Truth, Proof and Infinity: A Theory of Constructive Reasoning (Synthese Library, 276) 1998th Edition by P. Fletcher (PDF)

Description

Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms `construction’ and `proof’ has never been adequately explained (although Kriesel, Goodman and Martin-Löf have attempted axiomatisations). This monograph develops precise (though not wholly formal) definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of Heyting arithmetic and constructive analysis are given. The philosophical basis of constructivism is explored thoroughly in Part I. The author seeks to answer objections from platonists and to reconcile his position with the central insights of Hilbert’s formalism and logic. Audience: Philosophers of mathematics and logicians, both academic and graduate students, particularly those interested in Brouwer and Hilbert; theoretical computer scientists interested in the foundations of functional programming languages and program correctness calculi.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is not a traditional mathematics book. In the first part, an informal discussion on exactly the three subjects that make up the title, no theorems are formulated and then proven. In the second part, a “Term Language for Constructions” and subsequently a “Protologic expressing the relations between constructions” are worked out.As such, the entire book is implicitly an answer to the question formulated in the first sentence of Chapter 1: “Geology is the study of rocks; astronomy is the study of stars; what is mathematics the study of ?” The answer seems to be: “Mathematics is what mathematicians do, and as such a highly enjoyable, intellectual exercise, the usefulness of which is only subject to pointless questioning”.The book only gets three stars out of five, however, for voicing in many places an author’s point of view, and for the frequent use of the word “I”: “I quarrel with two features of lambda calculus: extensionality and normal-order reduction…”, p 142, as an example. Other sections are nothing but the author’s critique of Brouwer’s Intuitionism, which reduce the value of the book.

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