Abstract
In this article I respond to Heathcote’s “On the Exhaustion of Mathematical Entities by Structures”. I show that his ontic exhaustion issue is not a problem for ante rem structuralists. First, I show that it is unlikely that mathematical objects can occur across structures. Second, I show that the properties that Heathcote suggests are underdetermined by structuralism are not so underdetermined. Finally, I suggest that even if Heathcote’s ontic exhaustion issue if thought of as a problem of reference, the structuralist has a readily available solution.
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Notes
I will assume throughout that the concept of structure is generally unproblematic, except for the explicit problems I will be considering here.
This terminology is not common in the literature, where this problem is normally referred to as simply the identity problem for ARS, or the problem of indistinguishable objects for ARS. I use this terminology to distinguish it from Heathcote’s ontic exhaustion problem, which is also a problem of identity and indistinguishability.
Just what it means to be a structural relation has occupied much of the discussion which followed the introduction of the OIP. In particular, it is often asked whether the identity relation is a structural relation. Several authors (see, for example, Ladyman and Leitgeb (2008)) have postulated different accounts of distinguishability to solve the OIP. Because the application of this solution to Heathcote’s problem does not rely on any of these accounts of distinguishability, or any of the subtleties of the debate about what counts as a structural relation, I will put these issues to one side.
Heathcote is not clear about whether the axioms for the complex field can be satisfied for more than one system (up to isomorphism). If there is only exactly one system which satisfies them, then there might be more to the complex field than he suggests, since on first glance the axioms specified do not seem to pick out a unique system. If there is more than one system which satisfies the axioms of the complex field, then Heathcote’s ontic issue might be more widespread then he lets on (it might affect all realist philosophers of mathematics, for example). This would be because any philosopher of mathematics would have to answer which of the distinct systems is the complex field.
Presumably, if there is any sense in which the complex field contains \(i\) and \(-i\), they will be indistinguishable as well. However, the complex field will likely not admit of the complex conjugate automorphism.
It is quite important for Shapiro’s ARS that we have use of second order Peano arithmetic rather than first order (see footnote 9). It is not clear whether Heathcote intends to say that the natural number structure is underspecified by the first or second order Peano axioms. I will give the characterization of his argument using the term “Peano axioms”, but the reader should note that part of the solution to Heathcote’s ontic issue requires that it is the second order axioms which are being used by Shapiro.
The arguments which follow can be adapted to show that the meanings of “the complex numbers” and “the natural numbers” are also not underdetermined by ARS.
For Frege, “it is not possible to extend the domain of the cardinal numbers to that of the real numbers; they are simply completely separate domains” (Grundgesetze, Sect. 157). Cardinal numbers are extensions of concepts, while real numbers are measures of magnitudes. For Russell, real and natural numbers occur at different places in the type hierarchy, and thus cannot be identical.
This is an example of why it is so important for Shapiro that we work with second order Peano arithmetic. If we were to take as the arithmetic properties of numbers all and only those properties which could be derived from the first order Peano axioms, then we would have issues with incompleteness and non-standard models. For example, given the first order axioms, and in light of Gödel’s incompleteness theorems, there may be (arithmetic) properties of the natural numbers which are not derivable from those axioms. We have a similar issue with non-standard models of the first order axioms: it would not be clear whether the properties of the non-standard numbers ought to be considered arithmetical properties. Thanks to an anonymous reviewer for pointing this out.
The explanation of casting out nines rests of the fact that in base ten, nine is one less than the base. This can generalize. Assume that we are working on base \(b\), and we would like to cast out \(b-1\hbox {s}\) for some number \(n\). Then we will be able to show that \(n \equiv\) the sum of the base-\(b\) digits of \(n (mod (b-1))\), which is what is required to show that casting out \(b-1\) works for any base \(b\).
Here, we must exclude identity as part of the range of relations in PII, otherwise the problem is solved trivially. It is normally thought that PII ranged over exactly the structural relations, where identity is not thought to be a structural relation.
The first solution proposed to the AHP does not require a dismissal of PII, so if this solution to the OIP is unsatisfactory, there is still a way out.
A second solution, from Kouri (2010), has similar results, but uses different tools. Rather than introducing parameters, we might also make use of some sort of choice function. It is possible that mathematicians are implicitly using a choice function in their reasoning when they refer to indistinguishable objects. In effect, this solution just makes explicit the feeling of “picking one arbitrarily” that Shapiro seeks to capture with parameters. This second solution likely comes to the same thing as Shapiro’s.
There is no need to reject PII to use this solution.
This reasoning also applies to the choice functions in Kouri (2010).
References
Burgess J (1999) Book review: Stewart Shapiro. Philosophy of mathematics: structure and ontology. Notre Dame J Formal Logic 40(2):283–291
Heathcote A (2013) On the exhaustion of mathematical entities by structures. Axiomathes 24:167–180
Keränen J (2001) The identity problem for realist structuralism. Philos Math 9(3):308–330
Kouri T (2010) Indiscernibility and mathematical structuralism. Master’s thesis, University of Calgary
Ladyman J, Leitgeb H (2008) Criteria of identity and structuralist ontology. Philos Math 16(3):388–396
Shapiro S (2000) Philosophy of mathematics: structure and ontology. Oxford University Press, Oxford
Shapiro S (2008) Identity, indiscernibility, and ante rem structuralism: the tale of i and \(-\)i. Philos Math 16(3):285–309
Acknowledgments
I am grateful to Stewart Shapiro for encouragement and valuable feedback on several earlier drafts, and to an anonymous reviewer for helpful comments throughout.
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Kouri, T. A Reply to Heathcote’s: On the Exhaustion of Mathematical Entities by Structures. Axiomathes 25, 345–357 (2015). https://doi.org/10.1007/s10516-014-9241-z
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DOI: https://doi.org/10.1007/s10516-014-9241-z