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Ontological Dependence and Grounding for a Weak Mathematical Structuralism

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Objects, Structures, and Logics

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 339))

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Abstract

In the philosophy of science, Weak Structural Realism (WSR) offers a promising priority-based strategy to avoid the main objection to eliminative Ontic Structural Realism (OSR). On that view, quantum particles depend for their identity on quantum entanglement structures but are defined as not entirely structural thin physical objects. A similar approach can be applied to mathematical structuralism, where Weak Mathematical Structuralism (WMS) provides a novel, more moderate interpretation of ante rem structuralism. WMS is articulated in terms of grounding: numbers are grounded for their identity in the abstract structure they belong to. However, they are not completely reduced to their structural features and are re-conceptualized as thin mathematical objects, endowed with both structural and non-structural properties. The introduction of such objects in the structural ontology allows to escape some typical objections to ante rem structuralism without abandoning the priority of structures.

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Notes

  1. 1.

    Cf. French and Krause (2006).

  2. 2.

    French specifically refers to Tahko and Lowe’s (2016) analysis of dependence, in which different accounts are illustrated.

  3. 3.

    Tahko and Lowe (2016, sec. 2.1.) consider as an example the relation between Socrates and its life, which are said to be dependent on each other.

  4. 4.

    Such view is generally supported by a relational interpretation of properties and by a discussion of quantum entanglement in terms of non-separability.

  5. 5.

    Analogous considerations apply to the mathematical framework and meet with similar difficulties (cf. sec. 5).

  6. 6.

    Lowe (2012) raises a more specific objection, according to which a coherent structuralist ontology should include at least some self-individuating entities.

  7. 7.

    Cf. Keränen (2001, p. 313)

  8. 8.

    Symmetry groups transformations are specifically presupposed by Quantum Field Theory (QFT) and represent the mathematical counter-part of quantum entanglement states. This interpretation traces back to Cassirer, Born, Weyl and Eddington.

  9. 9.

    Several attempts to apply this reduction have been performed (cf. Wigner’s, 1939, original characterization of properties; more recently, Castellani 1998). However, these structuralist strategies proved to be largely unsuccessful.

  10. 10.

    Still, this is controversial; for a discussion about kind properties and intrinsic properties see McKenzie (forthcoming).

  11. 11.

    On the one hand, ontological realism embraces a convincing semantics, whereby mathematical statements are interpreted at face value. Still, the abstract nature of these objects – which makes them not located in space and time and not causally effective – introduces serious epistemological problems. On the other hand, anti-realism about objects ensures a more straightforward epistemology but cannot account for a corresponding semantics, which would require objects to refer to.

  12. 12.

    Shapiro (2006, 2008) has more recently developed a more moderate position about objects, according to which they possess non-structural properties as well (the property of being abstract, non-spatio-temporal, of not entering in any causal relation, etc., 2006, p. 116). Still, he has not really developed a view that accounts for mathematical objects in these terms.

  13. 13.

    See Leitgeb (2020, part B sec.1–3) for this useful distinction.

  14. 14.

    This worry has been introduced in Russell (1903), Benacerraf (1965) and Kitcher (1983).

  15. 15.

    Internal symmetries that are not identity mappings.

  16. 16.

    This means to reject the grounding-reduction link formulated by Rosen (2010), according to which if p reduces to q, then q grounds p: assuming grounding as a reductive notion would commit to an identity relation between the two facts – and not just to the claim that the grounded facts are less fundamental than or ontologically secondary to the groundees grounding them, which is the view here proposed.

  17. 17.

    Both claims are implicitly presupposed by Shapiro (2000, p. 253): «the number 2 is no more and no less than the second position in the natural number structure; and 6 is the sixth position. Neither of them has any independence from the structure in which they are positions, and as positions in this structure, neither number is independent of the other».

  18. 18.

