SpringerLink
Log in

Algebraic structuralism

  • Published:
Philosophical Studies Aims and scope Submit manuscript

Abstract

This essay is about how the notion of “structure” in ontic structuralism might be made precise. More specifically, my aim is to make precise the idea that the structure of the world is (somehow) given by the relations inhering in the world, in such a way that the relations are ontologically prior to their relata. The central claim is the following: one can do so by giving due attention to the relationships that hold between those relations, by making use of certain notions from algebraic logic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. There are other motivations, of course, such as metaphysical underdetermination (Ladyman 1998; French 2006, 2011), or certain features of particle physics (Roberts 2011; McKenzie 2014a, b). I only outline two here, because they are the two which connect most directly with what I want to say; for a comprehensive overview of structuralism and the motivations for it, see Frigg and Votsis (2011).

  2. Worrall (1989).

  3. Worrall (1989, p. 157).

  4. Dasgupta (2009, 2011).

  5. This is an instance of the general argument discussed by Roberts (2008).

  6. Here, I’m employing Melia (1999)’s definition of haecceitism as “the view that there are distinct possibilities that differ only over which objects play which roles.”

  7. cf. Pooley (2006) and Greaves (2011).

  8. cf. McSweeney (2016) and Sider (2017, chapter 5).

  9. Esfeld and Lam (2010).

  10. French and Ladyman (2003, 2010).

  11. cf. Kaplan (1975).

  12. I read Weatherall (2016) as advocating that haecceitism should be modelled in this latter fashion.

  13. French and Ladyman (2010, p. 33).

  14. Wolff (2012, p. 607).

  15. Chakravartty (1998, p. 399).

  16. The structure of what? For the purposes of this discussion, either the world, or some model-theoretic representation thereof.

  17. Thus, this proposal—especially if taken in an eliminativist vein—may naturally be viewed as a form of the bundle theory.

  18. Isomorphic, in this context, meaning that there is a bijection f between the two algebras which preserves the algebraic operations (so \(f(a \cdot b) = f(a) \cdot f(b)\), \(f(- a) = - f(a)\), etc.).

  19. This corresponds to the well-known observation that the bundle theory is (at least prima facie) committed to the identity of indiscernibles.

  20. I stress that the language is without identity in order to stress that the elementary equivalence here is with respect to that language: i.e., that \({\mathcal {M}}^+\) and \({\mathcal {N}}^+\) will not (in general) satisfy the same identity-sentences.

  21. Halmos (1954, 1962); for interesting reflections on the development of polyadic algebras, see Halmos (2000).

  22. See Monk (2000) and Henkin et al. (1971, 1985).

  23. As this suggests, one can think of cylindric algebras as polyadic algebras enriched with an operation of identity: more precisely, one can show that the “locally finite” polyadic algebras with identity are equivalent to the “locally finite” cylindric algebras (see Galler (1957) or (Plotkin 2000, §1.2)).

  24. cf. Leitgeb and Ladyman (2008).

  25. Thus, the elements of the algebra are sets of infinite sequences, not—as one might have thought—sets of n-tuples. This just avoids certain technical wrinkles that we would have to deal with if working with sequences of different lengths: see the discussion in Németi (1991, §4).

  26. The cylindric set algebras of the form \(C({\mathcal {M}})\), for some model \({\mathcal {M}}\) of predicate logic, are exactly the \(\omega \)-dimensional cylindric set algebras which are “regular” and “locally finite”: see (Monk 2000, §11).

  27. Strictly speaking, the representable cylindric algebras are those which are isomorphic to some subdirect product of cylindric set algebras. But one can show that any such algebra is directly representable as a cylindric set algebra (see (Németi 1991, p. 503)).

  28. Monk (1969); see (Németi 1991, Theorem 4).

  29. See Henkin et al. (1986) for a survey.

  30. (Dasgupta 2009, pp. 52–53).

  31. Dasgupta (2009, p. 66).

  32. Here’s a possible answer: elsewhere, Dasgupta writes that “it is arguably analytic of the existential quantifier that existentially quantified facts are grounded in their instances.” (Dasgupta 2011, p. 131) The problem is that what makes this an analytic feature of the existential quantifier is surely the form of the semantics for it—as Dasgupta then goes on to say, “this understanding of the quantifier is arguably implicit in the standard Tarskian semantics for PL.” (Dasgupta 2011, pp. 131–132). If so, then it will equally well be analytic of the operator \(\mathbf{c}\) that facts of the form \((\mathbf{c}\mathbf {\Phi }^1)^0 \, \mathbf {obtains}\) are grounded in instantiations of \(\varvec{\Phi }^1\).

