This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and (...) to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional intensions of epistemic two-dimensional semantics solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between topic-sensitive epistemic two-dimensional truthmaker semantics, the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{11} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. Chapter \textbf{12} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory. The multi-hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, \textbf{11}, \textbf{12}, and \textbf{14}.}. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely (...) logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical and metamathematical vocabulary. (shrink)

This paper examines the philosophical significance of the consequence relation defined in the $\Omega$-logic for set-theoretic languages. I argue that, as with second-order logic, the hyperintensional profile of validity in $\Omega$-Logic enables the property to be epistemically tractable. Because of the duality between coalgebras and algebras, Boolean-valued models of set theory can be interpreted as coalgebras. In Section \textbf{2}, I demonstrate how the hyperintensional profile of $\Omega$-logical validity can be countenanced within a coalgebraic logic. Finally, in Section \textbf{3}, the philosophical (...) significance of the characterization of the hyperintensional profile of $\Omega$-logical validity for the philosophy of mathematics is examined. I argue (i) that $\Omega$-logical validity is genuinely logical, and (ii) that it provides a hyperintensional account of formal grasp of the concept of `set'. Section \textbf{4} provides concluding remarks. (shrink)

In this paper we explain our pretense account of truth-talk and apply it in a diagnosis and treatment of the Liar Paradox. We begin by assuming that some form of deflationism is the correct approach to the topic of truth. We then briefly motivate the idea that all T-deflationists should endorse a fictionalist view of truth-talk, and, after distinguishing pretense-involving fictionalism (PIF) from error- theoretic fictionalism (ETF), explain the merits of the former over the latter. After presenting the basic framework (...) of our PIF account of truth-talk, we demonstrate a few advantages it offers over T-deflationist accounts that do not explicitly acknowledge pretense at work in the discourse. In turning to the Liar Paradox, we explain how the quasi-anaphoric functioning that our account attributes to truth-talk provides a diagnosis of the Liar Paradox (and other instances of semantic pathology) as having no content—in the sense of not specifying any of what we call M-conditions. At the same time, however, we vindicate the intuition that we can understand liar sentences, thereby avoiding one standard objection to “meaningless strategy” responses to the Liar Paradox. With this diagnosis in place, we then, by way of treatment, introduce a new predicate, ‘semantically defective’, and show how the explanation we give for its application allows for a consistent, yet revenge-immune, (dis)solution of the Liar Paradox, and semantic pathology generally. (shrink)

We discuss the concept of a cardinal number and its history, focussing on Cantor's work and its reception. J'ay fait icy peu pres comme Euclide, qui ne pouvant pas bien >faire< entendre absolument ce que c'est que raison prise dans le sens des Geometres, definit bien ce que c'est que memes raisons. (Leibniz) 1.

Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed in (...) terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many. (shrink)

In the introduction, following the formulation of the theses on the concept ‘philosophy of science’, on interdisciplinarity in modern science, and on foundational studies in science, and on the bases of their content, a thesis on the interdisciplinary approach to foundations of science is formulated. In accordance with the latter, together with the canonical approach to foundations of science, which consists in an elaboration of the foundations of mathematics, physics and other fundamental canonical sciences, also an interdisciplinary approach to foundations (...) of science is realised, one that consists in the elaboration of four foundational complex sciences containing the most significant productive factors for the elaboration of various interdisciplinary theories: Methodology of S-modelling, Logic of science, Definitics, a complex theory of definite structures, aggregates, processes and systems in reality and in knowledge, Indefinitics, a complex theory of indefiniteness in its various forms in reality and in knowledge. Taking into account the internal connection between the first two complex sciences enlisted above, it is expedient in the process of investigations of their foundations that they be unified in yet another complicated complex science—logistics. (shrink)

We investigate two notions of independence— independence and complete independence—applied to the Peano axioms for the sequence of natural numbers. We review the results that, although they...

