Against Ontology: A Naturalist Critique on Two Varieties of Mathematical Structuralism
Much of the work dedicated to structuralism as part of a philosophical attempt to understand mathematical ontology has been focused on developing and articulating specialized varieties of the view. As with familiar, more historical debates between platonists and nominalists seeking to illuminate the opaque character of mathematical objects, most forms of structuralism center around the proposal of specific ontological claims such as the eliminative structuralism of Parsons, sui generis structuralism, and the modal structuralism of Hellman, and other such variants. This paper seeks to move away from the focus on ontology in discussions of structuralism by invoking a scientific naturalist perspective, and more particularly to claim that this effort should be directed towards philosophical reflection on existing mathematical practice.
Structuralism is the view that mathematical objects are not the numbers, sets, shapes, and so forth that are commonly identified in mathematical practice, but are instead something called structures or models. A structure is a potential set of any objects combined with relations and functions between them. When those objects are considered inside the structure, they are said to lack intrinsic nature and all other properties besides the narrowly “structural” ones that describe the relations among them. Although structures can be obtained from a wide range including groups, models, ordered sets, and topological spaces, for the sake of simplicity this paper will focus primarily on debates directly concerning the nature of numbers. Numbers are fundamental to both mathematical and philosophical interests and that further discussion involving other complex structures like functions and sets can only be practically approached once the core example of numbers is settled.
In the midst of so many formulations of structuralism and keeping in mind that many of these distinctions are arguable, this paper will adopt the in re versus ante rem differentiation with its terminology attributed to Shapiro. A large inventory of structuralist literature is devoted to settling the significant ontological disagreement that exists between these two varieties of in re and ante rem structuralism. Ante rem structuralists believe in structures and their elements as property-deficient entities that exist independently of whether systems exemplifying them exist or not. For instance, ante rem structuralists consider natural numbers as objects that are places of the unique natural number structure, and this is what Shapiro terms as the “places-are-objects” realist perspective. Regardless of which object occupies that place, whether it is a pineapple, a boat, or Alexander the Great, the place is an object in its own right, and mathematical statements are made in reference to those places without any particular exemplification in mind. Ante rem structuralists share platonist considerations in arguing for the ontology of structures because the truth of mathematical statements seems to invoke the existence of mathematical objects.
In re structuralists believe that mathematical statements are non-specific generalizations over all isomorphic systems of a certain kind, such as progressions with a successor function and a set containing an initial element, next-to-initial element, next-to-next-to-initial element, and so on. They eliminate the ontology of mathematical objects, in that they believe structures and places in structures do not exist because no single system out of the many that exemplify the structure has the special right to be named the structure. In this line of reasoning, the statement “number ‘2’ exists,” is in reality saying “every natural-number progression has an object in its next-to-initial place.” In re structuralists typically cite this view for motivations that seem analogous to Burgess’ hermeneutic nominalism. Proponents of both doctrines decline to assert the often problematic ontology of abstract objects, which conflicts with providing a satisfactory epistemological account of them. Instead, by asserting that the surface semantics of mathematical statements is misleading, proponents can legitimize their stance against mathematical ontology and preserve the validity of true statements without invoking problematic ontological assumptions involving actually nonexistent objects.
Unless philosophers understand what constitutes the nature of the very objects we are discussing, substantive responses to more complex questions involving semantics for mathematical language, epistemology behind mathematical knowledge, and the relationship between mathematics and empirical science cannot be provided. Contrary to the widespread concern present across structuralist works, this paper seeks to reform the current focus in the field away from the ontological debate by adopting scientific naturalism. Scientific naturalism, the doctrine that philosophical arguments judging scientific practices outside of science’s own prior interests are methodologically unjustified, is an overarching orientation that applies to the whole relationship between philosophy and the sciences broadly construed (natural and mathematical). Mathematics remains a central practice of inquiry in discovering facts about our world because the subject is largely successful in addressing the problems it sets for itself, including an impressive array of issues in pure mathematics as well as a wide variety of applications. As compared to philosophy, mathematics saw more success in answering its own fundamental questions, and it would be a methodological error to replace mathematics’ proper methods with those of philosophy, placing the unjustified concerns of less successful philosophical endeavors ahead of well-justified considerations from a robustly successful field like mathematics. The goals of philosophy are different from those of mathematics and it would be equally wrong to replace philosophers’ methods with mathematical ones. And besides the obvious setbacks both subjects would experience if they were to adopt each other’s methodologies and values, mathematicians are simply not interested and convinced enough to reform their practice in accordance with outside philosophical concerns. Working mathematicians have already directly witnessed the efficacy of their methods, and the idea that philosophy should stand before them and dictate the course of mathematical development is unappealing to them, not to mention to naturalists. This style of naturalist argumentation citing the already proven success of mathematical practices has been famously adopted by Maddy in arguing that the utility and justifications of axioms should be studied inside of mathematics itself, not by philosophical reflections on mathematics.
