1. Introduction.

The general model of theory reduction presented by Kenneth Schaffner (Reference Schaffner1967, Reference Schaffner1969, Reference Schaffner and Cohen1976, Reference Schaffner1993a, Reference Schaffner1993b) has played an important role in the literature on reduction. It provided a clear logical empiricist account of theory reduction. It also served as a counterpoint for critics to develop alternative views of reduction and antireduction. In this paper I wish to complement previous literature on Schaffner's model in two ways. First, I articulate a previously unexplored critique that arises from noticing a tension internal to the model. The model fails on its own terms. Indeed, previous external criticisms regarding the model's logical empiricist assumptions about laws and reduction functions provide an inadequate and incomplete picture of the weaknesses of the model.

Second, I advocate reconstructing a Schaffner-like model. Some types of mathematical laws and reduction functions are pertinent to theoretical biology. Moreover, theoretical biology (e.g., parts of evolutionary theory and systems theory) obeys important regulative ideals concerning the search for derivational, and embedding, deductive relations among abstract mathematical structures. In this context, distinct levels of biological organization are often associated with different types of mathematical structures (e.g., models of gene regulatory networks and models of morphological development). Since Schaffner's model captures laws, reduction functions, and the regulative ideal better than other available accounts of theory reduction, reconstructing something like it is worthwhile. Such a model would also shed light on the epistemology of mathematical practices.

This paper is structured as follows. For the remainder of this section, I document the importance of Schaffner's model. I also present a skeleton version of my critique. In Section 2, I elucidate the model. Section 3 explores the internal critique. Section 4 evaluates whether there is any escape for Schaffner's model. Section 5 assesses the relevance of a reconstructed mathematical account of theory reduction for contemporary theoretical biology.

2. The Basics of Schaffner's Model.

The basic logical empiricist account of reduction views scientific explanation as a deductive relation between theories (e.g., Hempel and Oppenheim Reference Hempel and Oppenheim1948; Nagel Reference Nagel and Stauffer1949, Reference Nagel1961, Reference Nagel1979; Hempel Reference Hempel1966). For a theory T ₁ to reduce a theory T ₂ is for T ₂ to be deducible from T ₁:

Following convention, we call T ₁ the ‘reducing theory’ and T ₂ the ‘reduced theory’. Nagel distinguishes between homogeneous and heterogeneous reduction. Homogeneous reduction “generates no special logical puzzles” (Reference Nagel1961, 339) because the two theories have similar formalizations and also contain directly connectable, even identical, terms. Terms are the objects, predicates, and relations pertinent to the theory. But heterogeneous reduction does elicit philosophical problems since the two theories differ in their formalizations and terms. Moreover, they are conceptually distant.

The basic logical empiricist account holds that a theory consists roughly of theoretical and observational components: (1) axiomatic theoretical laws, and experimental-observational laws, (2) theoretical and observational vocabularies, and (3) bridge principles connecting theoretical and observational components (e.g., Hempel Reference Hempel1966). Nagel makes the existence of (1) laws and (2) vocabularies the first two “formal conditions for reduction” (Reference Nagel1961, 345–351).Footnote ¹ Nagel's third and last explicit condition for reduction requires the background structure given by the first two conditions. This third condition consists of two “formal requirements that must be satisfied for the reduction of one science to another” (352): derivability and connectability. Bridge principles are the “assumptions” (352–353) that ground these two formal requirements. Again, the requirements are trivially satisfied for homogeneous reductions. However, for heterogeneous reductions, satisfying them is challenging; communal effort over historical time is required.

The “condition of derivability” (354) requires that the laws and theoretical assumptions of the reduced science must be deductively derivable from those of the reducing science (Nagel Reference Nagel1961, 354). Indeed, there are many important similarities between Hempel's Deductive-Nomological model of explanation and Nagel's account of theory reduction.

