Philosophy

Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/150 Structure and Coherence of Two-Model-Descriptions of Technical Artefacts Ulrich Krohs Department of Philosophy, University of Hamburg, and Konrad Lorenz Institute for Evolution & Cognition Research, Altenberg Abstract A technical artefact is often described in two ways: by means of a physicalistic model of its structure and dynamics, and by a functional account of the contributions of the components of the artefact to its capacities. These models do not compete, as different models of the same phenomenon in physics usually do; they supplement each other and cohere. Coherence is shown to be the result of a mapping of role-contributions on physicalistic relations that is brought about by the concept of function. It results a sandwich-like structure of the two models, which can be reconstructed as a two-sorted theory element. Keywords: Theory structure, Theory element, Function, Transistor A technical artefact may be described in physicalistic and in functional terms. The physicalistic description accounts for structure and dynamics of the entity, while the functional description is based upon a design-and-use-centred view of the artefact. Both descriptions belong to different areas of knowledge and the question is how they cohere. Regarding the different descriptions as different models of the same entity, this question for coherence becomes a question of theory relation, i.e., of the relation between both models. To investigate this relation, I am splitting the problem of coherence into two questions. The first one is the question for consistency of the models. Though small inconsistencies would not completely corrupt coherence, they would diminish it and we should not presuppose that descriptions of technical artefacts are usually inconsistent. Consistency is reflected in theory structure, so the task is to reconstruct the structure of the "sandwich" of a physicalistic and a functional model and look whether it is consistent. Consistency alone, however, does not bring about coherence. Obviously, the junction of two isolated sets of propositions is consistent, but it would be strange to call this junction, which falls in two, coherent. Isolation has to be regarded as a feature that diminishes coherence (BonJour 1985, p. 95). Some relation must hold between the elements of physicalistic and functional descriptions that brings coherence about. Laurence BonJour regards mainly inferential relations between the elements of a class of propositions as increasing its coherence (op. cit.). According to Paul Thagard, among the candidates for such relations are explanatory and deductive relations as well as other mappings (e.g., Thagard et al. 2002). So the second task is to inquire into the kind of relation that holds between a physicalistic and a functional model and show how this relation accounts for the coherence of the description that is made up by a combination of both models. The required concepts for the reconstruction are introduced in the first section of this paper. The second section develops the structure of consistent two-model-descriptions. The third section is dedicated to the question of coherence of physicalistic and functional models. In the fourth section, I will demonstrate the application of my reconstructive approach to the case of the Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/151 transistor. I will then discuss how this approach deals with the ambiguity of structure-function relations, and finally sum up my results. 1. Theory relations and sorts of models The case of physics Most investigations of theory relations have concentrated on physical theories. When two different models are given for the same process, in physics the question is usually: which is the better one? "Better", of course, needed to be qualified and may be relative to some epistemic goal. Even if there is no ranking among the models, they are in most cases alternatives, not supplements. A description may use one model or the other, but one cannot combine both without running into inconsistency. For example, classical and relativistic mechanics cannot be mixed up. However, one of the models may be reduced to the other. This means that it is possible either to express all that can be said by the reduced model in terms of the reducing one, or at least to map all results of one model onto the results of the other. In terms of Scheibe's synthetic concept of reduction (Scheibe 1993, p. 266), we have a partial reduction in the latter case.1 Nevertheless, since such a mapping of results may hold even with inconsistent models, reducibility is not a guarantee for the consistency of two theories. There is a case in physics in which the competition between two models is set aside and instead two seemingly incompatible models are used in parallel without a reduction relation holding between them: the case of the dualism of wave and particle descriptions of elementary particles. Neither model is sufficient for the description of all phenomena observed. Therefore both are used, as a workaround, in parallel. But in this case, both models have different ranges of application, which prevents inconsistency. So this is not a paradigm for cases like those that shall now be discussed. The case of biology and of technology If we look at biology, things turn out different. Here, we have two different sorts of models: physicalistic models that describe the physicochemical mechanisms of biological processes and functional models that describe the various mechanisms as realizing physiological functions. The functional account is normative since it allows for judgments about whether a component of a system behaves as it is supposed to do. So function modules are of another sort than physicalistic ones. While this characterization of the model sorts classifies models according to the involvement of functional terms, I have proposed an independent delineation criterion elsewhere (Krohs 2008). The case of technology is