The Logic of Number

Neil Tennant's Natural Logicist account of the foundations of natural, rational, and real numbers is based on the logical system free Core Logic. It formulates rules of natural deduction governing variable-binding number-abstraction operators and other logico-mathematical expressions, enabling single-barreled abstraction. It imposes four conditions of adequacy: showing how the various kinds of number are applicable, deriving schemas for them, showing how they sit among the rationals and reals, and demonstrating that there are uncountably many reals.

Format: Hardback
Length: 384 pages
Publication date: 25 February 2022
Publisher: Oxford University Press

In his groundbreaking work, "The Logic of Number," Neil Tennant presents a comprehensive and intricate account of the foundations of natural, rational, and real numbers. Building upon the framework of free Core Logic, Tennant's central approach revolves around formulating rules of natural deduction that govern the behavior of variable-binding number-abstraction operators and other logical-mathematical expressions, including zero and successor. This innovative methodology allows for a single-barreled abstraction, in contrast to the double-barreled abstraction achieved through principles such as Frege's Basic Law V or Hume's Principle.

To ensure the adequacy of his Natural Logistic account, Tennant imposes four crucial conditions upon it. Firstly, he must demonstrate how the various kinds of numbers find application in our broader thought and discourse about the world. This is accomplished by deriving all instances of three distinct schemas: Schema N for the naturals, Schema Q for the rationals, and Schema R for the reals. These schemas provide truth-conditions for statements that involve terms referring to numbers of the respective kind.

Secondly, Tennant must establish the relationship between the naturals and the rationals, as well as between the rationals and the reals. This involves revealing sufficient aspects of the metaphysical nature of numbers to derive the mathematician's fundamental laws governing them.

Thirdly, Tennant aims to unveil the sheer abundance of reals by demonstrating that there exist an uncountably infinite number of them. While Natural Logicism acknowledges the limitations of logicism when dealing with the real numbers, it recognizes the need for recourse to geometric intuition for deeper starting points beyond which logic alone can yield the desired results.

The culmination of Tennant's program is a comprehensive framework that enables the principled delimitation of number theory's aspects that are generated by the Kantian u. This program provides a solid foundation for understanding the fundamental properties and behaviors of natural, rational, and real numbers, offering a profound insight into the intricate nature of mathematical inquiry.

Weight: 704g
Dimension: 242 x 163 x 25 (mm)
ISBN-13: 9780192846679