Philosophical Uses of Categoricity Arguments

This Element examines the viability of categoricity arguments in philosophy, comparing and contrasting earlier uses with more recent work. It concludes that categoricity arguments have been more effective in historical cases that reflect on internal mathematical matters than in recent questions of pre-theoretic metaphysics.

Format: Paperback / softback
Length: 64 pages
Publication date: 21 December 2023
Publisher: Cambridge University Press

This Element delves into the viability of categoricity arguments in philosophy, offering a thoughtful exploration of the specific conclusions drawn by prominent figures. It begins by questioning the conventional interpretations of Dedekind, Zermelo, and Kreisel, casting doubt on their readings and highlighting their achievements in attaining their intended goals. Subsequently, these earlier uses of categoricity arguments are compared and contrasted with the more recent work of Parsons and his co-authors, Button, and Walsh. The Element emphasizes the roles of first- and second-order theorems, external and internal theorems, and the distinction between pre-theoretic metaphysics and historical cases that reflect on internal mathematical matters. Through this analysis, the Element concludes that categoricity arguments have demonstrated greater effectiveness in addressing historical cases that delve into philosophical inquiries related to internal mathematical concepts compared to contemporary questions of pre-theoretic metaphysics.

Introduction:
Categoricity arguments have played a significant role in philosophical discussions, particularly in the realm of mathematics. These arguments aim to establish the validity of certain mathematical statements or theorems based on their logical structure and the principles of logic. While categoricity arguments have been widely studied and debated, their viability and effectiveness in different contexts have been subject to scrutiny. This Element aims to explore the extent to which categoricity arguments can be successful in supporting specific conclusions in philosophy.
Categoricity Arguments in Philosophy:
Categoricity arguments typically involve the use of logical principles and theorems to establish the validity of mathematical statements or theorems. These arguments rely on the idea that certain logical relations hold between mathematical statements and other mathematical statements or theorems. For example, a first-order theorem is a mathematical statement that can be proved using other mathematical statements or theorems within the same mathematical system. A second-order theorem is a mathematical statement that can be proved using first-order theorems within the same mathematical system.
Theorems of Dedekind, Zermelo, and Kreisel:
Dedekind, Zermelo, and Kreisel were influential figures in the development of set theory and logic. They made significant contributions to the field, including the development of the concept of a Dedekind cut and the Zermelo-Fraenkel axiom system. However, their work on categoricity arguments has been subject to criticism and debate.
Doubt on Received Readings:
One of the criticisms of Dedekind, Zermelo, and Kreisel's work on categoricity arguments is that their interpretations of certain theorems have been disputed. For instance, some scholars argue that their readings of the axiom of choice and the continuum hypothesis are incorrect. These scholars suggest that the axiom of choice is not necessary for the existence of infinite sets, and that the continuum hypothesis is false.
Success of Categoricity Arguments:
Despite these criticisms, Dedekind, Zermelo, and Kreisel were successful in achieving their intended goals. They demonstrated that certain mathematical statements or theorems can be proved using other mathematical statements or theorems within the same mathematical system. This achievement is significant because it provides a foundation for the validity of mathematical statements and theorems.
Comparison with Parsons and Co-authors:
In recent years, Parsons and his co-authors have made significant contributions to the study of categoricity arguments. They have developed new methods and techniques for proving categoricity theorems and have applied these methods to various mathematical systems.
Conclusion:
In conclusion, categoricity arguments have been a subject of debate and discussion in philosophy. While some criticisms have been raised against the work of Dedekind, Zermelo, and Kreisel, their success in proving certain mathematical statements or theorems using other mathematical statements or theorems within the same mathematical system is significant. Parsons and his co-authors have made valuable contributions to the study of categoricity arguments, and their work has demonstrated the potential of categoricity arguments in addressing philosophical questions related to mathematics. However, further research and exploration are needed to fully understand the implications and limitations of categoricity arguments in philosophy.

Weight: 110g
Dimension: 150 x 228 x 6 (mm)
ISBN-13: 9781009432924