Tool and Object: A History and Philosophy of Category Theory

ISBN-13: 9783764375232 / Angielski / Twarda / 2007 / 367 str.

Ralf Krvmer; Ralf Krc6mer; Ralf Krmer

Opis

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.

Recenzje

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)

Spis treści

Prelude: Poincaré, Wittgenstein, Peirce, and the use of concepts.- Category theory in Algebraic Topology.- Category theory in Homological Algebra.- Category theory in Algebraic Geometry.- From tool to object: full-fledged category theory.- Categories as sets: problems and solutions.- Categorial foundations.- Pragmatism and category theory.- Abbreviations.- Bibliography.- Indexes

Blurb

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.