Semantics and the Ontology of Number

The meaning of number expressions and their implications for the existence of numbers is discussed in this Element. It highlights a debate between substantivalists and adjectivalists and proposes a comprehensive polymorphic semantics for number expressions. This semantics strengthens the argument for realism by accounting for the various meanings and relationships of number expressions.

Format: Paperback / softback
Length: 75 pages
Publication date: 20 May 2021
Publisher: Cambridge University Press

The meaning of number expressions and their significance in the philosophy of mathematics is a complex and ongoing debate. This Element aims to contribute to this discussion by exploring a recent debate between substantivalists and adjectivalists regarding the semantic function of number words in numerical statements.

Substantivalists argue that number expressions have a substantive, or real, meaning, referring to concrete entities or quantities. They hold that numbers exist independently of human perception or thought and that they have properties and attributes of their own. Adjectivalists, on the other hand, argue that number expressions do not have a substantive meaning but rather serve as adjectives or modifiers that describe or modify other entities or quantities. They contend that numbers are not independent entities but are created and defined by human conventions and mathematical structures.

The motivations and challenges of substantivalists and adjectivalists are significant. Substantivalists argue that numbers have a fundamental existence that is independent of human perception or thought, providing a basis for mathematical reasoning and scientific inquiry. They argue that accounting for the numerous meanings and how they are related leads to a strengthened argument for realism, one that renders familiar forms of nominalism highly implausible.

On the other hand, adjectivalists argue that number expressions do not have a substantive meaning but rather serve as adjectives or modifiers that describe or modify other entities or quantities. They contend that numbers are not independent entities but are created and defined by human conventions and mathematical structures. Adjectivalists argue that accounting for the numerous meanings and how they are related leads to a weakened argument for realism, one that is open to alternative interpretations and explanations.

To address these challenges, I develop a comprehensive polymorphic semantics for number expressions. Polymorphic semantics is a framework that allows for the analysis of number expressions in terms of their various meanings and how they are related. It recognizes that number expressions can have different meanings depending on the context and the intended interpretation.

In my proposed semantics, I argue that number expressions have a multifaceted meaning that includes both substantive and adjectival aspects. Numbers can refer to concrete entities or quantities, but they can also be used as adjectives or modifiers to describe or modify other entities or quantities. This multifaceted meaning allows for a more nuanced and accurate analysis of number expressions and their role in mathematical statements.

By accounting for the numerous meanings and how they are related, my proposed semantics strengthens the argument for realism. It renders familiar forms of nominalism highly implausible, as they require the existence of abstract or non-existent entities that are created and defined by human conventions and mathematical structures.

In conclusion, the meaning of number expressions and their significance in the philosophy of mathematics is a complex and ongoing debate. Substantivalists argue that number expressions have a substantive, or real, meaning, referring to concrete entities or quantities. Adjectivalists argue that number expressions do not have a substantive meaning but rather serve as adjectives or modifiers that describe or modify other entities or quantities. By developing a comprehensive polymorphic semantics for number expressions, I argue that accounting for the numerous meanings and how they are related leads to a strengthened argument for realism, one that renders familiar forms of nominalism highly implausible. This approach provides a more nuanced and accurate analysis of number expressions and their role in mathematical statements.

Weight: 150g
Dimension: 151 x 227 x 12 (mm)
ISBN-13: 9781108456258