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The Epsilon-Reconstruction of Theories and Scientific Structuralism

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Abstract

Rudolf Carnap’s mature work on the logical reconstruction of scientific theories consists of two components. The first is the elimination of the theoretical vocabulary of a theory in terms of its Ramsification. The second is the reintroduction of the theoretical terms through explicit definitions in a language containing an epsilon operator. This paper investigates Carnap’s epsilon-reconstruction of theories in the context of pure mathematics. The main objective here is twofold: first, to specify the epsilon logic underlying his suggested definition of theoretical terms and a suitable choice semantics for it. Second, to analyze whether Carnap’s approach is compatible with a structuralist conception of mathematics.

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Notes

  1. See Andreas (2013) for an overview of the discussion and relevant literature.

  2. For a discussion of this indefinite character of theoretical terms see, in particular, Andreas (2010), Cei and French (2006), Friedman (2011), and Psillos (2000). Carnap’s work on the so-called “partial” or “indirect interpretation” of theoretical terms in Carnap (1939) and Carnap (1956) can be considered as an early systematic attempt to deal with this indefinite character of such terms.

  3. Work on this topic has so far focused mainly on the Ramsification of theories (to be described below) and its ontological commitments. See, in particular, Hempel (1958), Carnap (1963), and Maxwell (1970) for early contributions as well as Psillos (2000), Demopoulos (2003), Friedman (2008) for more recent discussions.

  4. Remarks to a similar effect can be found in Psillos (2000) and Demopoulos (2013). Compare also Cei and French: “Carnap’s formulation leads to a structuralist treatment of the theoretical terms implying a form of multiple realisability.” (Cei and French 2006, p. 643)

  5. Structuralism will be understood here primarily as an epistemic thesis about the nature of scientific theories. Roughly put, it is the view that theories investigate only the structural properties of their subject fields while remaining ignorant about qualitative or intrinsic properties of the objects considered. Such an account of scientific knowledge has a long history and traces back to work by Poincaré, Russell, Ramsey, Maxwell, and Carnap. See, in particular, Demopoulos (2013) for an extensive investigation of the historical roots of scientific structuralism. Compare Worrall (1989) and Ladyman (1998) for formative contributions to the modern debate.

  6. Compare Lewis on this understanding: “I do not understand what it is just to be a theoretical term, not of any theory in particular, as opposed to being an observational term (or a logical or mathematical term). I believe I do understand what it is to be a T-term: that is, a theoretical term introduced by a given theory T at a given stage in the history of science. If so, then I also understand what it is to be an O-term: that is, any other term, one of our original terms, an old term we already understood before the new theory T with its new T-terms was proposed.” (Lewis 1970, p. 428)

  7. Consequently, old terms (in Lewis’ sense) will not be relevant in the present context. See Sect. 2 for a further discussion of this point.

  8. See, however, Psillos (2000), Cei and French (2006), and Lewis (1970) for general discussions of Carnap’s approach.

  9. To keep the discussion simple, we will assume here that the theoretical vocabulary consists only of predicates of a specified arity. As a consequence, the existentially bound variables in the Ramsey sentence of a theory range over relations on the domain of \({\mathcal{L}}\).

  10. A central point of discussion in the literature is whether this fact commits one a realist conception of theoretical entities. For a detailed discussion of this and related issues concerning the proper interpretation of the Ramsey sentence reconstruction of theories see Psillos (2000), Demopoulos (2013) and Friedman (2011).

  11. While RS is supposed to capture the observational content of a theory, CS is usually considered to express its analytical content. See in particular Friedman (2011) and Demopoulos (2013) for further details on this point.