The “condition of connectability” requires that the terms of the reduced theory be connected to the terms of the reducing theory (Nagel Reference Nagel1961, 353–354). Derivability entails connectability, but not vice versa (355). Nagel (Reference Nagel1961) is, unfortunately, none too concerned with specifying the nature of these connections.Footnote ² He simply notes that “assumptions of some kind must be introduced” in order to connect the terms of the two theories (353). Such “assumptions” have been named “bridge principles,” “bridge laws,” and “correspondence rules” (e.g., Nagel Reference Nagel1961, Hempel Reference Hempel1966, Kitcher Reference Kitcher1984, Sarkar Reference Sarkar1998). I follow Schaffner and call them ‘reduction functions’. This terminology reminds us that the principles and rules justifying the connections are intended to be functions in the technical sense, subject to the formal constraints of mapping elements in the domain (set of all inputs) to elements in the range (set of all outputs). Schaffner's view is even stronger. Each reduction function must be extensionally injective (one-to-one) and surjective (onto), that is, it must be bijective. By focusing on relations across theories, this terminology also disambiguates two senses of ‘bridging': (1) linking theoretical and observational terms of the same theory (e.g., Hempel's main use), or (2) connecting terms, of either vocabulary, across theories (e.g., Nagel's and Schaffner's primary use). In what follows, I employ Schaffner's sense of reduction functions.Footnote ³ I return to this point in Section 5.

Schaffner concurs with Nagel's two formal conditions. He supplements the formal account by (1) introducing corrected reduced theory and (2) clearly characterizing reduction functions.

First, theory correction. In response to Nagel's account of theory reduction, Feyerabend argues that the reducing and reduced theories are incommensurable. Specifically, the intensions of their respective terms, as well as their respective ontological commitments, are not commensurate. It is impossible to directly connect terms in a reduced science to terms in an uncorrected reducing science. Indeed, putative theory reduction is actually theory replacement (Feyerabend Reference Feyerabend, Feigl and Maxwell1962, Reference Feyerabend and Colodny1965). Schaffner addresses this problem by introducing the notion of a corrected reduced theory. Roughly, Schaffner holds that the original reduced theory, T ₂, and the corrected reduced theory, $T^{*}_{2}$, belong to a set, K, of versions of the same theory:

Between any pair of theories of the set K there are relations of “approximate equality, close agreement, and strong analogy” (Schaffner Reference Schaffner and Cohen1976, 617; see also Reference Schaffner1993a, 429). These are stringent constraints on the similarity between any two versions of the same theory. These closeness constraints define what it is to be part of the set K.Footnote ⁴ Schaffner's response to Feyerabend's incommensurability objection, then, is that the corrected reduced theory ($T^{*}_{2}$) serves as the commensurability link between the original reduced (T ₂) and reducing (T ₁) theory.Footnote ⁵

What about the correction of the reducing theory? For two reasons, I will not address this here. First, Schaffner rarely mentions it (e.g., Schaffner Reference Schaffner1969, 332; Reference Schaffner1993a; Hull Reference Hull and Cohen1976, 662; and Kitcher Reference Kitcher1984, 338, briefly discuss the necessity of reducing theory correction). Second, and more importantly, this kind of theory correction does not make any difference to my argument. In fact, considering it would only make the conceptual distance larger or maintain it. Moreover, we would now be concerned with simultaneously satisfying the closeness constraints (1) pertinent to deductive inference from $T^{*}_{1}$ to $T^{*}_{2}$ and (2) those relevant to theory correction of both T ₁ to $T^{*}_{1}$ and T ₂ to $T^{*}_{2}$.

Schaffner also supplements Nagel's account by characterizing reduction functions. According to Schaffner, reduction functions connect equivalent terms of the two theories through coextensional referencing (Reference Schaffner and Cohen1976, 614–615, 620–624; see discussion in Reference Schaffner1993a, 466–477). That is, the extension of a particular term of the corrected reduced theory is identical to the extension of the equivalent term in the reducing theory. Synthetic identities between the vocabularies of the two theories are established. Such reduction functions can be schematically represented thus (see Schaffner Reference Schaffner and Cohen1976, 618; Reference Schaffner1993a, 429):

Here, the terms—that is, the natural kinds—can be of entities, predicates, or relations, or combinations thereof. The total number of terms in the reduced theory is n. Each reduction function i has two further, technical properties. Each reduction function must be injective in that every element of the domain term i of the reducing theory maps uniquely to one element of the range term i of the reduced theory. That is, the mapping must be one-to-one, not many-to-one (n.b., one-to-many element relations are already prohibited by the very definition of function). Second, each reduction function must be surjective in that no element of the range term i is extensionally unmapped to an element of the domain term i, and vice versa. The reduction functions are thus bijective. The terms are extensionally equivalent. This bijective mapping structure guarantees that every term of the reduced theory has a one-to-one correspondence with a single term of the reducing theory. Now, since the reducing theory is more general, it may contain further terms that map bijectively to terms in other reduced theories. The existence of bijective reduction functions is necessary but not sufficient for the justification of the deduction from the reducing to the reduced theory. Moreover, these reduction functions impose closeness constraints.