    Linnebo refers to ‘abstract offices’ in algebraic structures as places in structures obtained by a process of Dedekind abstraction, mapping a system to its abstract structure. While in a system offices can be filled by different sorts of occupants, the corresponding abstract structure «is left with nothing but the offices themselves» (Linnebo 2008, p. 75).

  19. 19.

    Shapiro (2008, p. 302) himself has rejected a form of existential dependence, given that mathematical objects necessarily exist.

  20. 20.

    This distinction has been introduced by Fine (2012).

  21. 21.

    Among others, Dasgupta (2014) Raven (2015) and Thompson (2018) identify the two notions.

  22. 22.

    This is not the case for dependence, which can be reflexive (i.e. an entity ontologically depends on itself).

  23. 23.

    See Schiemer and Korbmacher (2017) for a distinction between Linnebo’s invariance account and Shapiro’s (2008) definability account of structural properties in ante rem structuralism.

  24. 24.

    Cf. Esfeld and Lam’s (2011, p. 144) definition in scientific WSR (Sect. 7.4).

  25. 25.

    This understanding of kind properties presupposes to interpret numbers as cardinals, rather than as ordinals.

  26. 26.

    In principle, ante rem structuralism is not inconsistent with platonism about objects (one can be committed to a background ontology of self-standing structures and yet admit objects, i.e. the natural numbers, which possess intrinsic properties and exemplify a specific structure); however, the same position appears quite odd if applied to Shapiro’s places as objects which – by definition – have no more than their structural relations.

  27. 27.

    Moreover, it is worth noting that intrinsic properties are consistent with other structuralist views, i.e. set-theoretic structuralism (in which sets have intrinsic properties of membership, making them ‘self-standing’) and in re structuralism (where any possible object can belong to the structures).

References

  • Audi, P. 2012. Grounding: Toward a theory of the in-virtue-of relation. Journal of Philosophy 109: 685–711.

    Article  Google Scholar 

  • Benacerraf, P. 1965. What numbers could not be. Philosophical Review 74: 47–73.

    Article  Google Scholar 

  • Burgess, J. 1999. Review of Stewart Shapiro (1997). Notre Dame Journal of Formal Logic 40: 283–291.

    Google Scholar 

  • Busch, J. 2003. What structures could not be. International Studies in the Philosophy of Science 17: 211–225.

    Article  Google Scholar 

  • Button, T. 2006. Realistic structuralism’s identity crisis: A hybrid solution. Analysis 66: 216–222.

    Article  Google Scholar 

  • Cao, T. 2003. Can we dissolve physical entities into mathematical structures? In Symonds (Ed.), Special issue: Structural realism and quantum field theory. Synthese 136 (1): 57–71.

    Article  Google Scholar 

  • Castellani, E. 1998. Galilean particles: An example of constitution of objects. In Interpreting Bodies: Classical and Quantum Objects in Modern Physics, ed. E. Castellani, 181–194. Princeton: Princeton Uni- versity Press.

    Google Scholar 

  • Chakravartty, A. 1998. Semirealism. Studies in History and Philosophy of Modern Science 29: 391–408.

    Article  Google Scholar 

  • Dasgupta, S. 2014. On the plurality of grounds. Philosopher’s Imprint 14 (20): 1–28.

    Google Scholar 

  • Dorato, M. 2000. Substantivalism, relationalism and structural spacetime realism. Foundations of Physics 30 (10): 1605–1628.

    Article  Google Scholar 

  • Esfeld, M. 2004. Quantum entanglement and a metaphysics of relations. Studies in the History of Philosophy of Physics 35B: 601–617.

    Article  Google Scholar 

  • Esfeld, M., and V. Lam. 2008. Moderate structural realism about space-time. Synthese 160: 27–46.

    Article  Google Scholar 

  • ———. 2011. Ontic structural realism as a metaphysics of objects. In Scientific Structuralism, ed. A. Bokulich and P. Bokulich, 143–160. Springer.

    Google Scholar 

  • Fine, K. 1995. Ontological dependence. Proceedings of the Aristotelian Society 95: 269–269.

