Structuralism holds that mathematics is ultimately about the shared structures that may be instantiated by particular systems of objects. Eliminative structuralists, such as Geoffrey Hellman (Mathematics Without Numbers Towards a Modal-Structural Interpretation), try to develop this insight in a way that does not assume the existence of abstract structures over and above any instances. But since not all mathematical theories have concrete instances, this brings a modal element to this kind of structuralist view: mathematical theories are viewed as being concerned with what would be the case in any system of objects satisfying their axioms. In Hellman’s version of the view, this leads to a reinterpretation of ordinary mathematical utterances made within the context of a theory. A mathematical utterance of the sentence S, made against the context of a system of axioms expressed as a conjunction AX, becomes interpreted as the claim that the axioms are logically consistent and that they logically imply S (so that, were we to find an interpretation of those axioms, S would be true in that interpretation). Formally, an utterance of the sentence S becomes interpreted as the claim:
◊ AX & □ (AX ⊃ S)
Here, in order to preserve standard mathematics (and to avoid infinitary conjunctions of axioms), AX is usually a conjunction of second-order axioms for a theory. The operators “◊” and “□” are modal operators on sentences, interpreted as “it is logically consistent that”, and “it is logically necessary that”, respectively.
This view clearly shares aspects of the core of algebraic approaches to mathematics. According to modal structuralism what makes a mathematical theory good is that it is logically consistent. Pure mathematical activity becomes inquiry into the consistency of axioms, and into the consequences of axioms that are taken to be consistent. As a result, we need not view a theory as applying to any particular objects, so certainly not to one particular system of objects. Since mathematical utterances so construed do not refer to any objects, we do not get into difficulties with deciding on the unique referent for apparent singular terms in mathematics. The number 2 in mathematical contexts refers to no object, though if there were a system of objects satisfying the second-order Peano axioms, whatever mathematical theorems we have about the number 2 would apply to whatever the interpretation of 2 is in that system. And since our mathematical utterances are made true by modal facts, about what does and does not follow from consistent axioms, we no longer need to answer Benacerraf’s question of how we can have knowledge of a realm of abstract objects, but must instead consider how we know these (hopefully more accessible) facts about consistency and logical consequence.
Stewart Shapiro’s (Philosophy of Mathematics Structure and Ontology) non-eliminative version of structuralism, by contrast, accepts the existence of structures over and above systems of objects instantiating those structures. Specifically, according to Shapiro’s ante rem view, every logically consistent theory correctly describes a structure. Shapiro uses the terminology “coherent” rather than “logically consistent” in making this claim, as he reserves the term “consistent” for deductively consistent, a notion which, in the case of second-order theories, falls short of coherence (i.e., logical consistency), and wishes also to separate coherence from the model-theoretic notion of satisfiability, which, though plausibly coextensive with the notion of coherence, could not be used in his theory of structure existence on pain of circularity. Like Hellman, Shapiro thinks that many of our most interesting mathematical structures are described by second-order theories (first-order axiomatizations of sufficiently complex theories fail to pin down a unique structure up to isomorphism). Mathematical theories are then interpreted as bodies of truths about structures, which may be instantiated in many different systems of objects. Mathematical singular terms refer to the positions or offices in these structures, positions which may be occupied in instantiations of the structures by many different officeholders.
While this account provides a standard (referential) semantics for mathematical claims, the kinds of objects (offices, rather than officeholders) that mathematical singular terms are held to refer to are quite different from ordinary objects. Indeed, it is usually simply a category mistake to ask of the various possible officeholders that could fill the number 2 position in the natural number structure whether this or that officeholder is the number 2 (i.e., the office). Independent of any particular instantiation of a structure, the referent of the number 2 is the number 2 office or position. And this office/position is completely characterized by the axioms of the theory in question: if the axioms provide no answer to a question about the number 2 office, then within the context of the pure mathematical theory, this question simply has no answer.
Elements of the algebraic approach can be seen here in the emphasis on logical consistency as the criterion for the existence of a structure, and on the identification of the truths about the positions in a structure as being exhausted by what does and does not follow from a theory’s axioms. As such, this version of structuralism can also respond to Benacerraf’s problems. The question of which instantiation of a theoretical structure one is referring to when one utters a sentence in the context of a mathematical theory is dismissed as a category mistake. And, so long as the basic principle of structure-existence, according to which every logically consistent axiomatic theory truly describes a structure, is correct, we can explain our knowledge of mathematical truths simply by appeal to our knowledge of consistency.
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