Abstract
Three metascientific concepts that have been the object of philosophical analysis are the concepts of law, model, and theory. The aim of this chapter is to present the explication of these concepts and of their relationships made within the framework of Sneedian or metatheoretical structuralism (Balzer et al., An architectonic for science. The structuralist program. Dordrecht: Reidel, 1987), and of their application to a case from the realm of biology: population dynamics. The analysis carried out will make it possible to support, contrary to what some philosophers of science in general and of biology in particular hold, the following claims: (a) there are “laws” in biological sciences, (b) many of the heterogeneous and different “models” of biology can be accommodated under some “theory,” and (c) this is exactly what confers great unifying power to biological theories.
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06 October 2020
The published version of the book has missed to include the correct figures in Chap. 10.
Notes
- 1.
“Over the last four decades, the semantic view of theories has become the orthodox view on models and theories” (Frigg 2006, p. 51).
- 2.
This idea has been developed in different particular ways, giving rise to different approaches, variants, or versions, which despite their differences constitute a family, the semantic family. For a characterization of this family, and of some of its members as well as a reference to many of them, see Lorenzano (2013) and Ariza et al. (2016).
- 3.
For skeptical positions about any notion of law and the substitution of the term “law” by other notion, such as “(fundamental) equations” or “(basic) principles,” see Cartwright (1983, 2005), Giere (1995), and van Fraassen (1989). By the way, Carnap himself had already considered the possibility of dispensing with the term “law” in physics (Carnap 1966, p. 207).
- 4.
The analysis of population dynamics is based on Díaz and Lorenzano (2017).
- 5.
- 6.
- 7.
- 8.
On the other hand, the expressions “fundamental law” and “special law” are not used here in Fodor’s sense (Fodor 1974, 1991)—the former for laws of basic or fundamental sciences, the latter for laws of special sciences—but rather in the sense used by structuralists, i.e., for different kinds of laws within a theory.
- 9.
The validity of laws can be regarded as exact—and thus as strict or non-interferable laws—or, rather, to the extent that they usually contain not only abstractions but also various idealizations, as approximate, as already pointed out by Scriven (1959) and more extensively by Cartwright (1983)—and so as non-strict or interferable laws, and compatible with various specific treatments of this situation, such as those referring to ceteris paribus clauses (Cartwright 1983), “provisos” (Coffa 1973 and Hempel 1988), or “normicity” (Schurz 2009).
- 10.
For more on the structuralist T-theoretical/T-non-theoretical distinction, see Sect. 10.3.
- 11.
By saying it in a model-theoretic way, fundamental laws determine the whole class of models of a theory while special laws determine only some of them, which constitute a subclass of the class of models.
- 12.
There are different ways of classifying the different types of factors that affect demographic processes. For instance, Krebs (2008) distinguishes two types of factors, external and internal, while Berryman (2003) distinguishes three types of factors, biotic, genetic, and abiotic. We follow this tripartite distinction but with a slightly different denomination. So, members of the set of population factors ((Fi)i ≤ k) are the so-called set of environmental factors (Famb) (such as physical and chemical conditions of the environment and resources), set of genetic factors (Fgene) (related to genetic properties of the distinct individuals that place them in different classes with respect to age, sex, or size), and set of biotic (or biological) factors (Fbio) (variables or “parameters” like carrying capacity, competition coefficient, capture efficiency, and conversion efficiency of predator, which measure the biotic interactions such as predation, parasitism, and intra- or interspecific competition). While the extension of the concepts environmental and genetic factors can be determined independently from population dynamics, and thus are PD-non-theoretical, the extension of most of biotic factors can only be determined by means of PD and in that sense a subset of them are PD-theoretical.
- 13.
The rate of population change (RPC) is a functional whose domain is the set of demographic processes (DP), the size of the population (Nt), and the whole of population factors ((Fi)i ≤ k) in an instant of time (T), and its co-domain is the set of real numbers: RPC: DP × Nt × (Fi)i ≤ k × T → ℝ. It is definitely a PD-theoretical concept: its extension can only be determined by applying the very same PD.