Thus, Schaffner's full account of theory reduction can be expressed as

Note that the deduction is to the corrected reduced theory (as he indicates in his formalizations in Reference Schaffner and Cohen1976, 618; Reference Schaffner1993a, 429), and that a large number and variety of bijective reduction functions are explicitly included in the deduction. Initial conditions such as background assumptions regarding ‘normal conditions’, must also be expressed and held fixed for the deduction to go through. In contrast, in Nagel's model, there is no explicit notion of a corrected reduced theory and the reduction functions are only implicit.

Before detailing my internal critique, I mention three external criticisms of Schaffner's model. These arose in the context of Schaffner applying his account to genetics. First, a number of commentators argue that molecular genetics does not have any strong, integrated theory (Hull Reference Hull1972; Wimsatt Reference Wimsatt and Cohen1976; Kitcher Reference Kitcher1984; Sarkar Reference Sarkar1992, Reference Sarkar1998). Therefore, Schaffner's account premised on deductive intertheoretical relations must fail. Second, Hull questions the very possibility of providing corrected reduced theories that satisfy Schaffner's closeness constraints. Hull argues that “similarity in substantive content … is not sufficient for individuating scientific theories. The substantive content of science changes too rapidly and sporadically for that” (Reference Schaffner and Cohen1976, 656). Instead, he endorses a view individuating scientific theories genealogically, rather than according to substantive content.

Third, it is argued that there are few, if any, one-to-one correspondence reduction functions between classical (T ₂) and molecular (T ₁) genetics (e.g., Hull Reference Hull1972, Reference Hull1974, Reference Hull and Cohen1976, Reference Hull1981, Reference Hull, Regenmortel and Hull2002; Rosenberg Reference Rosenberg1978, Reference Rosenberg1985, Reference Rosenberg, Regenmortel and Hull2002; Kitcher Reference Kitcher1984; Sarkar Reference Sarkar1998; see Ruse Reference Ruse and Cohen1976 for a response). There are two separate arguments for the absence of such functions:

1. 1. The reduction functions are extensionally one-to-many in that the elements of (at least some of) the single terms of T ₁ map to elements of two or more terms of T ₂ (no ‘type-type reduction'), and each element of T ₁ maps uniquely onto a single element of T ₂ (‘token-token reduction'). Since mappings exist across the terms of T ₂, such relations cannot be reduction functions in the sense of [3] (Brigandt and Love [Reference Brigandt and Love2008] call this “context-dependence of molecular features”).

2. 2. The reduction functions are extensionally many-to-one in that the elements of (at least some of) the single terms of T ₂ map to elements of two or more terms of T ₁ (no ‘type-type reduction') and we accept standard token-token reduction. Again, since mappings occur across terms, such relations cannot be reduction functions in the sense of [3] (this is the “multiple realizability argument”; see Fodor Reference Fodor1974, Kim Reference Kim1992; Sober Reference Sober1999).

Because of these two criticisms, neither the connectability condition nor the derivability condition can be satisfied (but see Waters Reference Waters, Fine, Forbes and Wessels1990, Reference Waters2000 on connectability). In short, Schaffner insists on the bijectivity of each ideal pair of terms and on type-type reduction. His critics deny bijectivity (and type-type reduction), while accepting token-token reduction. Antireductionist critics claim that there are one-to-many and many-to-one extensional mappings across pairs of actual terms in T ₁ and T ₂.

I bracket the first two external criticisms since my critique is internal. I will return to the third worry because even an internal critique must consider it explicitly.

3. The Internal Critique.

The internal critique of Schaffner's account arises from noticing a tension regarding closeness constraints. Given a large conceptual distance, if the corrected reduced theory ($T^{*}_{2}$) is of the same type as the original reduced theory (T ₂) (i.e., they both belong to the set K of [2]), thereby satisfying closeness constraints on theory correction, it will not be possible to articulate simple, bijective reduction functions. Conversely, in so far as suitable reduction functions can be expressed, thereby satisfying closeness constraints on deductive inference, the corrected reduced theory will not be of the same type as the original reduced theory. In short, both sets of closeness constraints cannot be met simultaneously when a large conceptual distance is to be covered. Satisfy one constraint, and you will not be able to satisfy the other. The constraints are in tension, even by the model's own standards. I justify my critique by further exploration of the two sets of closeness constraints and the idea of conceptual distance.

