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.
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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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