Foundations Without Foundationalism: A Case For Second-Order Logic

by Stewart Shapiro

Shapiro, Stewart

Genres: Philosophy, Logic, Mathematics

300 pages, Hardcover

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.