similar to that of biology. Physicalistic models of technical artefacts, i.e., descriptions of the physical processes going on within an artefact, and functional models that describe these processes as being implemented to follow a certain goal, complement each other. Both descriptions are required and technical artefacts, if described only in terms of their physical properties not also from the functional perspective, would become mere physical objects and would lose their status as technical entities (Kroes 2002, p. 294). Since the cases of modelling biological and technical systems both show the duality of physicalistic and functional views, many of the results that were obtained in philosophy of biology are also valid for the technological case. This holds in particular for the debate about reductionism. There was an extensive discussion of the question whether biology in general can be reduced to physics.2 However, up to now all attempts failed and reduction cannot be regarded as a general means to eliminate functional talk, one of the main obstacles being the multiple Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/152 realisability of functions (Putnam 1967; Fodor 1974).3 One way to escape from unsuccessful reductionism is to regard functions as emergent properties of technical and biological systems and base the analysis of functional models on strong concepts of emergence (see, e.g., Beckermann et al. 1992). Reference to emergent properties, however, would be a way out only if functionalities were adequately conceived as properties. But they are categorically different from properties in being normative: a function may still be correctly ascribed even if the function bearer is dysfunctional and thus unable to perform the function.4 I therefore treat functionalities not as properties but as theoretical terms of the functional models that we use to describe biological organisms. Functional models have consequently to be conceived as being of a different sort than physicalistic models. Biological theories, then, are composites of models of both sorts. The strategy to investigate the interplay of models of different sort is to reconstruct the structure of such composite theories (Krohs 2004, chs. 7-8). It is important to notice that a functional model of a technical artefact is not only an incomplete version of a physicalistic one. It makes use of classifications (like: being a signal or a forcegenerator) that cannot occur in a physicalistic model but are genuinely technical, i.e., related to design, use, or technical function of an artefact. Only the different carriers of a signal can occur in the physicalistic model. And only a functional model may be used to refer to malfunction, since this requires some kind of norm that discriminates function from malfunction – there is no such normativity in a physicalistic model. We have two non-exclusive possibilities by which to model these devices: a physicalistic model, and a functional model. Neither of them alone covers all that can be known about a technical artefact. 2. Theory structure: Consistency To examine the coherence and consistency of mixed physicalistic/functional theories, the different models have to be described uniformly, i.e., they have to be reconstructed to some degree. For the present purpose, the structuralist version of model theory as developed by Sneed (1971) and refined by Stegmüller and co-workers (Balzer et al. 1987) seems to be the most adequate method of reconstruction. It starts from the assumption that the reconstructed models are not universally applicable but only locally, in an intended range of applications (which may be modified whenever new and adequate applications are found). This spares a lot of caveats, which were needed with a global approach like van Fraassen's constructive empiricism (van Fraassen 1972). So my decision to use structuralism is not based on any special "structuralist theory conception" that would claim that some theories have a structuralist structure (Stegmüller 1986, p. 3). I regard structuralism just as one of several tools to reconstruct theories and theory relations, and feel that it is for my purpose the most convenient of the presently available tools. The approach I will use a few concepts of structuralism only and will confine myself to introducing only these instead of the whole apparatus. Moreover, my version is a simplified version of structuralism. Talking about models in the sense of structuralism, a second concept of a model has to be used that differs from the one used in science and in technology. A model in the structuralist sense is a mathematical structure, i.e., a set of objects with relations defined on it. (We can ignore for the present purpose that, strictly speaking, we would have to deal not with structures but with classes of isomorphic structures.) The structure itself, i.e., the mathematical model, is not yet the reconstruction of a technical description or of a scientific model, since it does not by itself apply to any phenomenon; it is not a model of anything. What is needed in addition is the specification of the intended phenomena the mathematical model has to be applied to. Therefore, the model (or Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/153 class of models) in the structuralist sense plus a class of intended applications makes up the smallest piece of theory or of description. This piece is called a theory element. A theory element, not a model in the mathematical sense, is the structuralist equivalent to a scientific model or to a technical description. For example: the mathematics of the Kepler laws defines a structure only. To turn this into a model of the dynamic of components of the solar system, we have to add that this structure is intended to describe this very dynamic, which is done in the definition of the variables. Moreover, we will not reject the Kepler laws just because they fail when applied to raindrops or to billiard balls. Instead, we simply leave these possible applications