  33. The hermeneutic/revolutionary distinction is borrowed from Burgess (1983).

  34. My thanks to an anonymous referee for raising this concern.

  35. Indeed, in modern differential geometry one standardly defines a vector field as a derivation on the smooth algebra (e.g. (Baez and Muniain 1994, chap. 3)).

  36. For further discussion of Einstein algebras, see Rosenstock et al. (2015).

  37. See Ruetsche (2011) for more on this question.

  38. See (Landsman 2017, chap. 7).

  39. Although, as discussed above, the application to scientific theories requires more detailed work.

  40. Lam and Wüthrich (2015, p. 624).

  41. Psillos (2001, p. S22).

  42. Not to be confused with the distinction between radical and moderate ontic structuralism! After all, as I argue here, algebraic structuralism is a radical form of ontic structuralism, but a moderate form of anti-individualism.

  43. Sider (2017, p. 93).

  44. Sider gives Turner (2016) as an example; Rayo (2017) also outlines such a project.

  45. It remains an open question, to my mind, whether there is or could be such an apparatus: the best hope is likely something category-theoretic, but the details remain to be made out.

  46. This is, essentially, the point that Rynasiewicz (1992) makes in the context of spacetime algebras.

  47. See e.g. Hawthorne (2001) and Schaffer (2005).

  48. One might have thought that characterising the structure of a theory would just be a matter of characterising the structure of each of its models. However, that risks neglecting those aspects of a theory’s structure concerning the relationships between models, which may be important for determining the theory’s content (see Halvorson (2012, §7) for an elaboration of this claim).

References

  • Baez, J. C., & Muniain, J. P. (1994). Gauge fields, knots, and gravity. Singapore: World Scientific.

    Book  Google Scholar 

  • Burgess, J. P. (1983). Why I am not a nominalist. Notre Dame Journal of Formal Logic, 24(1), 93–105.

    Article  Google Scholar 

  • Chakravartty, A. (1998). Semirealism. Studies in History and Philosophy of Science Part A, 29(3), 391–408.

    Article  Google Scholar 

  • Dasgupta, S. (2009). Individuals: An essay in revisionary metaphysics. Philosophical Studies, 145(1), 35–67.

    Article  Google Scholar 

  • Dasgupta, S. (2011). The bare necessities. Philosophical Perspectives, 25(1), 115–160.

    Article  Google Scholar 

  • Esfeld, M., & Lam, V. (2010). Holism and structural realism. In R. Vanderbeeken & B. D’Hooghe (Eds.), Worldviews, science and US. Studies of analytical metaphysics. A selection of topics from a methodological perspective (pp. 10–31). Singapore: World Scientific.

    Google Scholar 

  • French, S. (2006). Structure as a weapon of the realist. Proceedings of the Aristotelian Society, 106(1), 169–187.

    Google Scholar 

  • French, S. (2011). Metaphysical underdetermination: Why worry? Synthese, 180(2), 205–221.

    Article  Google Scholar 

  • French, S., & Ladyman, J. (2003). Remodelling structural realism: Quantum physics and the metaphysics of structure. Synthese, 136(1), 31–56.

    Article  Google Scholar 

  • French, S., & Ladyman, J. (2010). In defence of ontic structural realism. In A. Bokulich & P. Bokulich (Eds.), Scientific structuralism (pp. 25–42). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Frigg, R., & Votsis, I. (2011). Everything you always wanted to know about structural realism but were afraid to ask. European Journal for Philosophy of Science, 1(2), 227–276.

    Article  Google Scholar 

  • Galler, B. A. (1957). Cylindric and polyadic algebras. Proceedings of the American Mathematical Society, 8(1), 176–183.

    Article  Google Scholar 

  • Greaves, H. (2011). In search of (spacetime) structuralism. Philosophical Perspectives, 25(1), 189–204.

    Article  Google Scholar 

  • Halmos, P. R. (1954). Polyadic boolean algebras. Proceedings of the National Academy of Sciences of the United States of America, 40(5), 296–301.

    Article  Google Scholar 

  • Halmos, P. R. (1962). Algebraic logic. Hartford: Chelsea Publishing Company.

    Google Scholar 

  • Halmos, P. R. (2000). An autobiography of polyadic algebras. Logic Journal of IGPL, 8(4), 383–392.

    Article  Google Scholar 

  • Halvorson, H. (2012). What scientific theories could not be. Philosophy of Science, 79(2), 183–206.

    Article  Google Scholar 

  • Hawthorne, J. (2001). Causal structuralism. Noûs, 35, 361–378.

    Article  Google Scholar 

  • Henkin, L., Monk, J. D., & Tarski, A. (1971). Cylindric algebras: Part I. Amsterdam: North-Holland.

