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Philosophy and Model Theory
Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics.
But these wide-ranging uses of model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers.
The first aim of this book, then, is to explore the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures.
Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs.
But these wide-ranging uses of model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers.
The first aim of this book, then, is to explore the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures.
Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs.
512 pages, Paperback
First published March 1, 2018
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About the author
Tim Button
6 books13 followersTim Button is a Senior Lecturer, and a Fellow of St John's College, at the University of Cambridge. His first book, The Limits of Realism (OUP 2013) explores the relationship between words and world; between semantics and scepticism. His main research interests lie in meta(meta)physics, logic, mathematics, and language. In 2014 he received a Philip Leverhulme Prize.
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Displaying 1 - 2 of 2 reviews
August 2, 2019
A book which summarises the major themes of model theory from a philosophical point of view. While knowledge of mathematical logic is not precisely assumed, the pace is such that the book is unlikely to make much sense to anyone without a postgraduate background in mathematical logic.
The aim is to make clear some of the implications of the work that has been done in model theory for the philosophy of mathematics. The presentation is Socratic, in that naive views of how models in mathematical logic might work are shown to be insufficient by work in model theory. While this is an excellent way to point out the importance of model theory to the rest of mathematics, it does introduce an element of repetitiveness to the reader, one reason why it took me a long time to work through the book.
I would recommend this book to those with an interest in its topics and the background to follow it; it is clear, precise and concise - the last perhaps to a fault, as it would be hard to follow, for instance, the summary proof given of Gödel's Completeness Theorem if it is a result for which a reader had not already seen a more detailed proof.
The book concludes with an appendix, describing the history of model theory by Wilfrid Hodges, whom I knew as dean of the mathematics department where I studied as a postgraduate. This is an admirable description of the history of this branch of mathematics, both as seen from the inside and in relation to the rest of mathematics, with a personal touch.
The aim is to make clear some of the implications of the work that has been done in model theory for the philosophy of mathematics. The presentation is Socratic, in that naive views of how models in mathematical logic might work are shown to be insufficient by work in model theory. While this is an excellent way to point out the importance of model theory to the rest of mathematics, it does introduce an element of repetitiveness to the reader, one reason why it took me a long time to work through the book.
I would recommend this book to those with an interest in its topics and the background to follow it; it is clear, precise and concise - the last perhaps to a fault, as it would be hard to follow, for instance, the summary proof given of Gödel's Completeness Theorem if it is a result for which a reader had not already seen a more detailed proof.
The book concludes with an appendix, describing the history of model theory by Wilfrid Hodges, whom I knew as dean of the mathematics department where I studied as a postgraduate. This is an admirable description of the history of this branch of mathematics, both as seen from the inside and in relation to the rest of mathematics, with a personal touch.
October 23, 2024
10/10 - There are some bits I haven’t read yet but others I’ve read 10 times. Amazing resource, sets an incredibly high standard for textbooks in terms of scope, style and completion.
Displaying 1 - 2 of 2 reviews