Consider theory correction. Schaffner does not suggest methods, formal or otherwise, for such transformations. Sarkar observes that a “controversial element” in Schaffner's model is that “there seemed to be no context-independent characterization of the relation between T ₂ and $T^{*}_{2}$ …, let alone a formal one. In such a circumstance, no model of reduction that relies only on formal criteria for delineating theories and relations between them can be constructed” (Sarkar Reference Sarkar1998, 27; formalism changed for consistency). Indeed, for Schaffner the relation between T ₂ and $T^{*}_{2}$ is “informal” (Reference Schaffner and Cohen1976, 617). It remains unclear whether any characterization of theory correction consistent with logical empiricist assumptions could be given. Despite this external criticism, Schaffner insists that theory correction can be done and that there are closeness constraints. He states: “traditional genetics (T ₂) is not radically reanalyzed; it is rather corrected and enriched” (Reference Schaffner1993a, 445). Original and corrected theories must be strongly similar, in both formal and substantive respects.Footnote ⁶

Now consider reduction functions of the form of [3]. In discussing the reduction of classical to molecular genetics, Schaffner provides the example of linking the classical genetic term “dominance” to coextensive combinations of molecular genetic (1) predicates: “DNA sequence” and “amino acid sequence” and (2) relations: “chemically causes” (see Reference Schaffner and Cohen1976, 623; Reference Schaffner1993a, 442 for details).Footnote ⁷ This bijective reduction function, with a molecular genetic term that combines molecular predicates and relations in a particular way, would indeed satisfy the closeness constraints on deductive inference. However, it is questionable whether this is a legitimate reduction function. Indeed, Schaffner simply assumes that the molecular basis of dominance will only involve the production of amino acid sequences from DNA, and will furthermore be robust to other causes and background conditions (for criticisms of these assumptions, see Sarkar Reference Sarkar1998, 168–173; Bagheri Reference Bagheri2005). Despite this external criticism, Schaffner insists on the availability of reduction functions to ground deductions of all the terms of $T^{*}_{2}$. Such functions impose one-to-one correspondence extensional mappings, and thereby force $T^{*}_{2}$ and T ₁ to be close.

Now, the tension between the two closeness constraints arises because the conceptual distance between T ₁ and T ₂ is large. After all, this fact is what makes—or at least, would make—a reduction between them interesting and useful. As Sarkar contends in a more general context: “Unless there is some such difference between where the explanans and the explanada come from, there is little sense in calling an explanation a reduction” (Reference Sarkar and Singh2001, 238). Moreover, Nagel observes that heterogeneous reductions, and not homogeneous reductions, are problematic: “Difficulties are frequently experienced in comprehending the import of a reduction as a consequence of which a set of distinctive traits of some subject matter is assimilated to what is patently a set of quite dissimilar traits. In such cases, the distinctive traits that are the subject matter of the secondary (reduced) science fall into the province of a theory that may have been initially designed for handling qualitatively different materials and that does not even include some of the characteristic descriptive terms of the secondary science in its own set of basic theoretical distinctions” (Nagel Reference Nagel1961, 339–340). The two theories have “quite dissimilar traits,” handle “qualitatively different materials,” “include … [different] descriptive terms,” and have their own “theoretical distinctions.” They are conceptually distant. Given the proximity of Schaffner's analysis to Nagel's model, and the general force of these arguments, Schaffner would have to agree that heterogeneous reductions imply a large conceptual distance.

Furthermore, the two theories will be strikingly dissimilar in most actual, historical cases. They have distinct formalizations, key concepts, ontological commitments, historical origins, and developmental routes. Any theory reduction of one to the other would be a nonautomatic, communal accomplishment. This is an empirical argument for the significant conceptual distance between T ₁ and T ₂.

How exactly is my critique internal? Hull, Kitcher, and Rosenberg argue that much of the original meaning (intension) of the term ‘dominance’ in uncorrected classical genetics is lost when any effective reduction function is articulated. In fact, terms of any T ₂ lose much of their intension as they get transformed and restricted into terms that can ground the deduction of $T^{*}_{2}$ from T ₁. Can I then legitimately grant Schaffner the possibility of stating sufficiently simple reduction functions? If I concede this possibility, then how do I respond to Hull's, Kitcher's, and Rosenberg's worries? If I do not grant it, then how is my critique internal?

