Abstract
A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante-rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism.
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05 April 2018
In the original publication of the article, footnote 17 was incorrectly published. The corrected footnote is given below.
Notes
In principle, however, one can be a non-eliminativist structuralist without thereby being an ante-rem structuralist who holds that abstract structures are made up of positions. See Isaacson (2011).
From now on, I shall drop ‘ante-rem’ unless otherwise indicated. The term ‘ante-rem’ is due to Shapiro (1997). With an explicit allusion to Platonic universals, it intends to suggest that ante-rem structures are ontologically independent from their instantiating systems. To accord with the literature, I will also use this term. However, a more apt candidate is ‘positional structures’. The common feature between positional and ante-rem structures is that they are made up of positions. The notion of a positional structure, however, is neutral as to whether structures are ontologically independent from their corresponding systems.
See, for example, Cameron (1999, p. 29).
See Leitgeb (forthcoming) for taking this suggestion at face value. On his view, structures are unlabeled graphs. We shall come back to his proposal.
See Shapiro (1997, pp. 10–11, and p. 100).
My account here draws on Linnebo and Pettigrew (2014, §4).
On pain of the Burali-Forti paradox, it cannot be the case that for each system, there is a structure isomorphic to it. See Hodes (1984, p. 138). One option for dealing with this problem is to take systems to be set-sized. Hence, we would not take all isomorphic systems in the domain of our quantification; rather, it is enough to consider those systems which are accommodated by the set-theoretical hierarchy.
The thesis that each structure itself is a system can be found in Dedekind (1888), where he asks us to ‘entirely neglect’ or ‘abstract away’ from the elements of any \(\omega\)-sequences, ordered under \(\phi\), any property and relation other than their being ordered under \(\phi\). The resulting entity is the sequence of elements which are called ‘natural numbers’ or ‘ordinal numbers’. The thesis has been also reflected in Shapiro’s distinction between two perspectives about positions: ‘positions-as-offices’ and ‘positions-as-objects’ (1997, p. 77). He writes that the ‘places of a given [ante-rem] structure—considered from the places-are-objects perspective—are objects. As is characterized here, then, each structure is also a system’ (1997, p. 94). Resnik’s notion of a structure or pattern is itself a system: for him, a structure ‘is a complex entity consisting of one or more objects [positions] standing in various relationships’ (1981, p. 530), and also talks of isomorphism between structures. For example, he identifies the natural number sequence with a ‘pattern with a single binary relation (successor) and the natural numbers to be its positions’ (1997, p. 203).
The restriction of the purity thesis in terms of constitutive properties has been espoused by Reck (2003, §13) and Shapiro (2006, §1). Nodelman and Zalta (2014, §3) employ the encoding-exemplifying distinction, and Linnebo and Pettigrew (2014, §3) and Schiemer and Wigglesworth (forthcoming) defend two different versions of the last strategy.
Some other variants of mathematical structuralism also accommodate this principle. For example, Leitgeb (forthcoming) identifies ante-rem structures with unlabelled graphs, where the criterion of identity for unlabelled graphs \(G_{1}\) and \(G_{2}\) is this: \(G_{1} = G_{2}\leftrightarrow G_{1}\cong G_{2}\). See also Leitgeb and Ladyman (2008). In the context of the Univalence Axiom, Awodey (2014) writes that the fundamental principle of structuralism is the thesis that ‘isomorphic objects are identical’.
In §5, I will discuss the semantic roles of arbitrary constants.
In addition, if the platonist construal of mathematical structuralism requires the uniqueness thesis, then the failure of uniqueness rules out platonism. We will not discuss this issue here.
This is Leitgeb’s view. He holds that permuting the nodes of a structure, which he identifies with an unlabeled graph, induces a labeled, set-theoretic graph. See Leitgeb (forthcoming).
