Structuralism and the notion of dependence

Philosophical Quarterly

Volumn 58, Issue 230, 2008, Pages 59-79

  Linnebo, Øystein ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

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[No Author keywords available]

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EID: 60949152238     PISSN: 00318094     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1467-9213.2007.529.x     Document Type: Article

References (42)

1. See W.V. Quine, 'Structure and Nature', Journal of Philosophy, 89 (1992), pp. 5-9, for an example of a semantic form of global structuralism
2. See J. Ladyman, 'What is Structural Realism?', Studies in the History and Philosophy of Science, 29 (1998), pp. 409-24, for an example of structuralism about the physical world
3. H. Putnam, 'The Thesis that Mathematics is Logic', repr. in his Mathematics, Matter and Method (Cambridge UP, 1975), pp. 12-42
4. An early example is Putnam, 'Mathematics without Foundations', Journal of Philosophy, 64 (1967), pp. 5-22
5. The most developed version is G. Hellman, Mathematics without Numbers (Oxford: Clarendon Press, 1989)
6. M. Resnik, 'Mathematics as a Science of Patterns: Ontology and Reference', Noûs, 15 (1981), pp. 529-50, at p. 530
7. a passage quoted approvingly in a large number of defences of structuralism, e.g., C. Parsons, 'The Structuralist View of Mathematical Objects', Synthese, 84 (1990), pp. 303-46, at p. 303
8. S. Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford UP, 1997), p. 75
9. Shapiro, Thinking about Mathematics (Oxford UP, 2000), p. 258
10. Resnik, Mathematics as a Science of Patterns (Oxford UP, 1997), p. 90; see also p. 210
11. Shapiro, Philosophy of Mathematics: Structure and Ontology, p. 79; see also pp. 80-1, 258ff
12. See, e.g., Parsons, 'Mathematical Intuition', Proceedings of the Aristotelian Society, 80 (1979/80), pp. 145-68, at §II
13. and 'The Structuralist View of Mathematical Objects', pp. 334-5
14. Resnik, Mathematics as a Science of Patterns, pp. 90, 211
15. Resnik, 'Mathematics as a Science of Patterns: Ontology and Reference', p. 529
16. Parsons, 'Structuralism and Metaphysics', The Philosophical Quarterly, 54 (2004), pp. 56-77, at p. 57
17. See J.P. Burgess, review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology, Notre Dame Journal of Formal Logic, 40 (1999), pp. 283-91, at p. 286
18. Ø. Linnebo, Critical Notice of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology, Philosophia Mathematica, 11 (2003), pp. 92-104, at pp. 97-9
19. F. MacBride, 'Structuralism Reconsidered', in S. Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic (Oxford: Clarendon Press, 2005), pp. 563-89, at pp. 583-4
20. See Shapiro, 'Structure and Identity', in F. MacBride (ed.), Identity and Modality (Oxford: Clarendon Press, 2006), pp. 109-45, at pp. 121-31
21. See Shapiro, 'Structure and Identity', p. 115
22. See B. Weatherson, 'Intrinsic vs Extrinsic Properties', in E.N. Zalta (ed.), Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/archives/ fall2006/entries/intrinsic-extrinsic
23. Resnik, 'Mathematics as a Science of Patterns: Epistemology', Noûs, 16 (1982), pp. 95-105, at p. 95
24. E.J. Lowe, in his 'Individuation', in M. Loux and D. Zimmerman (eds), Oxford Handbook of Metaphysics (Oxford UP, 2003), pp. 75-95
25. and 'Ontological Dependence', in Zalta (ed.), Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/archives/sum2005/entries/dependence- ontological, http://plato.stanford.edu/archives/sum2005/entries/dependence- ontological, defends the ban on cyclical relations of dependence
26. K. Fine, 'Ontological Dependence', Proceedings of the Aristotelian Society, 95 (1995), pp. 269-90, defends a qualified form of the principle (WF)
27. This principle is also implicit in Hellman, 'Three Varieties of Mathematical Structuralism', Philosophia Mathematica, 9 (2001), pp. 184-211
28. and in MacBride, 'What Constitutes the Numerical Diversity of Mathematical Objects?', Analysis, 66 (2006), pp. 63-9, to be discussed below
29. Hellman, 'Three Varieties of Mathematical Structuralism', pp. 193-4
30. Another similar passage is found in Hellman, 'Structuralism', in Shapiro (ed.), Oxford Handbook of the Philosophy of Mathematics and Logic, pp. 536-62, at p. 545
31. Indeed, in his 'Structuralism Reconsidered', in Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic, pp. 563-89, at p. 581, MacBride expresses serious reservations about this kind of argument
32. See MacBride, 'What Constitutes the Numerical Diversity of Mathematical Objects?', p. 67
33. See H. Allison, Kant's Transcendental Idealism (Yale UP, 1986), p. 43
34. See Parsons, 'The Structuralist View of Mathematical Objects', pp. 337-8
35. See G. Boolos, 'The Iterative Conception of Set', Journal of Philosophy, 68 (1971), pp. 215-32
36. Lowe, 'Ontological Dependence'
37. See Parsons, 'Structuralism and the Concept of Set', in W. Sinnott-Armstrong (ed.), Modality, Morality, and Belief (Cambridge UP, 1994), pp. 74-92
38. It will allow us to derive Burali-Forti's paradox: see A.P. Hazen, review of Crispin Wright, Frege's Conception of Numbers as Objects, Australasian Journal of Philosophy, 63 (1985), pp. 250-4, at pp. 253-4
39. For a more systematic and satisfactory solution, see Ø. Linnebo, 'Bad Company Tamed', forthcoming in Synthese, 2008
40. What about the positions of non-rigid abstract structures? Are they too objects? If so, how are they individuated? For the purposes of establishing my compromise view, I need not take a stand on these difficult questions. For discussion, see J. Keränen, 'The Identity Problem for Realist Structuralism', Philosophia Mathematica, 9 (2001), pp. 308-30
41. Shapiro, Philosophy of Mathematics, ch. 3
42. See Fine, 'Essence and Modality', inj. Tomberlin (ed.), Philosophical Perspectives, 8: Language and Logic (Atascadero: Ridgeview, 1994), pp. 1-16