Abstract
This paper considers the existing literature on what a categoricity theorem could achieve, and proposes that more clarity and explicit admission of background beliefs is required. The first claim of the paper is that formal properties such as categoricity can have no philosophical significance whatsoever when considered apart from informal, philosophical beliefs. The second is that we can distinguish two distinct types of philosophical significance for categoricity. The paper then highlights two consequences of this analysis. The first is a potential source of circularity in Shapiro’s wider project, that arises out of what appear to be arguments for both kinds of philosophical significance with respect to categoricity. Rather than a knock-down objection to Shapiro’s philosophy of mathematics, the discussion of this potential circularity is intended to demonstrate that implicit claims surrounding the significance of categoricity can lead to philosophical missteps without due caution. The second outcome is that, as an initial case study, categoricity has limited significance for the semantic realist.
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Notes
- 1.
- 2.
This is not intended strictly; certain of what many regard as axiomatisations predate a large body of mathematical practice, such as Euclid’s geometry. Still, we can suppose that Euclid informally reasoned about space before fixing his postulates.
- 3.
The latter claim is, of course, very controversial, and part of the controversy issues from the fact that there is no categoricity theorem for formalisations of set theory. Zermelo produced only a ‘quasi-’categoricity result for ZFC. On the other hand, however, a handful of philosophers have offered informal arguments that they take to secure the uniqueness of at least the concept of set (see McGee 1997; Martin 2001; Welch 2012).
- 4.
I am grateful to an anonymous reviewer for suggesting this natural example.
- 5.
The question of which of these semantics is ‘standard’ is itself a matter of debate. Shapiro (1985) takes full semantics to be standard for second-order logic, whilst Ferreirós (2011) takes restricted Henkin semantics to be ‘just as natural as the preferred [full] one, or even more natural’ (p. 383). One could think of full semantics as just a special case of Henkin semantics, in which we consider every subset of the powerset of the first-order domain.
- 6.
There may be additional conditions on being a formalisation; namely any condition on being a formal theory is necessarily a condition on being a formalisation, since every formalisation of an informal theory is a formal theory. For example, one might require that a formal mathematical theory, and consequently a formalisation, must be recursively axiomatisable.
- 7.
This would depend on what one takes to be the formal virtues of formal mathematical theories in general. For example, one might have a Quinean preference for first-order formal theories over high-order versions. One could also be more or less committed to various other (perhaps related) properties: expressiveness, ontological parsimony, simplicity, proof-theoretic completeness, etc.
- 8.
These are theories which seem to have an intended interpretation.
- 9.
Shapiro believes in communicability to the extent that he thinks it is observable and in need of explanation.
- 10.
I am indebted to an anonymous referee for suggesting this response to me.
- 11.
References
Antonutti Marfori, M. (2010). Informal proofs and mathematical rigour. Studia Logica, 96, 261–272.
Button, T., & Walsh, S. (2015). Ideas and results in model theory: Reference, realism, structure and categoricity. Draft.
Corcoran, J. (1980). Categoricity. History and Philosophy of Logic, 1, 187–207.
Dedekind, R. (1888). Was sind und was sollen die Zahlen. Braunschwig: Viewig.
Ferreirós, J. (2011). On arbitrary sets and ZFC. The Bulletin of Symbolic Logic, 17(3), 361–393.
Gaifman, H. (2004). Non-standard models in a broader perspective. In A. Enayat & R. Kossak (Eds.), Nonstandard models of arithmetic and set theory: AMS special session, January 15–16 2003 (Vol. 361, pp. 1–22). Contemporary Mathematics. Providence: American Mathematical Society.
Gödel, K. (1951). Some basic theorems on the foundations of mathematics and their implications. In S. Feferman et al., (Eds.), Kurt Gödel: Collected works. Unpublished essays and letters (Vol. 3, pp. 304–323). Oxford: Oxford University Press.
Goodman, N. (1983). Fact, fiction, and forecast (4th ed.). Cambridge: Harvard University Press.
Halbach, V., & Horsten, L. (2005). Computational structuralism. Philosophia Mathematica, 13, 174–186.
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon Press.
Hintikka, J. (2011). What is the axiomatic method? Synthese, 183, 69–85.
Horsten, L. (2007). Philosophy of mathematics. Stanford Encyclopedia of Philosophy.
Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Z. Novak & A. Simonyi (Eds.), Truth, reference, and realism (pp. 1–76). New York: Central European University Press.
Kripke, S. (1976). Is there a problem about substitutional quantification? In G. Evans & J. McDowell (Eds.), Truth and meaning (pp. 324–419). Oxford: Oxford University Press.
Lakatos, I. (1978). What does a mathematical proof prove? In J. Worrall & G. Currie (Eds.), Mathematics, science and epistemology: Philosophical papers (Vol. 2, pp. 61–69). Cambridge: Cambridge University Press.
Leitgeb, H. (2009). On formal and informal provability. In Ø. Linnebo & O. Bueno (Eds.), New waves in philosophy of mathematics (pp. 263–299). Palgrave Macmillan.
Linnebo, Ø. (2003). Review of Philosophy of mathematics: structure and ontology. Philosophia Mathematica, 11, 92–104.
Martin, D. A. (2001). Multiple universes of sets and indeterminate truth values. Topoi, 20, 5–16.
Martin, D. A. (2012). Completeness or incompleteness of basic mathematical concepts. Draft.
McGee, V. (1997). How we learn mathematical language. Philosophical Review, 106, 35–68.
Meadows, T. (2013). What can a categoricity theorem tell us? The Review of Symbolic Logic.
Melia, J. (1995). On the significance of non-standard models. Analysis, 55, 127–134.
Parsons, C. (1990). The uniqueness of the natural numbers. Iyuun, 13, 13–44.
Resnik, M. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.
Shapiro, S. (1985). Second-order languages and mathematical practice. The Journal of Symbolic Logic, 50(3), 714–742.
Shapiro, S. (1991). Foundations without foundationalism. Oxford: Oxford University Press.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford: Oxford University Press.
Shapiro, S. (2000). The status of logic. In P. Boghossian & C. Peacocke (Eds.), New essays on the a priori (pp. 333–366). Oxford: Oxford University Press.
Shapiro, S. (2006). Structure and identity. In F. MacBride (Ed.), Identity and modality (pp. 109–145). Oxford: Oxford University Press.
Shapiro, S. (2007). The objectivity of mathematics. Synthese, 156, 337–381.
Väänänen, J. (2001). Second-order logic and foundation of mathematics. The Bulletin of Symbolic Logic, 7(4), 504–520.
Walmsley, J. (2002). Categoricity and indefinite extensibility. Proceedings of the Aristotelian Society, New Series, 102, 217–235.
Wang, H. (1955). On formalization. Mind, 64(254), 226–238.
Welch, P. (2012). Conceptual realism: Sets and classes. Draft.
Williamson, T. (2007). The philosophy of philosophy. Oxford: Blackwell.
Wright, C. (1992). Truth and objectivity. MA: Harvard University Press.
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Pollock, S. (2016). The Significance of a Categoricity Theorem for Formal Theories and Informal Beliefs. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_17
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