In my view, Hull, Kitcher, and Rosenberg are justified in insisting on the necessary complexity of reduction functions if and only if they take the starting points of the reduction to be the uncorrected theories T ₂ and T ₁ (or theories that are extremely close to T ₁ and T ₂). But Schaffner need not accept this. Bijective reduction functions can be legitimately attained between $T^{*}_{2}$ and T ₁. Thus, Hull's, Kitcher's, and Rosenberg's criticism are external in that they dismiss Schaffner's two-step model and primarily consider deductive inference of T ₂ from T ₁ (see [1]). In contrast, my critique is internal in that it shows how his account then lacks the resources for dealing with the overall complexity of reducing an uncorrected theory to a reduced uncorrected theory, across a great conceptual distance. That is, contra Hull, Kitcher, and Rosenberg, Schaffnerian reduction functions, which are simple, effective, and bijective, are legitimately possible. However, because of conceptual distance, the simplicity in the reduction function comes at the cost of hiding complexity in (i) other unstated reduction functions not included in [4], (ii) the initial conditions of [4], or (iii) the transformation of $T_{2}$ to $T^{*}_{2}$. Put differently, the external criticisms can be justifiably bracketed for either deductive inference or theory correction, but not for both simultaneously. Indeed, by only considering deductive inference, Hull, Kitcher, and Rosenberg have not identified the full extent of the problem for Schaffner's model.

To summarize, we have seen that theory correction requires close proximity between T ₂ and $T^{*}_{2}$, whereas deductive inference necessitates a close fit between $T^{*}_{2}$ and T ₁. These respective closeness constraints cannot be satisfied for the same case of heterogeneous reduction.

4. Can the Internal Tension Be Eliminated?

There are three possible ways to eliminate the internal tension: (1) deny that there is a large conceptual distance, (2) weaken and qualify the constraints on theory correction, and (3) weaken and qualify the constraints on deductive inference. If we could meet one or more of these, then both sets of closeness constraints could be simultaneously satisfied even with a large conceptual distance. In general, however, none will work.

5. Reconstructing Schaffner's Model: Mathematical Reduction in Contemporary Theoretical Biology?

Let us take stock. I have shown that there is an internal tension in Schaffner's model of theory reduction. Basically, closeness constraints on theory correction and deductive inference cannot be met simultaneously with covering a large conceptual distance between the reduced and reducing theories. Only in one possible scenario of the OTRF interpretation might my internal critique turn out to be ineffective, but the jury is out on that possibility. Schaffner's model of theory reduction fails, even on its own terms.

My internal critique was explored primarily in the context of the reduction of classical to molecular genetics. But will this critique be relevant for every possible application? Given the general problems investigated in Section 3 and Section 4, it seems that Schaffner's account will fail in other domains as well. Moreover, philosophers of physics have recently provided powerful arguments belying the deductive closeness of phenomenological and statistical thermodynamics (e.g., Callender Reference Callender1999; Sklar Reference Sklar1999).Footnote ⁸ Furthermore, this and other paradigmatic cases of reduction in theoretical physics involve actual historical theories that have their own formalizations, key concepts, ontological commitments, historical origins, and developmental routes. Although further investigation is required, we suggest that the same internal tension would appear if we were to apply Schaffner's model to cases from the physical sciences.

Despite the deep problems with Schaffner's model, I believe that it deserves our continued attention. A reconstructed Schaffner model could assist us in making sense of the ongoing developments in theoretical biology, particularly in large parts of systems biology and evolutionary theory. There are currently few, if any, available philosophical analyses that can usefully and fairly analyze theory growth, theory reduction, and theory unification in mathematical biology, broadly construed. Rethinking Schaffner's account in light of other philosophical work on mathematical modeling—including the semantic view of theories—gives us a powerful perspective on derivation and embedding in mathematical theorizing and modeling. In what follows, I highlight three aspects of the proposed reconstructive project:

1. 1. The distinction between mathematical and mereological theory reduction.

2. 2. The generalization of deduction to encompass derivational and embedding mathematical practice.

3. 3. The articulation of a metric of conceptual distance.

I first explore (1). After discussing GRNs (gene regulatory networks) in systems biology, I turn to (2) and (3).