    Article  Google Scholar 

  • ———. 2012. Guide to ground. In Metaphysical Grounding: Understanding the Structure of Reality, ed. Fabrice Correia and Benjamin Schnieder, 37–80. Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • French, S. 2010. The interdependence of structure, objects and dependence. Synthese 175: 89–109.

    Article  Google Scholar 

  • ———. 2014. The Structure of the World: Metaphysics and Representation. Oxford: Oxford University Press.

    Book  Google Scholar 

  • French, S., and D. Krause. 2006. Identity in Physics: A Formal, Historical and Philosophical Approach. Oxford: Oxford University Press.

    Book  Google Scholar 

  • French, S., and J. Ladyman. 2003. Remodelling structural realism: Quantum physics and the metaphysics of structure. Synthese 136: 31–56.

    Article  Google Scholar 

  • Hellman, G. 1989. Mathematics Without Numbers: Towards a Modal Structural Interpretation. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2001. Three varieties of mathematical structuralism. Philosophia Mathematica 9 (3): 184–211.

    Article  Google Scholar 

  • Hellman, G., and S. Shapiro. 2019. Mathematical Structuralism. Cambridge: Cambridge University Press.

    Google Scholar 

  • Keränen, J. 2001. The identity problem for realist structuralism. Philosophia Mathematica (III) 9: 308–330.

    Article  Google Scholar 

  • Ketland, J. 2011. Identity and indiscernibility. The Review of Symbolic Logic 4 (2): 171–185. https://doi.org/10.1017/S1755020310000328.

  • Kitcher, P. 1983. The Nature of Mathematical Knowledge. Oxford: Oxford University Press.

    Google Scholar 

  • Ladyman, J. 1998. What is structural realism? Studies in History and Philosophy of Science 29: 409–424.

    Article  Google Scholar 

  • ———. 2005. Mathematical structuralism and the identity of indiscernibles. Analysis 65: 218–221.

    Article  Google Scholar 

  • ———. 2007. On the identity and diversity of individuals. The Proceedings of the Aristotelian Society 81: 23–43.

    Article  Google Scholar 

  • ———. 2020. Structural realism. In Stanford Encyclopedia of Philosophy, ed. E.N Zalta, https://plato.stanford.edu/archives/spr2020/entries/structural-realism/ .

  • Ladyman, L., and Leitgeb. 2008. Criteria of identity and structuralist ontology. Philosophia Mathematica 16: 388–396.

    Google Scholar 

  • Ladyman, J., and D. Ross. 2007. Every Thing Must Go: Metaphysics Naturalised. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Leitgeb, H. 2020. On non-eliminative structuralism: unlabeled graphs as a case study: Part B. Philosophia Mathematica. https://doi.org/10.1093/philmat/nkaa009.

  • Linnebo, Ø. 2008. Structuralism and the notion of dependence. Philosophical Quarterly 58 (230): 59–79.

    Google Scholar 

  • Lowe, E.J. 1994. Ontological dependency. Philosophical Papers 23 (1): 31–48.

    Article  Google Scholar 

  • ———. 2003. Individuation. In Oxford Handbook of Metaphysics, ed. M. Loux and D. Zimmerman, 75–95. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2005/2010. Ontological dependence. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, https://plato.stanford.edu/archives/spr2010/entries/dependence-ontological/.

  • ———. 2012. Asymmetrical dependence in individuation. In Metaphysical Grounding, ed. F. Correia and P. Schnieder, 214–233. Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • MacBride, F. 2006. What constitutes the numerical diversity of mathematical objects? Analysis 66 (1): 63–69.

    Article  Google Scholar 

  • McKenzie, K. forthcoming. Structuralism in the idiom of determination. British Journal for the Philosophy of Science: axx061.

    Google Scholar 

  • Menzel, C. 2018. Haecceities and mathematical structuralism. Philosophia Mathematica (III) 2: 84–111.

    Google Scholar 

  • Morganti, M. 2004. On the preferability of epistemic structural realism. Synthese 142: 81–107.

