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020 _a9783319004044
_9978-3-319-00404-4
024 7 _a10.1007/978-3-319-00404-4
_2doi
050 4 _aBD143-237
072 7 _aHPK
_2bicssc
072 7 _aPHI004000
_2bisacsh
082 0 4 _a120
_223
100 1 _aRodin, Andrei.
_eauthor.
245 1 0 _aAxiomatic Method and Category Theory
_h[electronic resource] /
_cby Andrei Rodin.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2014.
300 _aXI, 285 p. 63 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSynthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science ;
_v364
505 0 _aIntroduction -- Part I A Brief History of the Axiomatic Method -- Chapter 1. Euclid: Doing and Showing -- Chapter 2. Hilbert: Making It Formal -- Chapter 3. Formal Axiomatic Method and the 20th Century Mathematics -- Chapter. 4 Lawvere: Pursuit of Objectivity -- Conclusion of Part 1 -- Part II. Identity and Categorification -- Chapter 5. Identity in Classical and Constructive Mathematics -- Chapter 6. Identity Through Change, Category Theory and Homotopy Theory -- Conclusion of Part 2 -- Part III. Subjective Intuitions and Objective Structures -- Chapter 7. How Mathematical Concepts Get Their Bodies. Chapter 8. Categories versus Structures -- Chapter 9. New Axiomatic Method (instead of conclusion) -- Bibliography.
520 _aThis volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.
650 0 _aPhilosophy (General).
650 0 _aGenetic epistemology.
650 0 _aAlgebra.
650 0 _aLogic, Symbolic and mathematical.
650 1 4 _aPhilosophy.
650 2 4 _aEpistemology.
650 2 4 _aCategory Theory, Homological Algebra.
650 2 4 _aMathematical Logic and Foundations.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783319004037
830 0 _aSynthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science ;
_v364
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-319-00404-4
912 _aZDB-2-SHU
999 _c92475
_d92475