- 14.
It is worth noting in relation to the equations mentioned in these examples that different authors usually designate the same base sets or functions of population dynamics by different terms and symbols. For N and the demographic processes (b, d, i, and e) its use is generalized and uniform, but for the population factors there is much diversity of terms and symbols between the different authors, especially in what we label the “rate of population change” (and symbolize by RPC instead of by r as in Turchin 2001, p. 19) as well as in the presentation of the so-called “Lotka-Volterra equations.” This generates the coexistence of different terminology and symbology for the same factors or rates and the erroneous impression that the concepts are different as well as the distinct formulations of the equations in which they occur. In order to avoid this impression and ambiguity we have unified the use of terms and symbols.
- 15.
For an analysis of these theories from a structuralist point of view, see among others Balzer et al. (1987).
- 16.
In trying to be as precise as possible, metatheoretical structuralism prefers the use of (elementary) set theory—whenever possible—as the most important formal tool for metatheoretical analysis. However, this formal tool is not essential for the main tenets and procedures of the structuralist representation of science (other formal tools such as logic, model theory, category theory, and topology as well as informal ways of analysis are also used). Besides, there are also uses of a slight variant of Bourbaki notion of “structure species” in order to provide a formal basis of characterizing classes of models by means of set-theoretic predicates (Balzer et al. 1987, Ch. 1) and of a version of the von Neumann-Bernays-Gödel-type of language including urelements for providing a purely set-theoretical formulation of the fundamental parts of the structuralist view of theories (Hinst 1996). There is even a “categorial” version of metatheoretical structuralism that casts the structuralist approach in the framework of category theory rather than within the usual framework of set theory (see Balzer et al. 1983; Sneed 1984; Mormann 1996). The choice of one formal tool or another or of a more informal way of analysis is a pragmatic one, depending on the context which includes the aim or aims of the analysis and the target audience. Nonetheless, in standard expositions of metatheoretical structuralism, as well as in the presented here, models are conceived of as set-theoretical structures (or models in the sense of formal semantics), and their class is identified by defining (or introducing) a set-theoretical predicate, just as in the set-theoretical approach of Patrick Suppes (1957, 1969, 1970, 2002; McKinsey et al. 1953).
- 17.
In a complete presentation, we should include, besides the collection of so-called principal base sets D1,..., Dj or D1,..., Dk, also a second kind of base sets, namely, the so-called auxiliary base sets A1,..., Am. The difference between them is the difference between base sets that are empirically interpreted (the principal ones) and base sets that have a purely mathematical interpretation, like the set ℕ of natural numbers, or the set ℝ of real numbers (the auxiliary ones). Here, auxiliary (purely mathematical) base sets are treated as “antecedently available” and interpreted, and only the proper empirical part of the models is stated in an explicit way.
On the other hand, in philosophy of logic, mathematics, and empirical science has been intensively discussed what would be a better way of understanding the nature of sets occurring in the relational structures and of the models themselves. In relation to sets, according to the standard interpretation of “sets-as-one” (Russell 1903) or “the highbrow view of sets” (Black 1971) or “sets-as-things” (Stenius 1974) sets themselves, though not necessarily their elements which may refer to concrete entities, should be considered as abstract entities, while according to the interpretation of “sets-as-many” (Russell 1903) or “the lowbrow view of sets as collections (aggregates, groups, multitudes)” (Black 1971) or “sets-of” (Stenius 1974) sets have not to be interpreted that way. For theoretical models, even though they are usually considered as abstract entities, there is no agreement about what kind of abstract entities they are, i.e., what is the best way of conceive them—either as interpretations (Tarski 1935, 1936) or as representations (Etchemendy 1988, 1990), or as fictional (Godfrey-Smith 2006; Frigg 2010), or as abstract physical entities (Psillos 2011). However, due to space limitations, we will not delve into these issues.
- 18.