Moving on from the general relationship between philosophy and the sciences, the particular history of model theory proves mathematics to be epistemologically prior to corresponding philosophical interests. In 1930, Kurt Gödel demonstrated that it was possible to build a structure by “interpreting” the language of that structure in another structure, and through additional work by Mostowski, Tarski, Ershov, and others, this developed into what is known as “interpretation” in today’s model theory. The core techniques and languages of model theory started to develop during the first half of the 20th century, and from then on exhibited applications to different branches of mathematical practice, as exemplified through the various applications of the Löwenheim–Skolem–Tarski Theorem, model theory revealed its fruitful applications to non-Euclidean geometries in showing the independence of Euclid’s axiom of parallels from his four other axioms by finding a model in which the axiom of parallels is false. Many of the central applications of model theory to mathematics, however, were not realized in the philosophical community for sometimes decades after such developments had been made, and thus relevant philosophical discussions involving structuralism had to begin much later in the works of Benacerraf, Shapiro, Parsons, and Hellman, among others.
All such developments in model theory had been carried out independently of structuralist interests. It is impossible to imagine how exactly structuralism could be developed without model theory and algebra, as it heavily relies on model-theoretic notions of interpretation and isomorphism. Some could argue that the algebraic work on logical analyticity by Bernard Bolzano or logical consequence of Tarski, itself foundational to the development of model theory, could have been sufficient to craft structuralism. That is simply another mathematical dependency, however, confirming there is a requirement of significant mathematical input for a philosophical result in this area. Mathematics provides the relevant apparatus and enables philosophy to devise doctrines like structuralism that makes use of model theoretic notations. Therefore, the apparent lesson from the origins of structuralism indicates that mathematical practice is epistemologically prior to its philosophy. Philosophers should, as a point of methodology, conform closely to implications for mathematical practice by proposing principles reflecting on the contemporary methodologies of mathematicians, instead of suggesting drastic revisions to what has already proven successful in addressing its own goals.
In examining whether structuralism follows this epistemological lesson and serves the interests of current mathematical methodology, we can see that numbers are not actually treated as “structures” by working mathematicians as the structuralist picture of mathematical ontology suggests. As mentioned previously, structuralism sees numbers as elements or places in a structure that only have structural properties explaining their relationships to other elements in that structure. Concerning mathematical practice with a naturalist perspective, however, it would be strange to say that numbers are “treated as featureless positions in structures, lacking intrinsic nature.” Drawing from Burgess’ argument, since structuralists believe in numbers as strictly structural entities with no other substance, they would have to distinguish between 2 in structure N, +2 in structure Z, 2/1 in structure Q, 2.000 in structure R, and more. But, mathematicians in reality see no difference between saying “+2 is the first prime number,” “2/1 is the first prime number,” and “2.000 is the first prime number,” because they’re self-evidently referring to the same entity. The philosopher’s suggestion to the mathematician that he or she should, from now on, differentiate between the integer +2 and the rational number 2/1 cannot be convincing. Since structuralists want to say that the “2’s” are all different positions in unique structures and distinguish what is clearly considered as a same object to mathematicians, we see that the principle is strictly incompatible with mathematical methodology and goes against naturalist lessons, and therefore structuralism is not conforming to mathematical practice by neglecting a significant characteristic in mathematical practice.
Though structuralist philosophical debates often hinge on establishing definitional hierarchies, when there is disagreement in mathematics, it rarely involves arguing over which definition of a single entity is more “superior” or “truer” than others. An example that shows such indifference of mathematicians towards varying constructions of equivalent entities is the Euler’s number “e”: whichever definition of the irrational number the mathematician adopts between the limit definition of the irrational number, n (1+1n)n, and the series definition, 10!+11!+12!+13!+, there is no significant difference in the goals and results of mathematical practice. Another is that although Dedekind cuts and Cantor’s construction employ different methods, they both produce isomorphic results of complete ordered fields. Mathematicians are not really concerned in classifying which definition is better; they are solely interested in the characteristics of complete ordered fields.
