Recent decades have seen intense interest in models: What are they? How do they work? Should they be understood realistically? And so on. More recently philosophers of mathematics have come to focus on the nature of applied mathematics. Similar questions have arisen: How does mathematics apply? Or, more to the point, how does mathematics model the world? Is it, as Eugene Wigner suggested, a miracle? Can mathematics explain physical phenomena and, if so, does this mean mathematics must be true, as realists sometimes argue? The topics of models and applied mathematics come together in this very fine volume.

For a long time Otavio Bueno and Steven French have been addressing these issues. Sometimes they did so together, sometimes singly, sometimes with other coauthors. They see themselves now as having converged on a common outlook. Applying Mathematics is the outcome, a systematic account of their views on models and especially how mathematics figures in all this. Among other things, Bueno and French offer an alternative to Mark Steiner’s approach to the applicability of mathematics and a sensible response to Wigner’s puzzle about the unreasonable effectiveness of mathematics.

In this short review, we discuss a couple of topics, Bueno and French’s particular conception of mathematical models and how they see mathematics as explanatory. But it should be noted that there are lots of other interesting things in this book, such as the use of problematic mathematics (e.g., Dirac’s delta function), that we must pass over.

Making sense of applied mathematics is the principal aim of Bueno and French. Like many contemporary philosophers of science, they hold that the key to understanding science is to be found in models. Theories are hardly mentioned, except to be understood as a family of models, typically mathematical models. The range of opinion is great, but the authors prefer a “partial interpretation” version, which they see as the Goldilocks account, avoiding the excesses of formalists (e.g., Stegmüller) on the one hand and the too colloquial (e.g., Giere) on the other. Bueno and French’s inferential account is linked to partial set-theoretical structures and, they claim, has the virtue of accommodating both the open-endedness of theory change and idealizations in science.

A partial interpretation has three main ingredients: partial relation, partial structure, and quasi-truth (41). How do these come about? The pattern, according to Bueno and French, is used as their book’s subtitle: immersion (i.e., pick out a relevant mathematical model and map the physical structure into it), inference (i.e., calculations and derivations within the mathematical model), and interpretation (i.e., translate the result back into the physical realm). Partial interpretations are stressed in contrast to the usual ideas of an isomorphism and homomorphism, although there can be partial isomorphisms, and so on. Clearly a strict isomorphism is much too strong for most modeling purposes. They stress the fact that excess structure can be highly useful. So a partial interpretation will typically have a great deal of excess structure that is not initially used but might play a big role in future developments.

One of the things Bueno and French stress is the open-endedness of models. This is an important point, as it allows us to make sense of the historical development of something we might want to call the “same theory.” For instance, Bohr’s 1913 theory (or model) of the atom did not include electron spin, but it could be extended in that direction. Bueno and French are right to include open-endedness as a crucial feature of models, but inevitably, questions arise. Special relativity was open-ended and was extended by general relativity. Newtonian mechanics was open-ended but was not extended (but rather was rejected) by quantum mechanics. What is the difference? A sharp logical principle, say, logical compatibility, would not do, since general relativity overrules some aspects of special relativity. This is an area where “judgment,” for want of a better term, might play an irreducible role. Bueno and French might want to say something about this.

Perhaps the Bohm-Aharonov effect is an interesting example of excess structure. Classical electrodynamics modeled the electric and magnetic fields with the E and M vectors in a mathematical vector field. One of Maxwell’s equations says there are no magnetic monopoles, represented mathematically by $\nabla \cdot B=0$ . When the divergence of a field is zero, then there exists another field A such that $B=\nabla \times A$ . (This is secured by the theorem $\nabla \cdot (\nabla \times A)=0$ .) The vector potential A was long taken to be a mere mathematical entity without any physical counterpart. This could reasonably be called surplus structure. It proved to be linked to part of the physics later on, just as Bueno and French indicate sometimes happens. They suggest (38) that surplus structure could be understood differently by realists and by positivists. With some justice, Bueno and French doubt that this is at the heart of the issue. A realist might come to think that there is more to underlying physical reality than the electric and magnetic fields and that there is a third thing, the A field, as well. The antirealist thinks that by adding the A field to the E field and the M field, she is merely adding one more useful fiction to the other two. It does not seem to matter as far as scientific practice is concerned.