Sarkar (Reference Sarkar1998, 43ff.) distinguishes three substantive criteria for identifying different forms of reduction: (1) fundamentalism,Footnote ⁹ (2) abstract hierarchy, and (3) spatial hierarchy. My distinction between mathematical and mereological theory reduction is similar to his differentiation between (2) and (3). Mathematical reduction involves deriving the mathematical models of the reduced theory from those of the reducing theory; to put it in a different, and complementary, fashion, mathematical reduction entails embedding the models of the reduced theory into the models of the reducing theory. Mereological reduction is also theoretical. It occurs when a theoretical representation of higher-level parts of a system is explained in terms of a theoretical representation of lower-level parts (and lower-level relations) of that system.

These two forms of theory reduction are separable. An example of the former without the latter is the reduction of classical to relativistic mechanics. Cases of the latter without the former are the “mechanistic reductions” discussed by Craver (Reference Craver2007). Third, when morphogenetic models are reduced to GRNs, both formal and mereological reductions are attained. Schaffner's model simultaneously refers to both forms of reduction. Moreover, a reconstructed Schaffnerian model suggests a connection: the mathematical models of one theory representing a high level of organization are reduced to the mathematical models of another theory representing a mereologically-nested, lower level of organization. For instance, phenotypic mathematical models of development are (partially) reduced to GRNs.

Now that I have briefly characterized mathematical and mereological theory reduction, I turn to systems biology. This will motivate points (2) and (3) of the reconstructive project. The highly mathematical strands of systems biology have roots in earlier systems theory work by Rashevsky (Reference Rashevsky1954, Reference Rashevsky1961); von Bertalanffy (Reference von Bertalanffy[1933] 1962, Reference von Bertalanffy1968); Goodwin (Reference Goodwin1963); Rosen (Reference Rosen1964, Reference Rosen and Rosen1985); and Mesarovic (Reference Mesarovic1968). (For reviews, see Kitano Reference Kitano2002; O’Malley and Dupré Reference O’Malley and Dupré2005.) Much of this mathematical work was set aside and ignored starting in the 1970s, perhaps as a consequence of the ‘molecular revolution’ of biology. But as increasingly sophisticated data sets are made available and as the context dependency and complex nature of biological systems becomes ever more evident, a systems-oriented theoretical approach to biology is reemerging (Winther Reference Winther2008).

In the context of systems biology, GRNs are garnering significant attention (e.g., Smolen, Baxter, and Byrne Reference Smolen, Baxter and Byrne2000; Davidson Reference Davidson2001; De Jong Reference De Jong2002). A variety of modeling methodologies are available. Here I review two important ones: (1) Boolean networks and (2) coupled differential equations.

First, in a Boolean network, a number of elements (genes) are directly connected to one another (e.g., Thomas Reference Thomas1973; Kauffman Reference Kauffman1993; Mendoza and Álvarez-Buylla Reference Mendoza and Álvarez-Buylla1998). The state of each element of the system is either 0 (‘OFF’) or 1 (‘ON’) and depends on (i) its state one time unit earlier, (ii) the states, one time unit earlier, of the other elements that serve as input to it, and (iii) the Boolean function determining how its state changes as a function of its input elements. Most generally:

where $\hat{x}$_i represents the state of the ith element and $\hat{f}$_i expresses the Boolean functions mapping inputs to output for each element of the n-vector $\hat{x}$ of states. The n elements of the system determine a state space of n dimensions with $2\wedge n$ states. The total number of possible Boolean functions is $2\wedge 2\wedge k$, where k is the number of inputs. For example, an element with 2 inputs (k = 2) has 16 possible Boolean functions between its inputs, x ₁ and x ₂. Thus, there are 16 distinct binary Boolean rules, including AND, XNOR, NAND, and material implication. These rules can usefully be expressed in a Hasse diagram (Pemmaraju and Skiena Reference Pemmaraju and Skiena2003). Boolean networks readily capture the dynamics and topology of genetic connectivity. Here is an example of a single Boolean function (where n = 4 and k = 2):

By articulating the appropriate sets of Boolean functions, general dynamical regulatory behavior of gene networks can be represented.

