Doing Logic. The Continuum through Sets

This book on sets and the continuum has three parts. The first part lays the set-theoretical and logical groundwork; the second presents the analysis of the continuum itself; and third reflects on the key ontological and epistemological debates.

The first part offers a gentle introduction to sets, a basis for the discussions that follow. But the reader should be forewarned that there is more going on. From the empty set to the innocent abstraction principle, the text inculcates the principles of set theory.

The second part concerns the analysis of the continuum. The continuum is divisible infinitely. But there is a great dispute over whether the continuum is divisible all the way down. It might instead consist of indivisible points that are bonded together into a continuity.

The third part then turns to three related debates. The first debate is over set-theoretic ontology. Link argues that set theory is not superfluous. The second debate is between realism and nominalism.

Link finds a way to leave this debate open. The third debate concerns logical construction and the nature of scientific reduction. Reductionism has been called a dogma of empiricism. But mathematical and physical explanations often involve reductions. The overall argument lies within this third debate.

The overall argument is highly controversial. There is not one way or one best way to know the continuum. Any reduction qualifies, once it meets the formal and mathematical requirements, as a legitimate way of knowing the continuum. There are many ways to know it; however, as Link reminds the reader, the continuum is explained in set theory. The sets themselves are real, or so he argues. Indeed, that is what makes the book controversial.

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A clear and thought-provoking investigation exploring the Continuum through a philosophical lens. The argumentation is thorough, mining from a logical background to explore and analyze the ontological and epistemological debate surrounding the Continuum and its nature.

Akihiro Kanamori - Boston University

Doing Logic-The Continuum through Sets is an ambitious and intellectually distinguished work. With clarity, philosophical force, and unusual breadth, Montgomery Link reopens one of the deepest problems in modern thought: the nature of the continuum and its relation to sets, logic, and reality. Bringing formal argument into sustained contact with historical and philosophical reflection, he develops an original and provocative account of the relations among set-theoretic ontology, realism, nominalism, and logical construction. The result is rigorous, wide-ranging, and consistently stimulating. Scholars in philosophy of mathematics, logic, and the history and foundations of modern thought will find here a serious and rewarding contribution. It is a book that invites careful reading and repays it richly.Raffaele Pisano - University of Lille, France

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Montgomery Link is Associate Professor, Suffolk University. He lives with his wife and four children in Bullfinch Triangle in Boston and walks to work on Beacon Hill.