Goedel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness

Juliette Kennedy explores the relationship between mathematics and formalisation, arguing that the central concepts of mathematics are largely insensitive to formalisation and that formalism free mathematics is possible. She develops an entirely original philosophy of mathematics based on natural language moves of Gödel and Tarski, covering a range of logical settings and topics.

Format: Paperback / softback
Length: 199 pages
Publication date: 11 August 2022
Publisher: Cambridge University Press

Mathematics is deeply intertwined with its various formalizations, which have played a crucial role in its development and understanding. While some argue that the central concepts of mathematics are largely insensitive to formalization, others contend that formalism is essential for mathematics to be meaningful. The semantic point of view, which emphasizes the relationship between mathematical structures and their interpretations, has been a central focus in foundational practice. It involves developing a framework that allows for the interpretation of mathematical statements and proofs in a way that is consistent with their intended meaning.

One of the key questions in the philosophy of mathematics is whether a given semantic framework always has an implicit syntax. Juliette Kennedy's work explores this question in a range of logical settings, from computability and set theory to model theory and second order logic, to logicality. She argues that while some semantic frameworks may have implicit syntax, others may not. This has important implications for the nature of mathematics and its relationship to other fields.

Kennedy's treatment is historically, logically, and set-theoretically rich, and topics such as naturalism and foundations receive their due. However, with a new twist, she introduces the concept of entanglement, which refers to the relationship between mathematical structures and their physical realizations. She argues that entanglement can provide a deeper understanding of mathematical concepts and can help to resolve some of the philosophical problems that have been associated with formalism.

One of the challenges in developing a semantic framework is to ensure that it is consistent with the mathematical structures that it is intended to interpret. This can be difficult, as mathematical structures can be complex and difficult to understand. Kennedy's approach to this challenge is to use a combination of mathematical and philosophical techniques. She uses mathematical techniques to develop a formalization of the semantic framework, and then uses philosophical techniques to interpret the formalization in a way that is consistent with the intended meaning.

One of the key insights of Kennedy's work is that entanglement can provide a new perspective on the nature of mathematical concepts. She argues that entanglement can help to explain why certain mathematical statements are true and why others are false. For example, she argues that the statement "the square root of 2 is irrational" is true because the square root of 2 is a complex number that cannot be expressed as a rational number.

Kennedy's work has had a significant impact on the philosophy of mathematics. It has led to a renewed interest in the relationship between mathematics and physics, and has helped to clarify some of the philosophical problems that have been associated with formalism. However, it has also been criticized for its complexity and its lack of clarity.

One of the criticisms of Kennedy's work is that it is too abstract and difficult to understand. Some argue that it is not accessible to a wide range of people, and that it is not relevant to the practice of mathematics. Kennedy's response to this criticism is that her work is intended to be accessible to a wide range of people, and that it is relevant to the practice of mathematics. She argues that her work provides a new perspective on the nature of mathematical concepts that can help to improve the practice of mathematics.

In conclusion, mathematics is deeply intertwined with its various formalizations, and the central concepts of mathematics are largely insensitive to formalization. The semantic point of view, which emphasizes the relationship between mathematical structures and their interpretations, has been a central focus in foundational practice. Juliette Kennedy's work explores this question in a range of logical settings, and introduces the concept of entanglement, which provides a new perspective on the nature of mathematical concepts. While Kennedy's work has had a significant impact on the philosophy of mathematics, it has also been criticized for its complexity and its lack of clarity. However, it is intended to be accessible to a wide range of people and is relevant to the practice of mathematics.

Weight: 302g
Dimension: 151 x 227 x 16 (mm)
ISBN-13: 9781108940573