Foundations without Foundationalism : A Case for Second-Order Logic

ISBN-13: 9780198533917 / Angielski / Twarda / 1991 / 300 str.

Stewart Shapiro

Opis

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today.

Recenzje

Clearly written and well produced, making it a pleasure to read . . . Shapiro's excellent book is for just about everyone. Logicians and philosophers of mathematics will certainly want to read it; but metaphysicians and philosophers of science and of language will greatly benefit, too.

Spis treści

PART I: ORIENTATION; Terms and questions; Foundationalism and foundations of mathematics; PART II: LOGIC AND MATHEMATICS; Theory; Metatheory; Second-order logic and mathematics; Advanced metatheory; PART III: HISTORY AND PHILOSOPHY; The historical triumph of first-order languages; Second-order logic and rule-following; The competition; Index.