The Logic of Number

NOTE EDITORE

In The Logic of Number, Neil Tennant defines and develops his Natural Logicist account of the foundations of the natural, rational, and real numbers. Based on the logical system free Core Logic, the central method is to formulate rules of natural deduction governing variable-binding number-abstraction operators and other logico-mathematical expressions such as zero and successor. These enable 'single-barreled' abstraction, in contrast with the 'double-barreled' abstraction effected by principles such as Frege's Basic Law V, or Hume's Principle. Natural Logicism imposes upon its account of the numbers four conditions of adequacy: First, one must show how it is that the various kinds of number are applicable in our wider thought and talk about the world. This is achieved by deriving all instances of three respective schemas: Schema N for the naturals, Schema Q for the rationals, and Schema R for the reals. These provide truth-conditions for statements deploying terms referring to numbers of the kind in question. Second, one must show how it is that the naturals sit among the rationals as themselves again, and the rationals likewise among the reals. Third, one should reveal enough of the metaphysical nature of the numbers to be able to derive the mathematician's basic laws governing them. Fourth, one should be able to demonstrate that there are uncountably many reals. Natural Logicism is realistic about the limits of logicism when it comes to treating the real numbers, for which, Tennant argues, one needs recourse to geometric intuition for deeper starting-points, beyond which logic alone will then deliver the sought results, with absolute formal rigor. The resulting program enables one to delimit, in a principled way, those parts of number theory that are produced by the Kantian understanding alone, and those parts that depend on recourse to (very simple) a priori geometric intuitions.

SOMMARIO

1 - What is Natural Logicism?2 - Before and after Frege3 - After Gentzen4 - Foundations after Gödel5 - Logico-Genetic Theorizing6 - Introduction, with Some Historical Background7 - Denoting Numbers8 - Exact Numerosity9 - The Adequacy Condition Involving Schema N10 - The Rules of Constructive Logicism11 - Formal Results of Constructive Logicism12 - Reflections on Counting13 - Formal Results about the Inductively Defined Numerically Exact Quantifiers14 - What Would a Gifted Child Need in order to Grasp Fractions? The Case of Edwin15 - Past Accounts of the Rationals as Ratios16 - Mereology and Fraction Abstraction17 - Taking Stock and Glimpsing Beyond18 - The Trend towards Arithmetization19 - Resisting the Trend towards Arithmetization20 - Impurities and Incompletenesses21 - The Concept of Real Number22 - Geometric Concepts and Axioms23 - Bicimals24 - Uncountability25 - Back to Bicimals

AUTORE

Neil Tennant is Arts & Sciences Distinguished Professor in Philosophy at The Ohio State University. He has previously held positions at the University of Edinburgh, the University of Stirling, and the Australian National University. He is the author of Anti-Realism and Logic: Truth as Eternal (1987), The Taming of The True (1997), Changes of Mind: An Essay on Rational Belief Revision (2012), and Core Logic (2017), all published by OUP.

ALTRE INFORMAZIONI

• Condizione: Nuovo
• ISBN: 9780192846679
• Dimensioni: 242 x 25.0 x 163 mm Ø 704 gr
• Formato: Copertina rigida
• Pagine Arabe: 376