As mentioned in §3, the thesis that each structure is a system has been one of the recurring themes of structuralism. Shapiro is very explicit: once we treat positions as objects, each structure is a system instantiating itself. In the case of Leitgeb (forthcoming), things are a bit different. He identifies structures with unlabelled graphs whose isomorphism is sufficient for their numerical identity. But is an unlabelled graph itself a graph? Since in Leitgeb’s view, isomorphism among unlabelled graphs is sufficient for their identity, an unlabelled graph cannot be a graph. In other words, he cannot consistently add to his theory the standard mathematical definition of a graph as a set of nodes and edges, for this definition rules out the thesis that isomorphic graphs are identical. In fact, many graph theorists think of an unlabelled graph not as a graph, but as an equivalence class of isomorphic labelled graphs. For example, see West (2001, p. 9). Thus construed, there would be no move from isomorphism of unlabelled graphs to their identity. Thanks to Jeff Ketland for discussing this point with me.
Note that the structuralist is not in a position to simply reject the need for criteria of identity for positions, as is recommended by Shapiro (2008, p. 287) and Leitgeb and Ladyman (2008, p. 395). If positions are objects to which mathematical terms refer, the structuralist owes us an explanation of this reference relation, for reference can hardly taken to be primitive. We shall come back to this point.
Although the positions of [G] are not discernible by any one-place formula, they might still be ‘weakly’ discerned by the two-place formula ‘x is connected to y by an edge’. There are also structures whose positions are ‘utterly indiscernible’—for example, consider a structure just like [G], with two positions but with no edge connecting them. In either case, our question about the explanation of reference remains, whether the positions are just weakly discernible or utterly indiscernible.
See Pettigrew (2008, pp. 317–19).
See Breckenridge and Magidor (2012). Schiemer and Grazl (2016, §§4–5) apply it to a version of eliminative structuralism. See Woods (2014) for an alternative notion of arbitrary reference. In Woods’ ‘supervaluational’ view, what explains the behavior of parameters is not our epistemic ignorance, but their referential indeterminacy. Nevertheless, he holds that parameters function semantically as singular terms, where the involved notion of reference is taken to be primitive and sui-generis. Boccuni and Woods (ms.) apply this view to a logicist version of structuralism.
In the case of abstract objects such as numbers, our ability to use a term is unaccompanied by any causal interaction and perceptual capacity to recognize the supposed referent. I mention two adequacy conditions that codify aspects of our central uses of the arithmetical vocabulary: (1) The candidate objects of reference form an \(\upomega\)-sequence. (2) The candidate \(\upomega\)-sequence gives the cardinality relation that relates a collection of objects and one of the elements of the sequence. Although it is widely accepted that these two conditions are individually necessary, it is a controversial matter as to whether they are jointly sufficient, too. That they are jointly sufficient is a presupposition of Benacerraf’s (1965) argument, discussed in §2: since there are many \(\upomega\)-sequences each one of which meets (1) and (2), patterns of numerical usage would be accounted for even if we assign highly gerrymandered semantic values to the lexical items of our arithmetical language. Thus, no amount of reflection on our uses of the arithmetical vocabulary, either in pure arithmetic or in applications, will determine the references of numerals. Hence, by the metasemantic constraint, numerals are not genuine singular terms. (It is incumbent upon those who are unimpressed by this argument to show that there is a further aspect of our use of arithmetical vocabulary which gives us a reason to narrow down the intended interpretation of numerals to uniqueness. Another strategy is to say that there is no need to search for a further aspect of our arithmetical usage, for (2), once is properly analyzed, picks out a particular \(\upomega\)-sequence as the reference of numerals. That is, our counting practices privilege one \(\upomega\)-sequence over all others. See Russell (1919, pp. 9–10) and Hale (1987, pp. 223–4) for a defence of this strategy).
For the interpretation of mathematical terms along this line, see Pettigrew (2008). Shapiro (2012, p. 407) also interprets the semantics of parameters along the lines of free variables, but it should be noted that he applies this view only to terms standing for the structurally indiscernible positions of non-rigid structures.
