Tool and Object: A History and Philosophy of Category Theory (英語) ハードカバー – 2007/2/16
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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.
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)
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.