There are quite a few studies on late Bolzano’s notion of a collection (Inbegriff). We try to broaden the perspective by introducing the forerunner of collections in Bolzano’s early writings, namely the entities referred to by expressions with the technical term ‘et’. Special emphasis is laid on the question whether these entities are set-theoretical or mereological plenties. Moreover, similarities and differences to Bolzano’s mature conception are pointed out.

. This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiros identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiros claim that it is not possible to identify objects in a (...) structure except through the relations and functions that are defined on the structure in which the object has a place. I argue that, in the case of the definition of the so called direct image of a function, it is possible to individuate objects in structures. (shrink)

We expose some basic elements of a style of programming supported by functional languages like Haskell by relating them to a coherent set of notions and techniques from Curry’s work in combinatory logic and formal systems, and their algebraic and categorical interpretations. Our account takes the form of a commentary to a simple fragment of Haskell code attempting to isolate the conceptual sources of the linguistic abstractions involved.

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be (...) summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder. (shrink)

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language. (shrink)

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped (...) with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent. (shrink)

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating an account of numbers as properties of collections, work on the philosophy of absences, and recent (...) work in numerical cognition and ontogenetic studies. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.

A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...) fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism. (shrink)

ABSTRACT Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by (...) abstraction principles. (shrink)

I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination of (...) its formative period in the incompleteness results of Gödel, became modern logic, as distinct from the traditional logic of Aristotle, and why and how the logistic tradition that led from Frege through Russell, rather than the algebraic tradition that led from De Morgan and Boole through Peirce and Schröder, came, in his view, to define modern logic. (shrink)

Davide Bondoni, La teoria delle relazioni nell'algebra della logica schroderiana. Milan: LED Edizioni Universitarie Lettere Economia Diritto, 2007. 95 pp. [euro]13.00. ISBN 978-88-7916-349-1. Revie...

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...) contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...) of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. Work on the program progressed significantly in the 1920s with contributions from logicians such as Paul Bernays, Wilhelm Ackermann, John von Neumann, and Jacques Herbrand. It was also a great influence on Kurt Gödel, whose work on the incompleteness theorems were motivated by Hilbert's Program. Gödel's work is generally taken to show that Hilbert's Program cannot be carried out. It has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory. (shrink)

Our regimentation of Goodman and Myhill’s proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation.Here we revisit Zermelo’s requirement that the separating property be definite. The instance that Goodman and Myhill used is not constructively warranted. It is that principle, and not Choice alone, that precipitates Excluded Middle.Separation in various axiomatizations of constructive set theory is examined. We conclude that insufficient critical attention has been paid to how those forms of Separation fail, (...) in light of the Goodman–Myhill result, to capture a genuinely constructive notion of set. (shrink)

This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between (...) this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes. (shrink)

The book is devoted to the contemporary stage of quantum mechanics – quantum information, and especially to its philosophical interpretation and comprehension: the first one of a series monographs about the philosophy of quantum information. The second will consider Be l l ’ s inequalities, their modified variants and similar to them relations. The beginning of quantum information was in the thirties of the last century. Its speed development has started over the last two decades. The main phenomenon is entanglement. (...) The subareas are quantum computer, quantum communication (and teleportation), and quantum cryptography. The book offers the following main conceptions, theses and hypotheses: – dualistic Phythagoreanism as a new kind among the interpretations of quantum mechanics and information: arithmetical, logical, and metamathematical one; – Gödel ’ s first incompleteness theorem is an undecidable proposition, and consequently the second one,too. – a partial rehabilitation of Hilbert ’ s program for the self-foundation of mathematics; – the dual-foundation of mathematics; – Skolemian relativity between: Cantor ’s kinds of infinity, finiteness and infinity, discreteness and continuity, completeness and incompleteness, etc.; – information is a physical quantity representing the non-reducibility of a system to its parts, particularly nonaddtivity; – there exist pure relations «by itself», which cannot be reduced to predications; – energy conservation can and should be generalized; – Einstein’ s «general covariance» or «principle of relativity» can and should be generalized to cover discrete morphisms where the notion of velocity does not make sense. (shrink)

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed (...) by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)