the objective of inquiry itself is finite, or both of these points. The thesis that human understanding is finite may be found in Kant's proposal that human knowing is distinct from divine knowledge in the respect that human knowing is dependent on receptive intuition, and thus finite, while infinite knowledge is founded on a productive intuition. In the relevant sense, I argue, it may also be found in Heidegger's own thought. One of the major points of the present investigation is to demonstrate in what way a commitment to finitude is highly problematic, and that human knowing, human comprehension, and even the very character of what is known is not finite in any relevant sense. The motivation for such a departure is provided by the criticisms of Badiou, which are here treated as a moment of suspicion. I begin the work with a "Prolegomenon," which reviews in detail the specific challenge Badiou has posed for phenomenological hermeneutics, or any other philosophical position that is committed to the notion that human thought or understanding is finite. As a "Prolegomenon," however, nothing positive for my own position is accomplished there; instead the net result of the study is to produce: (a) an argument against Heideggerian finite hermeneutics, (b) a summary critique of the Badiou's own position, and (c) a clear statement on the eight separate tasks that I set out to accomplish in the argument that follows. The positive aspect of the text, the beginning of the movement of affirmation, thus occurs in "Part I: Infinite Hermeneutics," in which I present a defense of phenomenological hermeneutics as a viable philosophical method. In chapter three I begin by drawing on the work of Paul Ricoeur. My argument is that he is both the very first philosopher to articulate an infinite hermeneutics, and that this account, suitably elaborated throughout his career, is able to meet most of the specific challenges Badiou poses. There does remain, however, three separate points that Ricoeur's thought does not fully explore. In order to remedy those deficiencies, and in order to demonstrate the relative advantage of my hermeneutical position with respect to its competitors, I thus move to produce a new model for hermeneutical thought. Articulating the conditions for this model is the task for chapter four. My task here resolves into three parts. First, I argue for a Galoisian Revolution in phenomenological study, which sets forth a new between hermeneutics and phenomenology study. This relation, second, requires a rearticulation of phenomenological method such that it is "impersonal," as Jean-Paul Sartre's early work suggests. Additionally this relation, third, requires that one be attentive to the structures of consciousness, which is what completes the Galoisian Revolution. In order to support my account of an impersonal phenomenology I engage the contemporary Anglo-American discussions in the philosophy of mind concerning the character of first-person consciousness. In order to specify what is intended by a structure of first-person consciousness, provide a provisional phenomenology of eros. In chapter five I move to articulate the structure of consciousness that serves as the third model for phenomenological hermeneutics. It is at this point that I engage with the work of Bernard Lonergan. My central contention in chapter five is that it is possible to retrieve Longergan's work on cognitional structure as a phenomenology of inquiry for hermeneutical purposes. Taken together, these points, the Ricoeurean defense of hermeneutics, the development of an impersonal phenomenology, and the retrieval of a phenomenology of inquiry, form the hard core of my proposal for infinite hermeneutics. "Part II: On Worlds" concerns the fruits that I can reap from the harvest sown in Part I. In particular, I aim to develop an ecological sense of worlds in response to Badiou's category-theoretic and Heidegger's (early) existential world. My argument moves from an ecological account of natural worlds (chapter six), through a signifying account human worlds (chapter seven), to an account of human historical consciousness and a consideration of catastrophes such as the Shoah and the Encounter (chapter eight). In each of these chapters I focus on developing an account of different kinds of Events, with the aim not only of providing a more serviceable account than my rivals, but also with the hopes of providing a new and better picture of world process. The final section, "Part III: The Metaphysics of Excess" expresses the central Metaphysical claims of the work, especially those concerning Events and the peculiar form I call Emergence. This chapter, in short, constitutes the moment of affirmation in response to the moment of suspicion occasioned by Badiou's criticism of phenomenological hermeneutics. Additionally, however, I produce an argument for the intelligible relation of cosmic space and time with human (lived) space and time, a statement on the new forms of causation entailed by the possibility of Events, and a new account of Truth (to rival Badiou and Heidegger's). The work closes with a summary review of what I have achieved and what yet remains to be accomplished. Though as the title of the conclusion suggests, its main aim is to provide a new statement on the world-view that I work to articulate over the course of the investigation. That world-view, and this is the justification for the subtitle of the present work, is the trans-modern condition, which articulates the existential character of our modern globalized world.Thesis (PhD) — Boston College, 2011.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Philosophy

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