Abstract
The ante rem structuralist holds that places in ante rem structures are objects with determinate identity conditions, but he cannot justify this view by providing places with criteria of identity. The latest response to this problem holds that no criteria of identity are required because mathematical practice presupposes a primitive identity relation. This paper criticizes this appeal to mathematical practice. Ante rem structuralism interprets mathematics within the theory of universals, holding that mathematical objects are places in universals. The identity problem should be read as challenging this claim about universals. However, what mathematical practice presupposes is only relevant to what is true according to the theory of universals, if one takes it for granted such a theory offers the best interpretation of mathematics. In the current context, taking this for granted begs the question. Therefore, the appeal to mathematical practice in response to the identity problem is illegitimate.
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Notes
Strictly speaking, not all ante rem structures are structural universals. Some ante rem structures are relations; however, as MacBride notes (2005, p. 585) many interesting ante rem structures will be complex relations, i.e. structural universals.
But see MacBride (2008), which casts doubt on the supposed epistemological progress provided by the ante rem structuralist interpretation of mathematics.
Note: a structural property must be specified without naming any of the places in a structure, since the possibility of naming a place in a structure depends on the prior possibility of individuating each place in terms of its structural properties.
What Keränen calls the ‘identity problem,’ Button calls the ‘automorphism problem’. I follow Keränen’s terminology because it clearly distinguishes the problem of formulating a criterion of identity for places from the problem that in an automorphic structure some of the places are indiscernible.
A different view of the metaphysics of places would hold that places are ‘entities’ rather than genuine objects, in the sense introduced by Keränen (2001, p. 312). For example, phenomena such as waves are entities, as they are not ‘properly individuated entities’, in the sense that they do not have determinate identity conditions. Some of the early official statements of ante rem structuralists can be read as endorsing the view that places are entities. For example:
It makes no sense to pursue the identity between a place in the natural number structure and some other object, expecting there to be a fact of the matter. Identity between natural numbers is determinate; identity between a place in the natural number structure and other sorts of objects is not, and neither is identity between numbers and the positions of other structures. ... there is something odd about asking whether positions in patterns are identical to other objects. It is nonsense to ask whether the shortstop is identical to Ozzie Smith—whether the person is identical to the position (Shapiro 1997, p. 79).
This suggests that places are entities, so to speak, rather than objects. But elsewhere in the same work, Shapiro (1997, p. 92) endorses the thesis that objects need criteria of identity; moreover, in subsequent discussion of the identity problem, Shapiro has made clear that places should be taken to have determinate identity conditions (2006, p. 140).
This is because, according to the ante rem structuralist, places only have structural properties.
For an explanation of what it means for a structure to be formal see Shapiro (1997, pp. 98–99).
Traditional Platonism is not completely uninformative, it tells us that ‘1’ is a singular term and that 1 is an abstract object. But this minimal information does not help us answer any of the pressing questions about the place of mathematics in the world.
Of course, not everyone will agree that it is true; anti-realists and nominalists will dispute it. Relative to the question of realism vs. anti-realism, it is informative.
There are also many non-mathematical examples of structural universals that have non-trivial automorphisms, for example methane.
Shapiro places this sentence at the start of his (2008) paper as an epigraph.
For a more complicated example, imagine two instantiations of the universal methane; the first instantiation involves four hydrogen atoms H1–H4 and one carbon atom C1, the second involves four further hydrogen atoms H5–H8 and one further carbon atom C2. We can ask whether the place filled by H1 is the same place as the place filled by H8, and as with contiguity, intuitively there is no answer to this question.
This problem with the view that places are objects was first highlighted, as far as I know, by Kit Fine (2000, pp. 16–18).
I would like to thank an anonymous reviewer for prompting me to consider such a response.
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Normand, S. Criteria of identity and the hermeneutic goal of ante rem structuralism. Synthese 195, 2141–2153 (2018). https://doi.org/10.1007/s11229-017-1325-z
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DOI: https://doi.org/10.1007/s11229-017-1325-z