Let us turn now to the topic of mathematical explanation. Bueno and French resist those arguments that are in defense of the genuine, ineliminable explanatory role of mathematics in science. The implication of this resistance bears on at least three main philosophical views: Bueno and French claim (1) that, contrary to widespread opinion, it is false that an explanatory role of mathematics in sciences makes us ontologically committed to the existence of mathematical entities; (2) that their account can capture the explanatory role of infinite limits in the explanation of empirical phenomena such as phase transitions; and (3) that the assertion that some scientific features are hybrids is not well motivated. A word about each of these.

First, it is widely thought the explanatory role of mathematics in science has ontological significance. This means that we are justified to believe in the existence of mathematical objects because of their ineliminable explanatory role in sciences, a popular theme since Quine and Putnam and more recently Baker and Colyvan. Bueno and French argue that their representational three-step framework has sufficient resources to capture this putative explanatory role of mathematics; that is, the explanatory role is not distinct from the conceived immersion, inference, and interpretation roles of mathematics in their representational framework. In other words, they claim that there is no need to open any door to mathematical structure in the role of explanans. How so? They suggest two routes for embedding the explanatory role in their representational framework: (i) at the immersion step, understand mathematical structures such as symmetries to be embedded in the empirical world or, (ii) at the interpretation step, interpret mathematical facts to act as constraining physical principles. These, of course, will be debatable, and calling symmetries constraints leads to puzzlement. How, for instance, can the Pauli exclusion principle stop the collapse of white dwarf stars?

Second, Bueno and French aim to extend their account to accommodate highly idealized models such as renormalization group techniques that are claimed to play a genuine nonrepresentational role in the explanation of phase transitions. Why nonrepresentational? Batterman claims that there are no physical structures corresponding to divergent limits; hence, no structural similarity can relate the physical world to the mathematical model. Bueno and French’s solution is as follows: we can assume an iterated immersion step, composed of two (or more) immersion steps: first, there is a mapping between the physical structure S_p and the mathematical structure S_m and, then, there is a second immersion step from the S_m to the surplus mathematical structure S*m. In this way, their account can accommodate infinite limits. However, we do not find this proposal to be convincing. First, by their definition, the immersion step has been set between the physical and the mathematical phenomena: “The first step [of the inferential framework] consists in establishing a mapping from the physical (empirical/theoretical) situation to a convenient mathematical structure” (52). It is in virtue of this physical-mathematical similarity that the inferential conception of applied mathematics gets off the ground and is indeed representational as the authors have argued throughout the book.

Hence, we find it questionable whether establishing a relation between S_m and S*m really is an immersion step, in the same way that the relation between S_p and S_m is. It is true that the relation between S_m and S*m maps some structural similarities, but the domains that are mapped by similarity relations are mathematical, not physical. In other words, this step does not reflect any structural similarity between the physical and the mathematical and, therefore, is devoid of any physical reference. We suspect that this shortcoming might support some sort of nonrepresentational, explanatory role for mathematics.

Moreover, we think that the derivation step of this inferential framework does not contribute to the representational role of mathematics in relation to an empirical setup. Indeed, we think that the derivation and the interpretation steps of the representational framework have the potential to be counted as explanatory steps. Sometimes, formal derivations establish explanatory justification for why an interpretation can take place. So, this account does not fully disregard the explanatory role of mathematics. More needs to be said about the role of the derivation step and also the nature of this step.

Third, Bueno and French discuss a hybrid theory (due to Margaret Morrison). Electron spin is taken in her view to be a hybrid of mathematics and physics. They are critical of this on the grounds that we can causally explain a lot with electron spin (e.g., magnetism). But the mathematical aspect to this cannot be playing an explanatory role, since mathematics is causally inert. But we can see that Morrison is onto something, even if it cannot be exactly as stated. And it might figure in how we explain things. Perhaps we should see two kinds of explanation at work (e.g., “Alice explained how germs explain my fever”). The first explanation is pedagogical, while the second invokes a physical cause. Unlike most things, our concept of electron spin involves mathematics. Mathematics is pedagogically necessary; we need it to explain spin. But the mathematics cannot be explaining the phenomena that we cite electron spin to explain. We will not try to pursue this thorny issue.

We have only touched briefly on a couple of topics in Applying Mathematics. There is much more to be found in Otavio Bueno and Steven French’s important and richly detailed contribution to our better understanding of mathematical models, a topic that has taken on some urgency in recent years.