By Noel Weichbrodt
How to account for mathematics? In a disarrayed fashion, theories have passed before the philosopher’s eyes for many years. Platonism, Kantianism, Logicism, Formalism, Intuitionism, Realism, and Nominalism have dissolutely paraded across the stage set by Stewart Shapiro in Thinking About Mathematics (Oxford, 2000). In the ultimate chapter 10, Shapiro presents his own account of mathematics, called structuralism. Shapiro presents the arguments for structuralism as a series of acceptable circles. His honesty is admirable, and the circles are well-constructed, but the end of all his work seems meager in light of the vastly reduced, small and humble claims that structuralism makes for itself. If Platonism and Logicism, to pick two, have their failures, they at least fail boldly. Structuralism simply dimishes the goals of a proper account by contentedly keeping within the fence of circular reasoning and, ultimately, a purely Nominalist metaphysics and epistemology.
Structuralism posits that mathematics is the study of structures and their place-holders. Think of the branches of mathematics—arithmetic, real analysis, geometry, discrete math—as different wine racks. The wine racks have places to be filled by objects—wine bottles—such as positive integers, real numbers, and binary digits. The account of mathematics now branches out: what structures and objects are there, how do we know they are there, and what justifies our decisions about what is there and how we know they are there? Shapiro examines the outlines of structuralism by addressing first the ontological claims, and then the epistemological claims made by the account.
Ontologically, the question is what sorts of mathematical place-objects fill the structures of mathematics. In structuralism, there is “no ontological independence among the natural numbers” [Shapiro 258], or among any place-filling objects. The only things that exist are structures, and every significant branch of mathematics has its own unique structure. Structures give meaning and significance to the objects that fill them, “it is nonsense to contemplate numbers independent of the structure of which they are a part.” [Shapiro 261] A consequence of this approach is that it only allows questions to be asked of mathematical place-filling objects as they exist inside the system. So the number seven can be properly said to be a prime that is less than eight. As to whether seven is lucky, or what seven means, the structuralist simply replies that the questions have no meaning, because they inquire about the number outside the system within which it exists.
However, it is difficult to figure out just what is significant enough to constitute an ontologically independent structure. Shapiro states “(pure) mathematics is the deductive study of structures as such. The subject of arithmetic is the natural number structure, and the subject of Euclidean geometry is Euclidean space structure.” That seems most straightforward, but mathematics is commonly held to be the study of numbers and their manipulations. The structuralist accounts ontologically for numbers, but leaves out any story about manipulations. What is the ontological status of addition or of a diagonal set? The structuralist account works quite well for numbers, but seems inadequate to capture other parts of math.
Epistemologically, every place-filling object is independent, since you can grasp the number two “without knowing anything about 6 or 6,000,000” [Shapiro 258]. Mathematical place-filling objects are known by their roles, as places in a pattern [Shapiro 277]. Moving from objects to structures, structures coherently exist because we can discuss them—the indispensability argument [Shapiro 282]. Structures are defined implicitly, through direct description of them, as characterizations of them. The epistemology only works if the implicit definition successfully describes the structure, and that judgment is made by judging the coherence of the description.
The structuralist makes the idea of coherence properly basic in the account, for “…the [structuralist] must stop this infinite regress [of grounding structures]. Coherence of a description must be judged by set theory, but set theory is a structure. The final dilemma in Structuralism is that “we cannot ground mathematics in any domain or theory that is more secure than mathematics itself.” [Shapiro 288] Even so, “the final ontology is not to be understood in terms of structures, even if everything else in mathematics is.” [Shapiro 273, cf. 288]
I do not see how the structuralist account of mathematics makes any substantial improvement over the formalist account. Given the modern foundational crisis in math, the only accounts with any credibility left are the Realist (in some forms), Intuitionism, and Nominalist ones. Without any sort of metaphysical justification to back up appeals outside the system, formalist accounts, which Structuralism is a variation of, are left with an ultimately vacuous account of the way things are. When the players are limited to purely material or conceptual objects, the game isn’t very interesting or satisfying. That is why Platonism (as a form of Realism) and Intuitionism are still played around with today—they provide mathematics with a metaphysical meat and meaning denied by formalist accounts. Math must have significance, and Structuralism does not give it.
Mathematical objects must retain some sort of status outside the system of math itself. Mathematics itself needs to be able to affect or describe the features of reality in significant ways. Anything less, metaphysically, leaves an unsatisfying taste in the evaluation of an account. As a prime example, consider the Structuralist ontology of mathematical place-filling objects. In order to make the system cohere and tighten maximally, mathematical place-filling objects are not allowed to have any meaning outside of the structure that provides their architectonic. While this solves the Caesar problem, it strips mathematical objects of conceptual interest and natural relevance.
Shapiro sees this threat, it seems, and introduces a discussion about how to satisfyingly ground a “rich background ontology” [Shapiro 271]. However, since Structuralism refuses to go any further outside of structures in a metaphysical discussion of meaning, Shapiro simply ends quoting others, such as Hellman, who use modal systems to give structuralism a significance as possible systems. The result is a structuralism without structures, with possible background ontology. Shapiro contents himself with simply being like the rest of the formalist accounts—satisfiability in set theory is “sufficient for existence” [Shapiro 289], coherence is primitive, and any discussion of parts of math outside the area of math itself is nonsensical. A safe and tight place to be, perhaps, but constricting and stripped of significant existence.