Foundations without Foundationalism: A Case for Second-order Logic

by Stewart Shapiro

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.

Genres: Philosophy, Logic, Mathematics

300 pages, Paperback

First published November 7, 1991

Reviews

[TjeerdFokkens · Rating 3 out of 5]

In this book Shapiro makes a case for the use of second order logic in the foundations of mathematics. He outlines the generally accepted arguments for the use first-order logic and demonstrates that second order logic is not underperforming on many aspects. Also, he gives the categoricity argument in favour of second order logic. It's an interesting topic, but I think the book could have been a bit more concise and to the point. Some parts of it were not really intriguing.

[Mike · Rating 5 out of 5]

A powerful, pithy, and well-reasoned book. Disturbingly to me, more and more students and young scholars of supposed "philosophy" I meet have little grounding in math nor in neuroscience, which seems to exclude a vast amount of what philosophy as a field must consider. This book reconsiders the very structure of mathematics in terms of greater applications of logic and how study of math as a discipline has come to exclude some of the most crucial underpinnings of ontological epistemology.

There will be other scholars who argue with some of Shapiro's finer points, but that's expected: his understanding of semantics and his application of such to mathematics is astoundingly sure and resounding. This book should be read by anyone working in maths, philosophy, or linguistics.

[Harry Vincent · Rating 4 out of 5]

8.5/10 - Contains one of the clearest introductions to model theory that I have come across. His arguments for the increased use of second order logic are very persuasive and there’s a good mixture of more technical logic results and philosophical discussion. Feels a bit lacking in a conclusion and/or more thorough coverage of his general point of view and thesis of the book, just to tie it together as a read.

[J. · Rating 4 out of 5]

In the past, I've lacked the courage to teach ('teach') this book, but I'm resolved not to offer a logic course again without incorporating it.