  12. The use of the epsilon-term in the definiens of Carnap’s original formulation of (\(\epsilon\)-Def\(^{*}\)) is therefore redundant. The only relevant occurrence of an \(\epsilon\)-term is the one in the ‘global’ scheme (\(\epsilon\)-Def), where a particular \(\epsilon\)-representative for the tuple t of theoretical terms is fixed. Given this particular choice, the reference of the singular theoretical terms \(T_i\) can also be specified more easily in terms of a projection function. Carnap was in fact aware of the redundancy of the \(\epsilon\)-operator in (\(\epsilon\)-Def\(^{*}\)). Compare his remarks on (\(\epsilon\)-Def\(^{*}\)) in Carnap (1961): “Instead of the operator ‘\(\epsilon _{x}\)’ we could use here the customary description operator ‘\((\iota _{x})\)’, since the formula in square brackets fulfills the uniqueness condition with respect to ‘x’.” (Carnap 1961, p. 161)

  13. See, for instance, Friedman (2008). Friedman puts this point as follows: “For the Carnap sentence is now seen to take over the role of precisely existential instantiation from the Ramsey-sentence, and it then allows us to proceed with ordinary mathematical reasoning in the style of Hilbert without worrying about cumbersome restrictions on existential variables in natural deduction. Whereas existential instantiation, of course, is not a logically valid inference, the Carnap sentence (...), taken as a non-logical axiom of T, is now seen, nonetheless, as an analytic postulate—a conventional choice of (constant) names arbitrarily given to a sequence of values of the variables (...), which, by the Ramsey-sentence, must (synthetically) exist. ” (Friedman 2008, p. 397)

  14. It is possible, however, to speak also of old terms in the context of mathematics, at least if these terms are understood in Lewis’s sense. Such terms become relevant in cases where extensions of a mathematical theory are considered. Consider the example of Robinson arithmetic presented below: Axiom system (Q1-Q7) implicitly defines the terms \(S, +, *\). Thus, relative to theory (Q1-Q7), these count as theoretical terms in our understanding. However, relative to the extended axiomatic theory (Q1-Q8), expressions \(S, +, *\) can also be viewed as old terms, i.e. as already defined terms. This issue will not be pursued here any further.

  15. See, e.g., Hajék and Pudlák (1998, p. 28).

  16. See Zach (2003) for a detailed history of the epsilon logic. Compare Leisenring (1969), Zach (2009), and Moser and Zach (2006) for modern presentations of the epsilon calculus and its semantics.

  17. Given a second-order or type-theoretic language, “higher-order” epsilon terms can be constructed in a similar way. For instance, let A(X) be a formula with a free n-ary relation variable X. The term \(\epsilon _{X} A(X)\) will pick out an arbitrary n-ary relation on the domain of the model relative to which statement A is true. Epsilon terms built from function variables of any arity can be constructed in a similar way.

  18. Compare also the following related remark: “(...) the Hilbert \(\epsilon\)-operator belongs to a small class (...) of logical constants of a very particular kind. I will call them indeterminate. They are such that their meaning is not completely specified.” (Psillos 2000, p. 171)

  19. It should be noted here that Carnap was aware of the possibility of such a choice-functional interpretation of epsilon logic. He refers in Carnap (1961) to Asser’s paper “Theorie der logischen Auswahlfunktionen” (Asser 1957) were a first systematic treatment of different choice semantics for the epsilon calculus is given.

  20. We follow closely here Zach’s presentation of a choice semantics given in Zach (2009). For a different semantic account, see, e.g., Leisenring (1969).

  21. Given that our chosen background theory is set theory here, we can formulate choice functions for individual terms in the above sense without any loss of generality. Nevertheless, choice functions for higher-order epsilon terms can, in principle, be constructed in a similar way. Consider again the epsilon term \(\epsilon _{X} A(X)\) described in footnote 17. A suitable choice function for it is a function of the form \(\delta : \wp (\wp (D^{n})) \rightarrow \wp (D^{n})\) such that, for any set of relations \({\mathbf{R}} \subseteq \wp (D^{n})\), we have:

    \(\delta ({\mathbf{R}}) = {\left\{ \begin{array}{ll} X \in {\mathbf{R}}, &{\text{if}}\quad {\mathbf{R}} \not = \emptyset ;\\ X \in \wp (D^{n}) &\text{otherwise.} \end{array}\right. }\)

  22. See again Zach (2009).

  23. Note that the present explication of theoretical truth is based on a variable choice interpretation of the \(\epsilon\)-operator: \(\epsilon\)-terms (and thus sentences containing them) are evaluated relative to a all possible choice functions for a given model. The question whether such a flexible choice semantics is in conformity with Carnap’s original understanding of epsilon terms will not be addressed here.