In principle, it can be argued that there exists a “relativity” of definitions in subfields of mathematics, where one definition is more fitting for a certain mathematical enterprise than others. The relationship between the number π and computational theory is one such example. Listing some of the numerous definitions of π,
1. Geometric definition: “the ratio of any circle’s circumference to diameter”
2. Gottfried Leibniz’ series: 1–1/3+1/5–1/7+1/9–…=π/4
3. Buffon’s needle in probability theory: a statistical approximation of π by dropping needles on a grid of parallel lines
4. Gaussian integrals
5. Machin’s formula
and many more. The efficiency for computing out the decimal places of π is the fastest when using algorithms based on the Ramanujan-Sato series, while Leibniz’s Formula requires hundreds of terms to calculate few digits. Nevertheless, both are equally valid definitions of π in that in most of mathematics where only the properties of the number would be of concern, the “relativity” between definitions becomes unimportant. When engineers at NASA are using π to make calculations about orbits of spherical bodies, they are interested in the property of “3.141592…” at the moment, not the method that was used to derive it. Therefore, mathematicians should not devote their energy in arguing between different definitions when the outcomes are equivalent.
Given this phenomenon of mathematical indifference between equivalent definitions, the lessons from it can also be applied to current philosophical efforts directed towards deciding on the “true” definition of structuralism. What this implies is that if the revisions suggested by ante rem structuralism and in re structuralism produce the same picture of ideal mathematical practice, mathematicians and philosophers simply should not care about sparing so much energy arguing for that distinction. Assuming that mathematicians decide to adopt the structuralist practice into their mathematical methodologies, it is apparent that they will disagree when discussing metaphysical aspects of numbers. The observation, due to Burgess, that when the ante rem structuralist and in re structuralist both say that the missing mass problem in physics should not be attributed to numbers, shows that the two types make equivalent assertions. They both assert that numbers do not constitute mass, but with different lines of reasoning. The ante rem mathematician would justify numbers’ lack of spatial properties by reiterating their central principle that numbers only possess structural properties only pertaining to their place in the natural number progression. In contrary, the in re structuralist, who believes mathematical statements as generalizations, would concede that it cannot be said that numbers lack mass because not all objects that can possibly occupy places in a natural number progression do so, such as the Statue of Liberty occupying the next-to-next-to-initial place. Nevertheless, the in re structuralists’ belief in the nonexistence of numbers allows them to state that numbers lack spatial properties, along with any other characteristics.
The philosophical distinction between ante rem and in re varieties of structuralism clearly seems to be metaphysical, but it is hard to imagine how the difference will manifest in mathematical practice. One possible difference is that in re structuralists would have to include the phrase “…for all isomorphic structures of a progression” for every mathematical sentence said, as in writing “2+2=4 for all isomorphic structures of a progression” instead of “2+2=4.” However, this is only a play on words and does not mirror an actual difference in mathematical methodology. A class of mathematicians that decide to adopt ante rem structuralism and another group believing in in re structuralism will not disagree on the fundamental facts like “2+2=4.” Mathematics as a whole does not show any definite preference for one definition over another, whether it is Euler’s number, different constructions of complete ordered fields, or π. Therefore, the observance that the in re versus ante rem distinction is only metamathematical and philosophical without any apparent variance in mathematical methodology is a strong reason for a dismissal of this supposed distinction in structuralist ontology. With respect to the working and successful methods of mathematics, the distinction’s lacking influence on mathematical methodology should justify the statement that the energy invested in clarifying or choosing between the two should be redirected towards more fruitful areas of structuralist discussion.
A suitably naturalized structuralism faces two principal concerns. The first is the problem raised previously in this paper of how structuralism seems to differentiate between entities that are self-evidently equivalent in mathematical practice like the integer 2.0 and fraction 2/1. The second would be relocating the primary operative task of structuralism, in that we must shift our focus away from the existing distinction between ante rem and in re structuralism. The ontological presupposition of this structuralist debate is not justified by the interests of mathematical methodology, and the philosophical work put into resolving substanceless ontological disagreements could instead be directed towards mathematically productive considerations in line with naturalist methodological indications. Naturalist considerations strongly suggest that philosophers should reorient their critical practice to questions relating immediately to the problems of mathematics, rather than to the false distractions of ontology.
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