Second, coupled differential equations can also be used to model GRNs. Here, differential rate equations express the continuous rate of production of an element as a function of the concentrations of other elements of the system (e.g., Goodwin Reference Goodwin1963; Tyson and Othmer Reference Tyson, Othmer, Rosen and Snell1978; De Jong Reference De Jong2002). Most generally:

Here, x _ij is the concentration of the ith species at the jth level (e.g., a particular mRNA, protein or metabolite), x is the vector of concentrations of all the species at all levels, and g _ij is a function, often involving thresholds and sigmoid curves. In dynamical control systems, output often becomes input, both within and across types of elements. It is important to distinguish between feedback and coupling. Goodwin (Reference Goodwin1963) provided one of the first models of gene control in what he called “the epigenetic system.” He differentiated three levels of production in this system: (1) mRNAs, (2) proteins (usually enzymes), and (3) metabolites. Goodwin argued that gene regulation is a feedback loop: (1) → (2) → (3) → (1). A single idealized unit component (Reference Goodwin1963, 23) is one in which mRNA is linked only with its produced protein and metabolite. A general way of expressing the feedback loop of a unit component is

where a is a parameter of production, b is a parameter of degradation, r is a function mapping the concentrations of metabolites to the production of mRNA, and x _j is the concentration of the particular species at the jth level. Note that there is no indexed i since it has a single value for a given unit component. The function r is negative for the common case of repression. The concentration of species at each level (e.g., j = 2 for proteins, j = 3 for metabolites) depends on species concentrations at lower levels.

Moreover, we need to consider more realistic situations where multiple unit components are connected, and the equations are coupled. For instance, this would occur when metabolite species₁ (e.g., repressor₁) has some influence on mRNA₂, or when metabolite species₂ partially controls mRNA₁, or both. For these cases, the vector x of [7] has nonzero values in more than one column, where columns express species concentrations (i.e., there is now more than one i). The rate equations of each unit component are now coupled to other unit components and we would have to index i in [8] and [9]. In short, coupled differential equations capture (i) feedback within unit components, and (ii) connections across unit components.

GRNs modeled through Boolean networks, coupled differential equations, or other methods, capture changes of each element of a system as a function of what other elements are doing, over time. A number of contemporary systems theorists take GRNs to be a convenient and powerful reducing theory (T ₁ in Schaffnerian terms) that helps explain and predict development and physiology. Here is Davidson: “Someday we are going to be able to write down the regulatory network architecture [GRN] for key features of development that are characteristic of animals representing diverse regions of the phylogenetic diagram. … Differences in the regulatory network architecture are what cause phylogenetic diversification, and therein resides the ‘deep structure’ that underlies [‘phylogenetic'] diagrams” (Reference Davidson2001, 23). Davidson is not the only one with high expectations (e.g., Hartwell et al. Reference Hartwell, Hopfield, Leibler and Murray1999; Regev and Shapiro Reference Regev and Shapiro2002; Fell Reference Fell and Boogerd2007). While we must remain critical of the development and application of mathematical GRN theory and modeling, it does hold significant promise.

What exactly do GRNs explain? There are at least two sorts of potential theory reduction of phenotypic models to GRN models (T ₁): (1) qualitative (i.e., from T ₂) and (2) mathematically abstracted (i.e., from $T^{*}_{2}$). First, some GRN models explain well-defined but qualitative phenotypic characters, such as the spatial and temporal order of developmental cellular and tissue differentiation in sea urchin embryos and Arabidopsis flowers. In this case, T ₂ is a structured set of phenotypic characters. One example is the important ABC model of qualitative phenotype of the flowers of Arabidopsis and other plant species (e.g., Coen and Meyerowitz Reference Coen and Meyerowitz1991; Espinosa-Soto, Padilla-Longoria, and Álvarez-Buylla Reference Espinosa-Soto, Padilla-Longoria and Álvarez-Buylla2004). Second, other work on GRN abstracts explicit mathematical models of morphological and developmental phenotype (e.g., Salazar-Ciudad and Jernvall Reference Salazar-Ciudad and Jernvall2002; Prusinkiewicz et al. Reference Prusinkiewicz, Erasmus, Lane, Harder and Coen2007). Such phenotypic models can be thought of as corrected models on analogy with Schaffnerian theory correction. Mathematically abstracted phenotypic models (i.e., $T^{*}_{2}$) are directly derived from, embedded in, and, ultimately, reduced to GRN models (i.e., T ₁). Mathematical theory reduction may turn out to be only partial, yet it can produce explanations and predictions (Davidson Reference Davidson2001).