  24. It should be emphasized here that this semantic evaluation of a sentence like \(\psi\) is, in an important sense, theory relative. In Carnap’s approach, the t-terms are defined relative to a particular theory (expressed in the definiens of the \(\epsilon\)-term definition). It follows that the same terms are defined differently relative to different theories. For instance, the same algebraic signature can be defined relative to the theory of monoids or to group theory. These different theoretical contexts have direct implications for the semantic evaluation of the sentences expressed in terms of these theoretical terms. Sentence \(\psi\) is, as we saw, satisfied relative to the background theory of monoids, but obviously valid relative to the theory of groups.

  25. See in particular Carnap (1956). As has been pointed out in Psillos (2000) and Andreas (2010), this fact of the “non-uniqueness” of the interpretation of theoretical terms distinguishes Carnap’s \(\epsilon\)-definition from Lewis’ explicit definition of such terms by use of a logical \(\iota\)-operator (Lewis 1970).

  26. Psillos, for instance, states that: “(...) in a sense, the \(\epsilon\)-operator characterises an indefinite description (...).” (Psillos 2000, p. 157)

  27. For Russell’s remarks on ambiguous descriptions as opposed to definite descriptions, see his Russell (1919).

  28. An indefinite description can thus be constructed from the definite description “the A is a B”, expressed by \(\exists x (A(x) \wedge \forall y(A(y) \rightarrow x = y) \wedge B(x))\), simply by dropping the uniqueness clause, that is the claim that the description fixes the reference to one particular object in the domain.

  29. While Russell’s indefinites cannot be expressed in the language of EC, it turns out that they can be formulated in a natural extension of it, namely in a language of indexed epsilon terms (Heusinger 2000; Mints and Sarenac 2003). Another possible connection between the \(\rho\)-operator and epsilon logic concerns Bell’s work on the so-called “dependent” epsilon terms in the context of intuitionistic logic (Bell 1993). A closer comparison between the logic of the indexed \(\epsilon\)-operator, Bell’s work on dependent choice, and the Russellian \(\rho\)-operator will be given in a separate paper.

  30. See, in particular, Breckenridge and Magidor (2012) for the most extensive discussion of arbitrary reference in the literature.

  31. See Martino (2001) for an explicit discussion of arbitrary reference in the context of mathematics. Compare also Breckenridge and Magidor (2012) for an interesting discussion of instantial reasoning in geometrical reasoning.

  32. According to the latter view, instantial terms are best explained not as denoting particular objects but rather as ranging over a class of relevant objects just as variables do. Thus, terms or names are really variables in disguise, that is plurally referring expressions. For a detailed presentation of such a position, see, in particular, King (1991). Compare Breckenridge and Magidor (2012) for a critical discussion of this approach as well as a comparison of the different approaches to instantial terms.

  33. Fine’s account holds that in logical and mathematical reasoning, instantial terms can be taken to refer not to individual objects, but to objects of a different kind, namely arbitrary objects.

  34. Thus, one can say that while Fine assumes a classical semantics but a nonclassical ontology, the epsilon-logic is based on a nonclassical choice semantics, but presupposes a standard ontology of objects.

  35. Interestingly, Demolopous describes the “structuralist thesis” in direct comparison with a central method in modern structural mathematics, namely Hilbert’s formal axiomatics:

    The structuralist thesis is a simple extrapolation from Hilbert’s understanding of the essential generality of mathematical theories. Hilbert argued that the proper formulation of a mathematical theory should not be constrained by the demand that it preserve[s] preconceptions regarding the nature of the theory’s primitive notions. (Demopoulos 2013, p. 82)

  36. See, e.g., Shapiro (1997), Resnik (1997), and Linnebo (2008) for different versions of such a non-eliminative position. There are interesting parallels between non-eliminative mathematical structuralism and different versions of ontic structural realism discussed in philosophy of science. Compare, e.g., Busch (2003) and French (2010) for a closer discussion. Compare also Brading and Landry (2006) for a more general comparison of mathematical and scientific structuralism.

