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,

Geometric definition: “the ratio of any circle’s circumference to diameter” Gottfried Leibniz’ series: 1–1/3+1/5–1/7+1/9–…=π/4 Buffon’s needle in probability theory: a statistical approximation of π by dropping needles on a grid of parallel lines Gaussian integrals 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.