Arbitrary reference is significantly similar to Hilbert’s model of reference in terms of the introduction of an \(\epsilon\)-operator which functions as a logical term-forming operator: if \(\varphi (x)\) is a formula in which x occurs as a free variable, then \(\epsilon x(\varphi x)\) is a term where all occurrences of x are bound. Here, the metasemantic proposal is that any \(\epsilon\)-term refers but only arbitrarily. As a result, we can refer to the positions of \(\left[ S\right]\) by introducing the corresponding \(\epsilon\)-terms through the following stipulations: \(a=\epsilon x\)Position(x, \(\left[ S\right]\)); \(b=\epsilon y\)(Position (y, \(\left[ S\right]\)) \(\wedge y\ne b\wedge R(a,b))\); and \(c= \epsilon z\)(Position (\(z,\left[ S\right] ) \wedge z\ne a\wedge z\ne b)\). See Leitgeb (forthcoming) for appealing to this account of the reference relation in the context of ante-rem structuralism.
For arguments against the irreducibility of semantic facts, see, for example, Field (1975, p. 386) and Lewis (1984). In their defense of arbitrary reference, Breckenridge and Magidor (2012) reject the view that semantic facts supervene on use facts. It is worth noting here that one of the common objections to the epistemic view of vagueness defended by Williamson (1994) is that it severs a necessary connection between reference and use. The epistemicist thinks that there is a particular number of grains of sand that divides the last heap from the first non-heap but we cannot know which number it is. The epistemicist thus seems to be forced to say that it is just a primitive fact that the extension of ‘is heap’ is a particular precise set, and that the referent of ‘n’ is the precise number of grains of sand that divides the last heap from the first non-heap. In response, Williamson (1994, §7.5) agues that although semantic facts supervenes on use facts, they do not so in a straightforward way: ‘there is no algorithm for calculating the former from the latter’ (1994, p. 206). In his view, the relation between use and meaning is a complex relation such that we do not know what it is. Although there is nothing inconsistent about this reply, it is not satisfactory, either. For what is missing is exactly what we demand: an account of how it could be possibly the case that semantic facts supervene on use facts. What we demand is the meta-semantical task of explaining the link between them.
I owe this point to Tim Button. Thanks to him here.
References
Assadian, B. (2016). Displacing numbers: On the metaphysics of mathematical structuralism (Ph.D. Dissertation). Birkbeck College, University of London.
Awodey, S. (2014). Structuralism, invariance, and univalence. Philosophia Mathematica, 22(1), 1–11.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford: Oxford University Press.
Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74(1), 47–73.
Boccuni, F., & Woods, J. (ms.). Structuralist neo-logicism.
Breckenridge, W., & Magidor, O. (2012). Arbitrary reference. Philosophical Studies, 158(3), 377–400.
Burgess, J. P. (1999). Review of Stewart Shapiro’s philosophy of mathematics: structure and ontology. Notre Dame Journal of Formal Logic, 40(2), 283–291.
Burgess, J. P. (2015). Rigor and structure. Oxford: Oxford University Press.
Button, T., & Walsh, S. (2016). Structure and categoricity: Determinacy of reference and truth-value in the philosophy of mathematics. Philosophia Mathematica, 24(3), 283–307.
Cameron, P. J. (1999). Permutation Groups. London Mathematical Society Student Texts 45. Cambridge: Cambridge University press.
Carnap, R. (1961). On the use of Hilbert’s \(\varepsilon\)-operator in scientific theories. In Y. Bar-Hillel, et al. (Eds.), Essays on the foundations of mathematics (pp. 156–164). Jerusalem: The Magnus Press.
Dedekind, R. (1888). Was Sind und Was Sollen die Zahlen? Vieweg, Braunschweig. Translated by W. W. Behma as ‘the nature and meaning of numbers’. In R. Dedekind (Ed.), Essays on the theory of numbers (pp. 31–115). New York: Dover.
Dedekind, R. (1890). Letter to Keferstein. In J. van Heijenoort (Ed.), From Frege to Gödel (pp. 98–103). Cambridge MA: Harvard University Press.
Field, H. (1975). Conventionalism and instrumentalism in semantics. Noûs, 9, 375–405.
Fine, K. (1983). A defence of arbitrary objects. Proceedings of the Aristotelian Society, Supplementary Volumes, 57, 55–77+79–89.
Fine, K. (1998). Cantorian abstraction: A reconstruction and defense. Journal of Philosophy, 95(12), 599–634.
Hale, B. (1987). Abstract objects. Oxford: Blackwell.