  37. Compare Reck and Price (2000) for a detailed comparison of the two approaches.

  38. These semantic aspects related to structuralism, in particular the semantic indeterminacy of mathematical terms has been subject of recent discussions, in particular in Hodesdon (2014), Shapiro (2008), and Pettigrew (2008).

  39. See Reck and Price (2000) and Parsons (1990) for detailed discussions of this position.

  40. In the present example, \(\varphi ^{*}\) can be viewed as a structuralist translation of statement \(\varphi\) that is given in a “purified” (second-order) object language, i.e. a language with an empty signature. It can be viewed as an object-language translation of the underlying metatheoretical claim, namely that \(\varphi\) is a logical consequence of theory \({\mathsf{Q}}\). See Reck and Price (2000) for further details.

  41. This set-theoretic ontology assumed here is usually specified by an axiomatic set theory, for instance ZFC.

  42. Compare Pettigrew on this analogy:

    Thus, in a first-order system of natural deduction, free variables allow us to express generalities and to reason about them: in the case of universal introduction, we note that, if we can derive \(\Phi (a)\) without making any assumptions about a, then we may conclude that, for any a, \(\Phi (a)\); and, in the case of existential elimination, if we have that, for some a, \(\Phi (a)\), and, whatever a is, if \(\Phi (a)\), then B (where B says nothing of a), then we can infer B. For this reason, it is not surprising that mathematical discourse contains a great many expressions that behave in its non-formal discourse exactly as free variables behave in formal systems of natural deduction. After all, mathematics deals in generalities. (Pettigrew 2008, p. 313)

  43. Pettigrew therefore describes primitive terms as “contextually dedicated free variables”. The range of the variables \(0, 1, +, \times\) depends on the specific context in which they are introduced. Contexts can be theoretical constraints such as: (1) ‘\({\mathbb{R}}, 0, 1, +, \times\), and \(<\) satisfy the axioms for a complete ordered field’; ‘\({\mathbb{C}}, 0, 1, +, \times\), and \(<\) satisfy the axioms for a complex field’; ‘\(R, 0, 1, +\), and \(\times\) satisfy the ring axioms’. (Pettigrew 2008, pp. 316–317)

  44. The fact that mathematical primitive terms are taken to refer arbitrarily is not untypical of mathematical discourse. As pointed out in Martino (2001) and more recently, in Breckenridge and Magidor (2012), the use of arbitrary terms is quite common in mathematical reasoning, for instance in stipulative sentences occurring in mathematical proofs. What the present account shows is that this idea of arbitrary reference can be generalized to hold not only for uses of instantial terms in mathematical reasoning, but for mathematical discourse in general.

  45. We can thus say that this pure formula (or its Ramsification) functions similarly here to the way Pettigrew speaks of theoretical contexts and “contextually dedicated variables” in his account of universal structuralism. Unlike in Pettigrew’s account, however, theoretical constraints are not specified externally but internally, i.e. in the definiens of the relevant terms.

  46. Notice that in this structuralist reconstruction of mathematical statements, much depends on the choice of the theoretical context here. The semantic evaluation of a statement can vary significantly relative to different background theories. Say, for instance, that for a given statement expressed in a general algebraic terminology, we can specify as our background theory either the theory of monoids or the theory of abelian groups. The model selection given by the epsilon-operator can lead to different truth values of the statement in question.

  47. Consider, e.g., Becker Arenhart and Bueno (2015) and Ainsworth (2010) for excellent overviews of the current state of debate.

  48. Compare, for instance, Psillos (2000) on this point.

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Acknowledgments

The authors would like to thank two anonymous referees for their detailed comments on earlier drafts of this paper. They would also like to thank H. Leitgeb, M. Friedman, S. Morris, and O. Foisch for helpful discussions on the topics of this paper. Research on this article by the first author was funded by the Austrian Science Fund (FWF) (Project No. J3158-G17).

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  1. Department of Philosophy, University of Vienna, Vienna, Austria

    Georg Schiemer

  2. Munich Center for Mathematical Philosophy, LMU Munich, Munich, Germany

    Norbert Gratzl

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Schiemer, G., Gratzl, N. The Epsilon-Reconstruction of Theories and Scientific Structuralism. Erkenn 81, 407–432 (2016). https://doi.org/10.1007/s10670-015-9747-9

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