    Article  Google Scholar 

  • Parsons, C. 2008. Mathematical Thought and its Objects. Cambridge: Cambridge University Press.

    Google Scholar 

  • Psillos, S. 2001. Is structural realism possible? Philosophy of Science 68 (Supplementary Volume): S13–S24.

    Article  Google Scholar 

  • Quine, W.V. 1960. Word and Object. Harvard: Harvard University Press.

    Google Scholar 

  • Raven, M.J. 2015. Ground. Philosophy Compass 10 (5): 322–333.

    Article  Google Scholar 

  • Reck, E., and M. Price. 2000. Structures and structuralism in contemporary philosophy of mathematics. Synthese 125: 341–338.

    Article  Google Scholar 

  • Rosen, G. 2010. Metaphysical dependence: grounding and reduction. In Modality: Metaphysics, Logic, and Epistemology, ed. B. Hale and A. Hoffman, 109–136. Oxford: Oxford University Press.

    Chapter  Google Scholar 

  • Russell, B. 1903. The Principles of Mathematics. Cambridge: Cambridge University Press.

    Google Scholar 

  • Saunders, S. 2003. Physics and Leibniz’s principles. In Symmetries in Physics: Philosophical Reflection, ed. K. Branding and E. Castellani, 289–307. Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Schiemer, G., and J. Korbmacher. 2017. What are structural properties? Philosophia Mathematica (III) 26: 295–323.

    Google Scholar 

  • Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2000. Thinking About Mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2006. Structure and identity. In Identity and Modality, ed. F. MacBride, 34–69. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2008. Identity, indiscernibility, and ante rem structuralism: The tale of i and –i. Philosophia Mathematica (III) 16: 285–230.

    Article  Google Scholar 

  • Stachel, J. 2002. The Relations between things versus the things between relations. The deeper meaning of the hole argument. In Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, ed. D. Malament, 231–266. Chicago/LaSalle: Open Court.

    Google Scholar 

  • Stein, H. 1989. Yes, but… some skeptical remarks on realism and antirealism. Dialectica 43: 47–65.

    Article  Google Scholar 

  • Tahko, T., and E.J. Lowe. 2016.Ontological dependence. In Stanford Encyclopedia of Philosophy, ed. E.N. Zalta., https://plato.stanford.edu/archives/win2016/entries/dependence-ontological/.

  • Thompson, N. 2018. Metaphysical interdependence, epistemic coherentism, and holistic explanation. In Reality and Its Structure, Essays of Fundamentality, ed. Ricki Bliss and Graham Priest, 107–126. Oxford: Oxford University Press.

    Google Scholar 

  • Wigglesworth, J. 2018. Grounding in mathematical structuralism. In Reality and Its Structure: Essays in Fundamentality, ed. R. Bliss and G. Priest, 217–236. Oxford: Oxford University Press.

    Google Scholar 

  • Wigner, E. 1939. On unitary representations of the inhomogeneous Lorentz group. The Annals of Mathematics, Second Series 40: 149–204.

    Article  Google Scholar 

  • Wolff, J. 2012. Do objects depend on structures? British Journal for the Philosophy of Science 63 (3): 607–625.

    Article  Google Scholar 

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  1. University School for Advanced Studies (IUSS), Pavia, Italy

    Silvia Bianchi

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  1. Silvia Bianchi
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  1. Accademia Nazionale delle Scienze, Lettere ed Arti di Palermo, Università di Palermo, Palermo, Italy

    Gianluigi Oliveri

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain

    Claudio Ternullo

  3. Dipartimento di Filosofia e Beni Culturali, Università di Venezia Ca' Foscari, Trevio, Italy

    Stefano Boscolo

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Bianchi, S. (2022). Ontological Dependence and Grounding for a Weak Mathematical Structuralism. In: Oliveri, G., Ternullo, C., Boscolo, S. (eds) Objects, Structures, and Logics. Boston Studies in the Philosophy and History of Science, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-030-84706-7_7

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