Philosophical and Mathematical Logic

The author of this extremely interesting book, Harrie de Swart, is a disciple of the Dutch mathematician and philosopher, Johan de Iongh (1915–1999), to whom the present author has written a very poignant tribute presented on p. viii of the book. It is accordingly fair to say that this work’s philosophical framework includes influences from Platonism and intuitionism. The latter is not surprising, given that that de Iongh was a close friend and student of L. E. J. Brouwer. The former element is much more surprising, perhaps, since we do not really encounter such an explicit admission in most modern mathematical works. But de Iongh, in the tradition of Brouwer and Gerrit Mannoury, regarded mathematics as integrally tied to the act of doing and communicating mathematics, particularly with others. Says de Swart (loc.cit.):

 

Perhaps this evocative and idiosyncratic style of de Iongh can be credited with his pupil, de Swart, offering the book under review as both a textbook on mathematical logic (indeed on logic, as such, evolving into mathematical logic) and a discussion of something, for lack of a better word, cultural. “This book was written as an introduction to logic, with special emphasis on the interplay between logic and mathematics, philosophy, language and computer science.” Indeed, de Swart then goes on (p. xi) to give a compact but revealing account of what he is up to: 

 

That says a great deal: we have a very good idea of what to expect in the (over 500) pages that follow.

 

The book is very well-written and obviously includes some very serious mathematical logic — de Swart gives explicit credit Kleene’s classic books: Introduction to Metamathematics and Mathematical Logic. This is noteworthy in itself, perhaps, since there are so many other foundational texts on this material: I was taught some of this material at UCLA in the 1970s, and the texts by Enderton (A Mathematical Introduction to Logic) and Shoenfield (Mathematical Logic) dominated everything in those days. But that is neither here nor there. In any event, de Swart consistently goes very deep and happily includes a lot of exercises. For the reader for whom this subject, or these subjects with their particular philosophical connections, is (are) candy (even if he is a fledgling!) this book is a wonderful source indeed.

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.