Mathematical Determinacy

This chapter addresses the second major challenge for the extension of conventionalism from logic to mathematics: the richness of mathematical truth. The chapter begins by distinguishing indeterminacy from pluralism and clarifying the crucial notion of open-endedness. It then critically discusses the two major strategies for securing arithmetical categoricity using open-endedness; one based on a collapse theorem, the other on a kind of anti-overspill idea. With this done, a new argument for the categoricity of arithmetic is then presented. In subsequent discussion, the philosophical importance of this categoricity result is called into question to some degree. The categoricity argument is then supplemented by an appeal to the infinitary omega rule, and an argument is given that beings like us can actually follow the omega rule without any violation of Church’s thesis. Finally, the chapter discusses the extension of this type of approach beyond arithmetic, to set theory and the rest of mathematics.

Metamathematics versus Conventionalism

This chapter deals with objections to mathematical conventionalism, focusing especially on technical objections. To make things easier to follow and more self-contained, the chapter begins with a brief overview of arithmetization. Then a technical argument alleging incompatibility between the factuality of conventions and the conventionality of arithmetic is discussed followed by discussion of a related but more general worry about circularity. Then an anti-conventonalist argument from Pollock is discussed and related to Montague’s paradox. Finally, Gödel’s argument against conventionalism is critically discussed. Ultimately the chapter demonstrates that there is no incompatibility between conventionalism and metamathematical results and methods.

The Epistemology of Logic

This chapter shows that unrestricted inferentialism/conventionalism leads to a naturalistically satisfying account of our a priori knowledge of logical validity. The chapter first lays the groundwork by discussing the general question of what conditions arguments need to meet in order to lead to knowledge of their conclusions. Following Boghossian, the chapter then argues that inferentialism/conventionalism is particularly well posed to allow rule-circular arguments to lead to a priori knowledge of the validity of our basic rules. Restricted inferentialists were often forced to complicate and sometimes abandon their accounts of logical knowledge in the face of bad company. By contrast, unrestricted inferentialism has no problem at all with bad company. All told, conventionalism gives a naturalistic account of our a priori knowledge of logic.

Linguistic Conventions

What are linguistic conventions? This chapter begins by noting and setting aside philosophical accounts of social conventions stemming from Lewis’s influential treatment. It then criticizes accounts that see conventions as explicit stipulations. From there the chapter argues that conventions are syntactic rules of inference, arguing that there are scientific reasons to posit these rules as part of our linguistic competence and that we need to include both bilateralist and open-ended inference rules for a full account. The back half of the chapter aims to naturalize inference rule-following by providing functionalist-dispositionalist approaches to our attitudes, inference, and inference-rule–following, addressing Kripkenstein’s arguments and several other concerns along the way.

Metaontology, Existence, and Reference

This chapter presents and defends a conventionalist-friendly metaontology, thereby showing how conventionalism manages to vindicate trivial ontological realism in mathematics. After clarifying and demonstrating this entailment of conventionalism, it clarifies the metaontology involved. The chapter then defends metadeflationism about quantifiers, which entails a version of quantifier pluralism. This is a form of what has recently been called “modest quantifier variance” in joint work with Eli Hirsch. After laying out this view, it is defended from several objections. With this groundwork set out, the chapter then explains how this answers Kant’s challenge for trivial realism that was explained in the previous chapter. Finally, the chapter closes by discussing the metaphysics of mathematical objects, in conventionalist terms, addressing the Julius Caesar problem and structuralism, among other things.

Logical Conventionalism

This chapter argues that logical truth, validity, and necessity in any language can be fully explained in terms of the language’s linguistic conventions. More particularly, it is demonstrated that unrestricted logical inferentialism is a version of logical conventionalism by arguing for conventionalism in detail and answering various objections involving the role of metasemantic principles and semantic completeness in the conventionalist argument. The chapter then discusses how this account relates to the deflationist accounts offered by Field and others, before turning to the metaphysics and normativity of logic, which it discusses on conventionalist grounds. Overall, this chapter shows that conventionalism leads to a naturalistically acceptable and philosophically plausible theory of logic.

Unrestricted Logical Inferentialism

This chapter develops and defends an unrestricted inferentialist theory of the meanings of logical constants. Unlike restricted inferentialism, unrestricted inferentialism puts no constraints on which rules can determine meanings. The foundations of inferentialism are also discussed, including various types of holism and the distinction between basic and derivative rules. In order to develop and defend a detailed inferentialist theory of logic, this chapter provides an inferentialist account of the “logical” constants, solves Carnap’s categoricity problem for the meanings of logical constants, and provides inferentialist approaches to both the psychology and metaphysics of logic. Finally, the chapter briefly discusses the challenge to unrestricted inferentialism posed by tonk and related types of bad company. Building on the foundation provided by Part I (chapters 1-2) of the book, this chapter provides a freestanding development and defense of logical inferentialism.

The Facts of the Matter

This chapter concerns the status of the conventionalist theory developed, argued for, and defended throughout the book. It begins by discussing the views that historical conventionalists had about their own conventionalist theories and addresses a recent controversy about whether Carnap was truly a conventionalist. The chapter then argues that conventionalism is the best explanation of the logical and mathematical facts, assessing it according to a number of different theoretical virtues. Then two metaobjections are considered, one based on philosophical progress, and the other based on peer disagreement. Despite the chapter’s defense of conventionalism, it ends by expressing some very personal doubts.

Old Slogans, New Dogmas

This chapter steps back and provides a general overview. It begins by discussing how each of the classic conventionalist slogans (about truth in virtue of meaning, analyticity, tautologies, and more) fares in light of my conventionalist theory. Then the chapter discuses Carnap’s Logical Syntax-era theory of logic and mathematics in detail, before turning to Giannoni’s less well-known account in Conventionalism in Logic. Finally, the chapter briefly considers how it is that the rejection of conventionalism has turned into a new dogma, not just of empiricism, but of contemporary philosophy as a whole.

Truth, Paradoxes, Freedom, Applications, and Knowledge

This chapter begins by showing that with the problems of mathematical existence and determinate truth solved, a sophisticated inferentialist theory of mathematics leads to mathematical conventionalism. A philosophical worry harkening back to the Carnap/Quine debate is addressed before a number of issues in the philosophy of mathematics are given conventionalist treatments. The chapter discusses how conventionalists can handle the set-theoretic paradoxes, the freedom of mathematics, the many applications of mathematics to the physical world, and then provides a naturalistic epistemology of mathematics, even addressing the epistemology of consistency. In almost all of these cases, the discussion in the chapter shows that a conventionalist theory deals with these issues in a more satisfying way than other approaches to the philosophy of mathematics.