Tool and Object: A History and Philosophy of Category Theory (Science Networks. Historical Studies) [Hardcover]

Ralf Krömer (Author)

Book Description

ISBN-10: 376437523X | ISBN-13: 978-3764375232 | Publication Date: March 28, 2007 | Edition: 1

Category theory is a general mathematical theory of structures and of structures of structures. It occupied a central position in contemporary mathematics as well as computer science. This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.

Editorial Reviews

From the Back Cover

The book is first of all a history of category theory from the beginnings to A. Grothendieck and F.W. Lawvere. Category theory was an important conceptual tool in 20th century mathematics whose influence on some mathematical subdisciplines (above all algebraic topology and algebraic geometry) is analyzed. Category theory also has an important philosophical aspect: on the one hand its set-theoretical foundation is less obvious than for other mathematical theories, and on the other hand it unifies conceptually a large part of modern mathematics and may therefore be considered as somewhat fundamental itself. The role of this philosophical aspect in the historical development is the second focus of the book. Relying on the historical analysis, the author develops a philosophical interpretation of the theory of his own, intending to get closer to how mathematicians conceive the significance of their activity than traditional schools of philosophy of science. The book is the first monography exclusively devoted to the history of category theory. To a substantial extent it considers aspects never studied before. The author uses (and justifies the use of) a methodology combining historical and philosophical approaches. The analysis is not confined to general remarks, but goes into considerable mathematical detail. Hence, the book provides an exceptionally thorough case study compared with other works on history or philosophy of mathematics. The philosophical position developed here (inspired by Peircean pragmatism and Wittgenstein) is an interesting alternative to traditional approaches in philosophy of mathematics like platonism, formalism and intuitionism.

Product Details

• Hardcover: 403 pages
• Publisher: Birkhäuser Basel; 1 edition (March 28, 2007)
• Language: English
• ISBN-10: 376437523X
• ISBN-13: 978-3764375232
• Product Dimensions: 9.1 x 6.2 x 1.1 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #266,410 in Books (See Top 100 in Books)

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13 of 13 people found the following review helpful:

4.0 out of 5 stars Review of Krömer's Tool and Object, September 29, 2008

By 

Luis Estrada González (PhD Student, UAEM, Mexico) - See all my reviews
(REAL NAME)   

This review is from: Tool and Object: A History and Philosophy of Category Theory (Science Networks. Historical Studies) (Hardcover)

The main goal of the book under review is to provide a systematic and profound analysis of several important historical and epistemological aspects of category theory. The central theme of the book is to give an analysis of the remarkable fact that category theory gained position in daily mathematics as a useful and legitimate conceptual innovation, in spite of the difficulties of its set-theoretical foundations and the challenge it caused to this formerly well-established mathematical foundation and to some epistemological positions.
The philosophical stance to category theory developed here is inspired by the pragmatism of Peirce and by Wittgenstein's criticisms of reductionism, which represents a highly interesting alternative to more traditional approaches in philosophy of mathematics like logicism, intuitionism, formalism, realism, fictionalism, etc. In this vein, the author's philosophical position focusses on the "use" of concepts, instead of formal syntax and semantics, and on the thesis that that philosophical justification of mathematical reasoning is an accurate description of the way mathematicians work with categories.
I missed, in the context of a philosophy of category theory, more detailed discussions on some category-theorists philosophical positions, like Lawvere's "dialectical" philosophy of mathematics, different versions of structuralism and different "topos foundations" (for instance, those of Lambek, Bell, Mac Lane) and in this sense the book is more a history than a philosophy of category theory. Some passages are obscured rather than clarified by the philosophical tone, and a methodological fault is that the author sometimes regards spontaneous declarations of some mathematicians as well-elaborated philosophical conceptions or official historical explanations.
Nonetheless, this work is a serious attempt to discuss the history and a philosophy of category theory, and historians of mathematics, philosophers of mathematics, and also "working" mathematicians can profit to a large extent from Krömer's analysis.