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Oxford University Press.
Hellman, G. (2001). Three varieties of mathematical structuralism. Philosophia Mathematica, 9(3), 184–211.
Hodes, H. T. (1984). Logicism and the ontological commitments of arithmetic. Journal of Philosophy, 81(3), 123–149.
Horsten, L. ms. Generic structuralism.
Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Z. Novák & A. Simonyi (Eds.), Truth, reference, and realism (pp. 1–75). Budapest: Central European University Press.
Keränen, J. (2006). The identity problem for realist structuralism II: A reply to Shapiro. In F. MacBride (Ed.), Identity and modality (pp. 146–163). Oxford: Oxford University Press.
Ketland, J. unpublished ms. Abstract structure. https://www.academia.edu/7673229/Abstract_Structure.
Leitgeb, H. (forthcoming). On mathematical structuralism: A theory of unlabeled graphs as ante rem structures. Philosophia Mathematica.
Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology. Philosophia Mathematica, 16(3), 388–396.
Lewis, D. (1984). Putnam’s paradox. Australasian Journal of Philosophy, 62(3), 221–236.
Linnebo, Ø., & Pettigrew, R. (2014). Two types of abstraction for structuralism. Philosophical Quarterly, 64, 267–283.
MacBride, F. (2005). Structuralism reconsidered. In S. Shapiro (Ed.), The oxford handbook of philosophy of mathematics and logic (pp. 563–589). Oxford: Oxford University Press.
Nodelman, U., & Zalta, E. N. (2014). Foundations for mathematical structuralism. Mind, 123(489), 39–78.
Pettigrew, R. (2008). Platonism and Aristotelianism in mathematics. Philosophia Mathematica, 16(3), 310–332.
Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, 64(1), 5–22.
Reck, E. H. (2003). Dedekind’s structuralism: An interpretation and partial defence. Synthese, 137(3), 369–419.
Resnik, M. D. (1981). Mathematics as a science of patterns: Ontology and reference. Noûs, 15(4), 529–550.
Resnik, M. D. (1997). Mathematics as a science of patterns. New York: Oxford University Press.
Russell, B. (1919). An introduction to mathematical philosophy. London: George Allen & Unwin.
Schiemer, G., & Wigglesworth, J. (forthcoming). in British Journal for the Philosophy of Science. The structuralist thesis reconsidered.
Schiemer, G., & Gratzl, N. (2016). The epsilon-reconstruction of theories and scientific structuralism. Erkenntnis, 81, 407–432.
Shapiro, S. (1997). Philosophy of mathematics: structure and ontology. Oxford: Oxford University Press.
Shapiro, S. (2006). Structure and identity. In F. MacBride (Ed.), Identity and modality (pp. 34–69). Oxford: Oxford University Press.
Shapiro, S. (2008). Identity, indiscernibility, and ante remstructuralism: The tale of \(i\) and \(-i\). Philosophia Mathematica, 16(3), 285–309.
Shapiro, S. (2012). An “\(i\)” for an \(i\): Singular terms, uniqueness, and reference. Review of Symbolic Logic, 5(3), 380–415.
West, D. (2001). Introduction to graph theory (2nd ed.). Upper Saddle River, NJ: Pearson.
Woods, J. (2014). Logical indefinites. Logique et Analyse, 57(227), 277–307.
Williamson, T. (1994). Vagueness. London: Routledge.
Acknowledgements
This paper draws on Chapter 2 of my Ph.D. dissertation, Assadian (2016). For extremely helpful discussion and written comments, I am very grateful to Tim Button, Simon Hewitt, Keith Hossack, Daniel Isaacson, Jeff Ketland, Hannes Leitgeb, Øystein Linnebo, Jonathan Nassim, Richard Pettigrew, J. Robbie G. Williams, Jack Woods, Mohammad Saleh Zarepour, an anonymous referee of this journal, and the audiences at the universities of London, Manchester, and Munich.
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Assadian, B. The semantic plights of the ante-rem structuralist. Philos Stud 175, 3195–3215 (2018). https://doi.org/10.1007/s11098-017-1001-7
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DOI: https://doi.org/10.1007/s11098-017-1001-7