The suggested reconstructive project of Schaffner's model employs important philosophical tools surrounding the epistemology of mathematical practices: (1) the analysis of mathematical modeling by the semantic view of theories, and (2) the concept of mathematical embedding (e.g., Friedman Reference Friedman and Healey1981, Reference Friedman1983; van Fraassen Reference van Fraassen1989). Let us see how by returning to the second and third aspects of the project listed earlier in this section.

Derivational mathematical practice within and across theories can be legitimately interpreted as deductive. Thus, we can understand reduction functions in terms of mathematics rather than in terms of logic (e.g., Schaffner's extensional characterization of terms in different, logically formulated theories). Mathematical reduction functions can powerfully and justifiably: (1) map the same particular variables and parameters (i.e., term-like formalisms), and basic functions (“laws of the model”) across mathematical theories and models of different mereological levels, or (2) directly correlate variables, parameters, or basic functions of T ₁ models to different variables, parameters, or basic functions of T ₂ or $T^{*}_{2}$ models. The state space approach of the semantic view provides useful resources for constructing a mathematical account of reduction functions that simultaneously focuses on the importance of modeling (van Fraassen Reference van Fraassen1970, Reference van Fraassen and Colodny1972, Reference van Fraassen1989; Lloyd Reference Lloyd1988; Weisberg Reference Weisberg2009).

Moreover, embedding is a concept important to the generalization or mathematization, as it were, of deduction. In early work, Friedman argues that many scientific explanations “follow a common pattern”: “observational structure[s]” are “construe[d] as substructures” of “theoretical structure[s]”; moreover, theoretical structures have “complex and interesting mathematical properties” (Reference Friedman1983, 236; see also Reference Friedman and Healey1981). In discussing “embedding” as an “aspect” of theorizing, van Fraassen makes a similar point: “The phenomena are small but chaotic; they are treated as fragments of a ‘whole’ that is much larger but orderly and simple” (Reference van Fraassen1989, 230; entire sentence emphasized in the original). Embedding is here considered a mathematical subsumption relation between theoretical and observational formal structures within a single theory or set of models. I suggest a hierarchical extension of the concept: embedding can also be effected across mathematical theories representing different levels of organization (n.b., each theory has theoretical and observational components, which are potentially overlapping and interacting). The mathematical practice of embedding can lead to increased unification of mathematical structure. Even though embedding has limits and biases (Winther Reference Winther2006a; Wimsatt Reference Wimsatt2007), and although theoretical and observational structures of a mathematical theory cannot always be distinguished sharply, embedding across theories is an important reductionist research strategy and regulative ideal in science generally.

With a reconstructed model of mathematical theory reduction, light could also be shed on how to quantify conceptual distance. Metrics for measuring the formal similarity between mathematical theories representing different mereologically nested levels could be developed (e.g., “homeomorphism” in the sense of Lloyd Reference Lloyd1988, or the extent of variable or law sharing). Theories with higher degrees of formal similarity are conceptually closer.

In conclusion, two caveats are important. First, it is perhaps unsurprising that something like a reconstructed Schaffner model has not been previously attempted. Recent analyses of the biological sciences as “mechanistic” (e.g., Glennan Reference Glennan1996, Reference Glennan2002; Machamer, Darden, and Craver Reference Machamer, Darden and Craver2000; Bechtel Reference Bechtel2006; Craver Reference Craver2007; see also Wimsatt Reference Wimsatt and Cohen1976; Sarkar Reference Sarkar1998; Schaffner Reference Schaffner2006, Reference Schaffner and Boogerd2007), “compositional” or “part-whole” (Winther Reference Winther2006b, Reference Winther2009), and “experimental” (Weber Reference Weber2005) often diminish the relevance of mathematical practices for the life sciences. I do not here claim that mathematics will replace mechanisms, let alone compositionality or experiment. Rather, I see each of these as important, especially for the biological sciences. Second caveat: while mathematical derivation and embedding remain important regulative ideals, it is unclear whether we could ever expect an elegantly unified, global mathematical structure for the biological sciences. Indeed, cogent arguments have been given for the “dappled” structure of science (e.g., Cartwright Reference Cartwright1983, Reference Cartwright1999; Burian Reference Burian1993; Schaffner Reference Schaffner1993b; van der Steen Reference van der Steen1993). But the jury is still out. I end with a question. Should the clear importance of mechanisms, part-whole relations, and experiment, as well as the possibility of irreducible dappledness, continue to stop us from even trying to understand the roles mathematical reduction, derivation, and embedding play in the biological sciences?