A structure y is a substructure of another structure x (in symbols: y ⊑ x) when the domains of y are subsets of the domains of x and, therefore, the relationships (or functions) of y are restrictions of the relationships (or functions) of x. A structure y is a partial substructure of x (also symbolized by y ⊑ x) when, besides being a substructure of x, there is at least one domain or relationship (or function) in x that has no counterpart in y. The important thing is that the partial substructure y contains less components – domains or relationships (or functions) – than the structure x. Thus, structures x and y are of different logical types. If y is a substructure (either partial or not) of x, it is also said, inversely, that x is an extension of y.
- 19.
In this work a palynological study of the expansion of Pinus sylvestris in the geological stage after the last glaciation is carried out. The pollen grains contained in each sample are taken as organisms of the population.
- 20.
For a structuralist approach to features of approximation and a precise formal explication of the notion of the approximative empirical claim, see Balzer et al. (1987), chapter VII.
- 21.
This is the model-theoretic, semantic, in particular, structuralist version of what has been said in Sect. 10.2 about the testability and eventually refutability of particular hypotheses/terminal special laws. While in the classical approach of testing the particular hypotheses/terminal special laws are the entities to be tested, in the structuralist approach the “empirical claims” associated to terminal special laws are the entities that carry the weight of testing and to which it is able to direct “the arrow of modus tollens” (Lakatos 1970, p. 102).
- 22.
The theory-net of PD is a clear example of Levins’ following claim: “This is a modeling sequence from the general to the particular, adapting general knowledge to particular cases by successive specifications that might reach the end point of assigning numerical values. At each stage of specification the model is more precise and realistic but less general (Levins1993, p. 551):
- 23.
- 24.
The equations are named within ecology as “laws,”, “rules,” “models,” or “theories” in an indistinct and confused way.
- 25.
Nowadays, it can be considered a truism in philosophy of science that empirical science goes beyond “appearances,” “phenomena,” or “facts” in order to understand them better. Empirical science postulates, in addition, a realm of entities that are not directly empirically accessible, but they are accepted, at least inasmuch as the linguistic frameworks or theories in which they essentially occur are accepted as well (Carnap 1950). Thus, for instance, electric fields and wave functions are accepted, at least inasmuch as the theories of electromagnetism and quantum mechanics, respectively, are accepted. And scientists had good reasons to do so. Let us call that view on science “non-narrow inductivism” (inspired by Hempel 1966) or “non-restricted empiricism” (inspired by Carnap 1956). The analyses (explications) of (metascientific) concepts, such as law, model, or theory, can be considered as forming interpretative schemes or explanatory models—in the sense of Hintikka (1968) within epistemic logic, and of Stegmüller (1979) and Moulines (1991, 2002) within philosophy of science—of a philosophical nature, which propose or exhort us to “see the world” of science in a certain way. And a philosopher of science who uses one of these explanatory models overcomes narrow inductivism and restricted empiricism on a metascientific level in a similar way to what has been said and recommended for the case of the scientist. She/he interprets in a non-narrow inductive or non-restricted empiricist way what scientists do: not because we do not (directly) “see” their “fundamental laws/guiding principles,” they are not “(in some sense) there” to be seen. As Goodman says, “We see what we did not see before, and see in a new way. We have learned” (Goodman 1978, p. 173).
- 26.
For a systematic treatment of this less known metascientific concept, see Moulines (2014).
- 27.
For an application of metatheoretical structuralism to the reconstruction of systemic analysis as an empirical scientific theory and to the discussion of the relationships between systemic analyses and functional explanations, see Olmos et al. (2019, Chap. 11, this volume).
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This research was supported by the research projects PICT-2014-1741 (ANPCyT, Argentina), FFI2012-37354/CONSOLIDER INGENIO CSD2009-0056 (Spain), and FFI2016-76799-P (Spain).
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Lorenzano, P., Díaz, M.A. (2020). Laws, Models, and Theories in Biology: A Unifying Interpretation. In: Baravalle, L., Zaterka, L. (eds) Life and Evolution. History, Philosophy and Theory of the Life Sciences, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-39589-6_10
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