Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

This book is a defense of modal realism; the thesis that our world is but one of a plurality of worlds, and that the individuals that inhabit our world are only a few out of all the inhabitants of all the worlds. Lewis argues that the philosophical utility of modal realism is a good reason for believing that it is true.

Die Allgemeine Erkenntnislehre gilt als das Hauptwerk von Moritz Schlick. Hierin entwickelt Schlick in Auseinandersetzung mit zeitgenössischen Positionen seine einflussreichen Gedanken zum Wesen der Erkenntnis, zum Verhältnis zwischen Psychologie und Logik, zum Leib-Seele-Problem und zum erkenntnistheoretischen Realismusstreit. Der Text wurde während der frühen Rostocker Jahre Schlicks, von 1911 bis 1916, verfasst. Die Allgemeine Erkenntnislehre ist ein Meilenstein der wissenschaftlichen Philosophie und grundlegend für die spätere Entwicklung des Wiener Kreises des Logischen Empirismus.

"This book is valuable as expounding in full a theory of meaning that has its roots in the work of Frege and has been of the widest influence.

In this study two strands of inferentialism are brought together: the philosophical doctrine of Brandom, according to which meanings are generally inferential roles, and the logical doctrine prioritizing proof-theory over model theory and approaching meaning in logical, especially proof-theoretical terms.

Now in its fourth edition, this classic work clearly and concisely introduces the subject of logic and its applications. The first part of the book explains the basic concepts and principles which make up the elements of logic. The author demonstrates that these ideas are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The second part of the book shows the applications of logic in mathematical theory building with concrete examples that draw (...) upon the concepts and principles presented in the first section. Numerous exercises and an introduction to the theory of real numbers are also presented. Students, teachers and general readers interested in logic and mathematics will find this book to be an invaluable introduction to the subject. (shrink)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts, models, and mappings.

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...) slogan that mathematics is the science of structure? (shrink)

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

Much of The Reason’s Proper Study is devoted to defending the claim that simply by stipulating an abstraction principle for the “number-of” functor, we can simultaneously fix a meaning for this functor and acquire epistemic entitlement to the stipulated principle. In this paper, I argue that the semantic and epistemological principles Hale and Wright offer in defense of this claim may be too strong for their purposes. For if these principles are correct, it is hard to see why they do (...) not justify platonist strategies that are not in any way “neo-Fregean,” e.g. strategies that treat “the number of Fs” as a Russellian definite description rather than a singular term, or employ axioms that do not have the form of abstraction principles. (shrink)

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)

A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.

We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas (...) Hilbert dismisses elucidation and consequently treats the primitives as schematic. (shrink)

In this new book, the author of the classic Truth presents an original theory of meaning, demonstrates its richness, and defends it against all contenders. He surveys the diversity of twentieth-century philosophical insights into meaning and shows that his theory can reconcile these with a common-sense view of meaning as derived from use. Meaning and its companion volume Truth (now published in a revised edition) together demystify two central issues in philosophy and offer a controversial but compelling view of the (...) relations between language, thought, and reality. (shrink)

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation (...) of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation. (shrink)

The concept of implicit definition has played a central role in the controversies about the foundations of geometry. The history of this concept, however, exhibits several important confusions. The term has been used in at least three different senses—Gergonne's position, Hilbert's position of the so-called definitions by axioms , and that of Pasch and Dubislav in the sense of Russell's contextual definition. Frege's contribution to the explication of Hilbert's view has occasioned an adequate appraisal in recent years. A summary account (...) is given. The main purpose of the paper is to present some details about the historical development of the transfer of Gergonne's term ‘implicit definition’ to the Hilbert position, a transfer which goes back to the Peano School and was disseminated by Enriques. (shrink)

In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...) New Science that meets modern standards and to examine possible problems surrounding Frege’s original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege–Hilbert controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation of Frege’s proposal. In section 4 a more detailed reconstruction of Frege’s New Science is set out while section 5 examines what is left out. The concluding section is devoted to a discussion of Frege’s strategy and its significance from a broader perspective. (shrink)

This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...) of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert's Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program. (shrink)

This essay distinguishes between metaphysical and epistemological conceptions of analyticity. The former is the idea of a sentence that is ‘true purely in virtue of its meaning’ while the latter is the idea of a sentence that ‘can be justifiably believed merely on the basis of understanding its meaning’. It further argues that, while Quine may have been right to reject the metaphysical notion, the epistemological notion can be defended from his critique and put to work explaining a priori justification. (...) Along the way, a number of further distinctions relevant to the theory of analyticity and the theory of apriority are made and their significance is explained. (shrink)

Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

The daring idea that convention - human decision - lies at the root both of necessary truths and much of empirical science reverberates through twentieth-century philosophy, constituting a revolution comparable to Kant's Copernican revolution. This book provides a comprehensive study of Conventionalism. Drawing a distinction between two conventionalist theses, the under-determination of science by empirical fact, and the linguistic account of necessity, Yemima Ben-Menahem traces the evolution of both ideas to their origins in Poincaré's geometric conventionalism. She argues that the (...) radical extrapolations of Poincaré's ideas by later thinkers, including Wittgenstein, Quine, and Carnap, eventually led to the decline of conventionalism. This book provides a fresh perspective on twentieth-century philosophy. Many of the major themes of contemporary philosophy emerge in this book as arising from engagement with the challenge of conventionalism. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

This major publication is a history of the semantic tradition in philosophy from the early nineteenth century through its incarnation in the work of the Vienna Circle, the group of logical positivists that emerged in the years 1925-1935 in Vienna who were characterised by a strong commitment to empiricism, a high regard for science, and a conviction that modern logic is the primary tool of analytic philosophy. In the first part of the book, Alberto Coffa traces the roots of logical (...) positivism in a semantic tradition that arose in opposition to Kant's theory that a priori knowledge is based on pure intuition and the constitutive powers of the mind. In Part II, Coffa chronicles the development of this tradition by members and associates of the Vienna Circle. Much of Coffa's analysis draws on the unpublished notes and correspondence of many philosophers. The book, however, is not merely a history of the semantic tradition from Kant 'to the Vienna Station'. Coffa also critically reassesses the role of semantic notions in understanding the ground of a priori knowledge and its relation to empirical knowledge and questions the turn the tradition has taken since Vienna. (shrink)

The daring idea that convention - human decision - lies at the root of so-called necessary truths, on the one hand, and much of empirical science, on the other, reverberates through twentieth-century philosophy, constituting a revolution comparable to Kant's Copernican revolution. Conventionalism is the first comprehensive study of this radical turn. One of the conclusions it reaches is that the term 'truth by convention', widely held to epitomize conventionalism, reflects a misunderstanding that has led to the association of conventionalism with (...) relativism and postmodernism. Conventionalism, this book argues, did not contend that truths can be stipulated, but rather, that stipulations are often confused with truths. Their efforts were thus directed toward disentangling truth and convention, not reducing truth to convention."--Jacket. (shrink)

This chapter aims to provide materials with which to substantiate the claim that, under the appropriate circumstances, the notion of analyticity can help explain how one might have a priori knowledge even in the strong sense. It argues that Implicit Definition, properly understood, is completely independent of any form of irrealism about logic. The chapter defends the thesis of Implicit Definition against Quine's criticisms, and examines the sort of account of the apriority of logic that this doctrine is able to (...) provide. The chapter shows that, against the background of a rejection of indeterminacy, its insolubility cannot be conceded. It also argues that neither a non‐factualism about Frege‐analyticity, nor an error thesis about it, can plausibly fall short of an outright rejection of meaning itself. The chapter shows how the doctrine that appears to offer the most promising account of how we grasp the meanings of the logical constants. (shrink)

This article offers an overview of inferential role semantics. We aim to provide a map of the terrain as well as challenging some of the inferentialist’s standard commitments. We begin by introducing inferentialism and placing it into the wider context of contemporary philosophy of language. §2 focuses on what is standardly considered both the most important test case for and the most natural application of inferential role semantics: the case of the logical constants. We discuss some of the (alleged) benefits (...) of logical inferentialism, chiefly with regards to the epistemology of logic, and consider a number of objections. §3 introduces and critically examines the most influential and most fully developed form of global inferentialism: Robert Brandom’s inferentialism about linguistic and conceptual content in general. Finally, in §4 we consider a number of general objections to IRS and consider possible responses on the inferentialist’s behalf. (shrink)

Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...) the abstract concept of a simply infinite system, proves the existence of a “model”, insists on the stepwise derivation of theorems, and defines structure-preserving mappings between different systems that fall under the abstract concept. Crucial parts of these considerations were added, however, only to the penultimate manuscript, for example, the very concept of a simply infinite system. The methodological consequences of this radical shift are elucidated by an analysis of Dedekind’s metamathematics. Our analysis provides a deeper understanding of the essay and, in addition, illuminates its impact on the evolution of the axiomatic method and of “semantics” before Tarski. This understanding allows us to make connections to contemporary issues in the philosophy of mathematics and science. (shrink)