Abstract.
This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiro’s identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiro’s claim that it is not possible to identify objects in a structure except through the relations and functions that are defined on the structure in which the object has a place. I argue that, in the case of the definition of the so called direct image of a function, it is possible to individuate objects in structures.
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References
A. Awodey (1996) ArticleTitle‘Structure in Mathematics and Logic: A Categorical Perspective’ Philos. Math 4 209–237
P. Benacerraf (1965) ArticleTitle‘What Numbers Could Not Be’ Philosophical Review 74 47–73
A. Borel J. P. Serre (1958) ArticleTitle‘Le Theoreme de Riemann-Roch’ Bull. Soc. Math. France 86 97–136
Carter, J.: 2002, Ontology and Mathematical Practice, Ph.D-thesis, University of Southern Denmark.
D. Corfield (1998) ArticleTitle‘Beyond the Methodology of Mathematics Research Programmes’ Philos. Math 6 272–301
R. Dedekind (1996) ‘Was sind und was sollen die Zahlen’ W. B. Ewald (Eds) From Kant to Hilbert. A Sourcebook in the Foundations of Mathematics. Oxford University Press New York 787–833
E. Grosholz H. Breger (Eds) (2000) The Growth of Mathematical Knowledge Kluwer Academic Publishers Dordrecht
Grothendieck, A.: 1956/1957, ‘Sur les faisceaux algébriques et les faisceaux analytiques cohéherents’, Séminaire H. Cartan, E.N.S..
G. Hellman (2001) ArticleTitle‘Three Varieties of Mathematical Structuralism’ Philos. Math 9 184–211
J. Keránen (2001) ArticleTitle‘The Identity Problem for Realist Structuralism’ Philos. Math 9 308–330
K. Kodaira D. C. Spencer (1953) ArticleTitle‘On Arithmetic Genera of Algebraic Varieties’ Proc. Nat. Acad. Sci. USA 39 641–649
F.W. Lawvere (1966) ‘The Category of Categories as a Foundation of Mathematics’ Proc. of the Conference on Categorical Algebra (La Jolla 1965) Springer Verlag New York 1–20
S. Mac Lane (1996) ArticleTitle‘Structure in Mathematics’ Philos. Math 4 174–183
C. McLarty (1993) ArticleTitle‘Numbers Can be Just What They Have To’ Nous 27 487–498 Occurrence HandleMR1262549
P. Maddy (1997) Naturalism in Mathematics Clarendon Press Oxford
C. Parsons (1990) ‘The Structuralist View of Mathematical Objects’ W. D. Hart (Eds) The Philosophy of Mathematics Oxford University Press Oxford 272–309
M. Resnik (1997) Mathematics as a Science of Patterns Oxford University Press New York
S. Shapiro (1997) Philosophy of Mathematics. Structure and Ontology Oxford University Press Oxford
Shapiro, S.: forthcoming, ‘Structure and Identity’, to appear in F. Mac Bride and C. Wright (eds), Identity and Modality: New Essays in Metaphysics and the Philosophy of Mathematics, Oxford University Press.
G. Weaver (1998) ArticleTitle‘Structuralism and Representation Theorems’ Philos. Math 6 257–271
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Carter, J. Individuation of objects – a problem for structuralism?. Synthese 143, 291–307 (2005). https://doi.org/10.1007/s11229-005-0848-x
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DOI: https://doi.org/10.1007/s11229-005-0848-x