According to a familiar, simple argument, numbers exist because sentences like ‘Two is an even number’ are true. Whereas realists accept the argument as sound, anti-realists either reject that number words function referentially in such sentences (non-referentialism) or else that such sentences are true (fictionalism). We argue that this dialectic, though familiar, drastically underestimates the extent to which natural language supports realism. Indeed, if dominant accounts of number and measurement-related expressions within linguistic semantics are correct, then far more than just our overt talk about numbers as objects would not be true if numbers do not exist. The purpose of this paper is to sketch this strengthened argument for realism and to survey its consequences for extant forms of anti-realism.

Friedrich Waismann once suggested that mathematical concepts are not subject to open-texture; they are “closed”. This is not quite right, as there are some traditional mathematical notions that were, at least at one time, open-textured. One of them is the notion of “polyhedron” following the history sketched in Imre Lakatos’s Proofs and refutations. Another is “computability”, which has now been sharpened into an arguably closed notion, via the Church-Turing thesis.There are also some mathematical notions that have longstanding, intuitive principles underlying them, principles that later proved to be inconsistent with each other, sometimes when the notion is applied to cases not considered previously (in which case it is perhaps an instance of open-texture). One example is “same size”, which is or was governed by the part-whole principle (one of Euclid’s Common Notions) and the one-one principle, now called “Hume’s Principle". Another is the notion of continuity.The purpose of this paper is to explore the notion of “set” and other related notions like “class”, “totality”, and the like. I tentatively put forward a thesis that this notion, too, is or at least was subject to open-texture (or something like it) and has been sharpened in various ways.This raises some questions concerning what the purposes of a (sharpened) theory of sets are to be. And questions about how one goes about trying to give non-ad-hoc explanations or answers to various questions.

Smooth Infinitesimal Analysis (SIA) is a remarkable late twentieth-century theory of analysis. It is based on nilsquare infinitesimals, and does not rely on limits. SIA poses a challenge of motivating its use of intuitionistic logic beyond merely avoiding inconsistency. The classical-modal account(s) provided here attempt to do just that. The key is to treat the identity of an arbitrary nilsquare, e , in relation to 0 or any other nilsquare, as objectually vague or indeterminate—pace a famous argument of Evans [10]. Thus, we interpret the necessity operator of classical modal logic as “determinateness” in truth-value, naturally understood to satisfy the modal system, S4 (the accessibility relation on worlds being reflexive and transitive). Then, appealing to the translation due to Gödel et al., and its proof-theoretic faithfulness (“mirroring theorem”), we obtain a core classical-modal interpretation of SIA. Next we observe a close connection with Kripke semantics for intuitionistic logic. However, to avoid contradicting SIA’s non-classical treatment of identity relating nilsquares, we translate “=” with a non-logical surrogate, ‘ E ,’ with requisite properties. We then take up the interesting challenge of adding new axioms to the core CM interpretation. Two mutually incompatible ones are considered: one being the positive stability of identity and the other being a kind of necessity of indeterminate identity (among nilsquares). Consistency of the former is immediate, but the proof of consistency of the latter is a new result. Finally, we consider moving from CM to a three-valued, semi-classical framework, SCM, based on the strong Kleene axioms. This provides a way of expressing “indeterminacy” in the semantics of the logic, arguably improving on our CM. SCM is also proof-theoretically faithful, and the extensions by either of the new axioms are consistent.

According to singularism, 'the students' refers to a single collective entity, e.g. a sum or set. In contrast, according to pluralism, 'the students' plurally refers to multiple students at once, through the primitive relation of plural reference. Although it was originally designed exclusively for plural nouns, this paper addresses whether plural reference can be extended so as to provide an empirically adequate semantics for mass nouns, such as 'the furniture', as well, as certain pluralists have maintained. Our contention is that extant pluralist analyses of mass nouns suffer from two major setbacks. First, while they may or may not explain the semantic commonalities between plural and mass nouns, they do not explain their semantic differences. Secondly, pluralism is inconsistent with the possibility that mass nouns are gunky, i.e. fail to have atomic parts. However, we provide three kinds of semantic arguments showing that the gunkiness of at least some mass nouns is not only possible, but plausible.

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic HA, in a sense to be made precise. Our main target theory, the intuitionistic small set theory SST is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism.

The purpose of this paper is to present a genuinely potentialist account of Frege arithmetic. The (cardinal) numbers are not generated from Hume’s Principle, but rather from more or less standard principles of potentialism. The relevant version of Hume’s Principle is a principle stating a condition for numbers to be identical with each other. Essentially, (HP) tells us what we are generating— cardinal numbers—but the generation does not go through (HP) itself. We also develop an Aristotelian, potentialist set theory—in effect, a theory of hereditarily finite sets—a theory that is definitionally equivalent to Dedekind-Peano arithmetic.

Aristotle argued that paradoxes of the infinite can be avoided only by insisting that all infinities are potential, not actual. There is a long tradition of thinking that a Judeo-Christian God would collapse potential infinities to actual ones, thus removing the Aristotelian guard-rail against paradox. After all, does not God know all numbers, regardless of whether they are actual or merely potential? We analyze the Aristotelian guard-rail of potentiality, as well as challenges to it due to Augustine, Burley, Scotus, and Cantor. At the heart of the debate we find the metaphysical question of how the temporal or modal "dimension" that harbors potential infinities compares to ordinary spatial dimensions. Certain theological assumptions underwrite a strong analogy. The best Aristotelian response, we suggest, is to reject the analogy by refusing to analyze modality extensionally as a point-space of possible worlds. Our analysis thus reveals lessons for atheists as well as theists.

The aim of this chapter is to sketch John Corcoran's mathematical contributions and to relate them to his (and others') philosophical interests.

While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is "made true" at some such stage. The natural logic of this strict form of potentialism is semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic.

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege's Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject.

Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.

In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.

Perhaps the most pressing challenge for singularism-the predominant view that definite plurals like 'the students' singularly refer to a collective entity, such as a mereological sum or set-is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism-the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally .

One is often said to be reasoning well when they are reasoning logically. Many attempts to say what logical reasoning is have been proposed, but one commonly proposed system is first-order classical logic. This Element will examine the basics of first-order classical logic and discuss some surrounding philosophical issues. The first half of the Element develops a language for the system, as well as a proof theory and model theory. The authors provide theorems about the system they developed, such as unique readability and the Lindenbaum lemma. They also discuss the meta-theory for the system, and provide several results there, including proving soundness and completeness theorems. The second half of the Element compares first-order classical logic to other systems: classical higher order logic, intuitionistic logic, and several paraconsistent logics which reject the law of ex falso quodlibet.

In this paper, we outline and critically evaluate Thomas Hofweber's solution to a semantic puzzle he calls Frege's Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber's proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber's analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different solution to Frege's Other Puzzle is required.

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth-Kripke semantics for intuitionistic logic, we overcome those challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms.

Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.