    Google Scholar 

  • Henkin, L., Monk, J. D., & Tarski, A. (1985). Cylindric algebras: Part II. Amsterdam: North-Holland.

    Google Scholar 

  • Henkin, L., Monk, J. D., & Tarski, A. (1986). Representable cylindric algebras. Annals of Pure and Applied Logic, 31, 23–60.

    Article  Google Scholar 

  • Kaplan, D. (1975). How to Russell a Frege-Church. The Journal of Philosophy, 72(19), 716–729.

    Article  Google Scholar 

  • Ladyman, J. (1998). What is structural realism? Studies in History and Philosophy of Science Part A, 29(3), 409–424.

    Article  Google Scholar 

  • Lam, V., & Wüthrich, C. (2015). No categorial support for radical ontic structural realism. The British Journal for the Philosophy of Science, 66(3), 605–634.

    Article  Google Scholar 

  • Landsman, K. (2017). Foundations of quantum theory. Cham: Springer.

    Book  Google Scholar 

  • Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology. Philosophia Mathematica, 16(3), 388–396.

    Article  Google Scholar 

  • McKenzie, K. (2014a). On the fundamentality of symmetries. Philosophy of Science, 81(5), 1090–1102.

    Article  Google Scholar 

  • McKenzie, K. (2014b). Priority and particle physics: Ontic structural realism as a fundamentality thesis. The British Journal for the Philosophy of Science, 65(2), 353–380.

    Article  Google Scholar 

  • McSweeney, M. M. (2016). An epistemic account of metaphysical equivalence. Philosophical Perspectives, 30(1), 270–293.

    Article  Google Scholar 

  • Melia, J. (1999). Holes, haecceitism and two conceptions of determinism. The British Journal for the Philosophy of Science, 50(4), 639–664.

    Article  Google Scholar 

  • Monk, J. D. (1969). Nonfinitizability of classes of representable cylindric algebras. The Journal of Symbolic Logic, 34(03), 331–343.

    Article  Google Scholar 

  • Monk, J. D. (2000). An introduction to cylindric set algebras. Logic Journal of IGPL, 8(4), 451–496.

    Article  Google Scholar 

  • Németi, I. (1991). Algebraization of quantifier logics, an introductory overview. Studia Logica, 50(3–4), 485–569.

    Article  Google Scholar 

  • Plotkin, B. (2000). Algebra, categories and databases. In M. Hazewinkel (Ed.), Handbook of algebra (Vol. 2, pp. 79–148). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Pooley, O. (2006). Points, particles, and structural realism. In D. Rickles, S. French, & J. Saatsi (Eds.), The structural foundations of quantum gravity (pp. 83–120). Oxford: Oxford University Press.

    Chapter  Google Scholar 

  • Psillos, S. (2001). Is structural realism possible? Philosophy of Science, 68(3), S13–S24.

    Article  Google Scholar 

  • Rayo, A. (2017). The world is the totality of facts, not of things. Philosophical Issues, 27(1), 250–278.

    Article  Google Scholar 

  • Roberts, B. W. (2011). Group structural realism. The British Journal for the Philosophy of Science, 62(1), 47–69.

    Article  Google Scholar 

  • Roberts, J. T. (2008). A puzzle about laws, symmetries and measurability. The British Journal for the Philosophy of Science, 59(2), 143–168.

    Article  Google Scholar 

  • Rosenstock, S., Barrett, T. W., & Weatherall, J. O. (2015). On Einstein algebras and relativistic spacetimes. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52(Part B), 309–316.

    Article  Google Scholar 

  • Ruetsche, L. (2011). Interpreting quantum theories: The art of the possible. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Rynasiewicz, R. (1992). Rings, holes and substantivalism: On the program of Leibniz algebras. Philosophy of Science, 59(4), 572–589.

    Article  Google Scholar 

  • Schaffer, J. (2005). Quiddistic knowledge. Philosophical Studies, 123(1–2), 1–32.

    Article  Google Scholar 

  • Sider, T. (2017). The tools of metaphysics and the metaphysics of science (Unpublished manuscript).

  • Turner, J. (2016). The facts in logical space: A tractarian ontology. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Weatherall, J. O. (2016). Regarding the ‘hole argument’. The British Journal for the Philosophy of Science. https://doi.org/10.1093/bjps/axw012.

    Google Scholar 

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

    Article  Google Scholar 

  • Worrall, J. (1989). Structural realism: The best of both worlds? Dialectica, 43(1–2), 99–124.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich, Munich, Germany

    Neil Dewar

Authors
  1. Neil Dewar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Neil Dewar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dewar, N. Algebraic structuralism. Philos Stud 176, 1831–1854 (2019). https://doi.org/10.1007/s11098-018-1098-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11098-018-1098-3

Keywords

Search

Navigation