Structuralism (philosophy of mathematics)

The historical motivation for the development of structuralism derives from a fundamental problem of ontology. Since medieval times, philosophers have argued as to whether the ontology of mathematics contains abstract objects. In the philosophy of mathematics, an abstract object is traditionally defined as an entity that: (1) exists independently of the mind; (2) exists independently of the empirical world; and (3) has eternal, unchangeable properties. Traditional mathematical Platonism maintains that some set of mathematical elements—natural numbers, real numbers, functions, relations, systems—are such abstract objects. Contrarily, mathematical nominalism denies the existence of any such abstract objects in the ontology of mathematics. In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included intuitionism, formalism, and predicativism. By the mid-20th century, however, these anti-Platonist theories had a number of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historic context that the motivations for structuralism developed. In 1965, Paul Benacerraf published an influential article entitled "What Numbers Could Not Be". Benacerraf concluded, on two principal arguments, that set-theoretic Platonism cannot succeed as a philosophical theory of mathematics. Firstly, Benacerraf argued that Platonic approaches do not pass the ontological test. The fundamental epistemological problem thus arises for the Platonist to offer a plausible account of how a mathematician with a limited, empirical mind is capable of accurately accessing mind-independent, world-independent, eternal truths. It was from these considerations—the ontological argument, and the epistemological argument—that Benacerraf's anti-Platonist critiques motivated the development of structuralism in the philosophy of mathematics (though see below regarding Platonistic varieties of the latter). ==Varieties==

Stewart Shapiro divides structuralism into three major schools of thought. These schools are referred to as the ante rem, the in re, and the post rem. • '''Ante rem structuralism ("before the thing"), or abstract structuralism''' (particularly associated with Michael Resnik, Bob Hale and Crispin Wright, and Øystein Linnebo) has a similar ontology to Platonism (see also modal neo-logicism). Structures are held to have a real, albeit abstract and immaterial, existence. As such, it faces the standard epistemological problem, as noted by Benacerraf, of explaining the interaction between such abstract structures and flesh-and-blood mathematicians. • '''In re structuralism' (realism in truth value, but anti-realism about abstract objects in ontology). Structures are held to exist inasmuch as some concrete system exemplifies them; or, for the modal structuralist, inasmuch as it is possible that they could so exist. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures. The Aristotelian realism of James Franklin is also an in re'' structuralism, arguing that structural properties such as symmetry are instantiated in the physical world and are thus perceivable. In reply to the problem of uninstantiated structures that are too big to fit into the physical world, Franklin points to the fact that other sciences can also deal with uninstantiated universals; for example, the science of color can deal with a shade of blue that happens not to occur on any real object. • '''Post rem structuralism ("after the thing"), or eliminative structuralism' (particularly associated with Paul Benacerraf), is anti-realist about structures in a way that parallels nominalism. Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view, mathematical systems'' exist, and have structural features in common; if something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence. == See also ==