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Ralf Krömer
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Tool and Object: A History and Philosophy of Category Theory (Science Networks. Historical Studies, 32) 2007th Edition

by Ralph Krömer (Author)
4.5 4.5 out of 5 stars 3 ratings
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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.

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  1. ISBN-10
    376437523X
  2. ISBN-13
    978-3764375232
  3. Edition
    2007th
  4. Publisher
    Birkhäuser
  5. Publication date
    February 16, 2007
  6. Book 6 of 29
    Science Networks. Historical Studies
  7. Language
    English
  8. Dimensions
    6.75 x 1.25 x 9.5 inches
  9. Print length
    403 pages
  10. See all details
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Editorial Reviews

Review

From the reviews:

“Krömer provides a more complete historical study of category theory and its applications from the beginnings to about 1970. … It caught the attention of people interested in conceptual and philosophical issues … . I recommend Krömer’s work highly to anyone who seeks to gain an understanding of category-theoretic mathematics … . an interesting contribution to the historical philosophy of mathematical practice … . the book is more successful as a history. It is well produced, and the English is clear enough … .” (José Ferreirós, Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin, Issue 1, 2010)

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

  • Publisher ‏ : ‎ Birkhäuser; 2007th edition (February 16, 2007)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 403 pages
  • ISBN-10 ‏ : ‎ 376437523X
  • ISBN-13 ‏ : ‎ 978-3764375232
  • Item Weight ‏ : ‎ 1.9 pounds
  • Dimensions ‏ : ‎ 6.75 x 1.25 x 9.5 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 3 ratings

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Ralf Krömer

Ralf Krömer

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4.5 out of 5 stars
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Top reviews from the United States

  • Luis Estrada González
    4.0 out of 5 stars Review of Krömer's Tool and Object
    Reviewed in the United States on September 29, 2008
    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.
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  • A. da Veiga
    4.0 out of 5 stars Útil
    Reviewed in Spain on July 16, 2023
    Verified Purchase
    Um livro muito útil para perceber a história das categorias.
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