Abstract
The central topic of this article is (the possibility of) de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that (Peano) numerals are eligible for existential quantification in epistemic contexts (‘canonical’), whereas other names for natural numbers are not. In other words, (Peano) numerals are intimately linked with de re knowledge about natural numbers, whereas the other names for natural numbers are not. In this article I am looking for an explanation of this phenomenon. It is argued that the standard induction scheme plays a key role.
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Notes
In the main text of the section on the problem of quantifying into modal contexts Kaplan (1968, sec. VIII) talks exclusively about the numerals, but in a footnote Kaplan (1968, fn. 18) mentions Carnap (1956, p. 78)’s L-determinate designators, which are those designators \(d\) that are L-equivalent to a Peano numeral \(\overline{n}\), i.e. \(d = \overline{n}\) belongs to every state-description. (The Peano numerals are composed of the symbol for zero and the symbol for the successor function.) The set of L-determinate designators includes all purely arithmetical terms (in the form of individual descriptions), e.g. ‘the product of three and four’. After having proposed to allow only standard names in the set of terms that are exportable in modal contexts, Kaplan (1968, p. 197) suggests that ‘the same trick’ would work for doxastic contexts. Ackerman (1978, p. 147) thinks that that ‘the same trick’ refers to the restriction to standard names, but it may just as well have referred to a subset of the L-determinate designators, namely the Peano numerals.
Informally and roughly, one can prove that every natural number is denoted by a Peano numeral by invoking the second-order induction axiom and one can prove that every natural number is denoted by at most one Peano numeral by using the axiom that says that the successor function is a one-to-one function.
This theory is weaker than MEA, which was introduced in (Horsten 1994). Two important differences are, first, that MEA contains UIt, whereas \(\mathbf {MEP}^{\Diamond K-}\) contains only the restricted versions UI and RUIt, and second, that MEA contains a principle expressing closure of knowability under deducibility, whereas \(\mathbf {MEP}^{\Diamond K-}\) contains a weaker logical competence principle. The significance of the first difference is clear from the context here. The importance of the second difference is defended in (Heylen 2014).
The simplification consists in the fact that here it is not required that the explanans is a general empirical law. In the case at hand the explanans is a mathematical axiom scheme with an individual constant. Note also that I speak of an explanation relative to a background theory rather than include the background theory in the explanans.
Technically speaking, in what follows I will assume that the explanans and/or the explanandum are not added as (sets of) hypotheses but as axioms (or rather, axiom schemes) to the background theory.
The proof is almost identical to the proof in (Heylen 2013, p. 95), with the box operator replaced by the knowledge operator.
In (Horsten 2005, pp. 249–251) and (Heylen 2009, pp. 224–225) it is proved that in Carnap-style intensional arithmetic identity statements and distinctness statements are knowable if true. In (Horsten 2005) the second-order axiom of induction is used, whereas in (Heylen 2009) the first-order induction scheme is used. See (Heylen 2010, p. 370, fn. 9) for a more detailed comparison between the results. A major difference between the proofs in (Horsten 2005) and (Heylen 2009) on the one hand and the proof here is that the former make use of the postulate that Peano terms can be used in existential generalisation, whereas that is a theorem in the framework used here. Also interesting from the perspective of this paper are the discussions in (Horsten 2005, p. 256 fn. 11) and (Heylen 2009, pp. 222–229, 231–232) of the soundness of induction in intensional arithmetic.
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Acknowledgments
A previous version has been presented on the Workshop on Intensionality in Mathematics, which took place on 11–12 May, 2013 in Kungshuset, Lundagard, Sweden. I would like to thank Marianna Antonutti-Marfori, Carlo Proietti and Paula Quinon for organizing the workshop and for providing me with the opportunity to present my research and I am grateful to the audience for their feedback. Another version has been presented on May, 16, 2014 in Leuven, Belgium. Again I would like to thank the audience for their feedback. In addition, I would like to thank my colleagues Lorenz Demey, Markus Eronen and Harmen Ghijsen for reading and commenting on my paper. Also, I would like to thank two anonymous reviewers for their comments and suggestions. Finally, I would like to thank Marianna Antonutti-Marfori and Paula Quinon for editing this volume and for their careful reading of and useful feedback on various versions of this article.
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Heylen, J. The epistemic significance of numerals. Synthese 198 (Suppl 5), 1019–1045 (2021). https://doi.org/10.1007/s11229-014-0542-y
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DOI: https://doi.org/10.1007/s11229-014-0542-y