outside the class of intended applications of the laws. Only the model plus the intended applications allow for scientific explanation and make up a theory element. I will refer in my argument to the following constituents of a theory element: a basis O which is the set of objects the theory element deals with and a set R of relations defined over O. These two classes make up the core of a theory element and define the notions that are available within a particular element. O specifies what can be called the ontology of the theory, and R specifies the properties of the elements of O. In addition, there is the set I of intended applications of the theory core.5 So, a theory element can be characterised as (Balzer et al. 1987): T = <O, R, I>. Two-sorted theory elements A theory element as characterised by I, O, and R may be the reconstruction of any scientific model or technological description, be it of physicalistic or of functional sort. The sort depends on the elements of R, which may be physical or functional relations. The crucial question is how to represent the interaction of a physicalistic and a functional model. I have introduced the notion of a two-sorted theory element to analyse this case (Krohs 2004). A two-sorted theory element shall be conceived as a theory element that combines two "matching" models of the same phenomenon and has greater explanatory power than either of the two models standing alone. Not any arbitrary pair of models meets this requirement. The application of the concept has to be restricted to pairs of models that are (i) of different sort, that are (ii) models of the same entity or phenomenon and that (iii) describe the same aspect of this entity or phenomenon. I call two models that satisfy these requirements a pair of corresponding models. Satisfaction of condition (i) (sort-difference) is given in all cases where a physicalistic and a functional model are combined. Condition (ii) (identity of the described entity or phenomenon) is satisfied by any pair of sort-different models that share the same basis O. However, there will usually be some differences in the bases, with a large intersection. The basis of the two-sorted theory element is the union of both bases. The overlap is significant in all cases in which it can be assumed that the modellers do regard a physicalistic and a functional model as referring to the same entity or phenomenon (e.g., the description of an integrated circuit as a set of logic gates and as a collection of semiconductors with certain electrical characteristics that bring the logical functions about; the biochemical and the information-transfer model of protein biosynthesis). Condition (iii), asking for identity of the aspect described, rules out cases in which one aspect of the entity or phenomenon is described in a physicalistic way, and another aspect in a functional way. For example, a physicalistic model of the heat production of a radio receiver and a functional description of its capacity to receive radio programs should not be regarded as being corresponding models because the described aspects are different. Corresponding models, in contrast, would be a physicalistic and a functional model of the capacity of the radio receiver to receive radio programs, or, if you like, a physicalistic and a functional model of the heat production of a receiver. In terms of structuralism: condition (iii) is met only in such cases in Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/154 which both corresponding models share the application. We can now give the definition of the correspondence relation (corr): (corr) Two models of different sort correspond iff the intersection of the bases of both models is not empty and both are applied to the same shared element of the respective classes of intended applications. A two-sorted theory element then is a pair of corresponding models. In the simplest case, it is made up of the shared basis OT2, plus the classes of physicalistic and of functional relations, Rp and Rf. This results in a "sandwich-structure" of a two-sorted theory element: Two different slices of bread (the two sets of relations) share the butter (the set of objects): In the general case, as already mentioned, the classes of objects of both models differ but intersect. In addition, the class IT2 of intended applications has to be specified. Ip and If may be identical. In this case, the class IT2 of intended applications of the two-sorted theory element will be the same as for each of the separate models. The general case is that the two models have slightly different but overlapping intended applications. So, in general, IT2 is the intersection of the classes Ip and If of the classes of intended applications of the two separate models. According to the requirement of (corr), this intersection is never the empty set. So, the general structure of a two-sorted theory element is: T2 = <Op ∪ Of, Rp, Rf, Ip ∩ If>, with indices "p" and "f" for the physicalistic and functional model, respectively. Let me point out again that a two-sorted theory element requires that its models differ with respect to their sort. Otherwise they would represent either incompatible or redundant conceptions of the entity in question. In particular, the relations that belong to the different models are of different sort and are not shared among the models. This accounts for the consistency of the theory element, given that the models are themselves consistent: as long as two models share only objects and intended applications, but not relations, no contradictions can occur between them; their combination is consistent. The set of both models will of course contain any inconsistency that might occur within each of the isolated models, but the combination will not give rise to any further inconsistencies. functional relations: {R f1 , ..., R fo } Objects: {O 1 , ..., O n } physicalistic relations: {R p1 , ..., R pm } ] functional model ] physicalistic model Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/155 3. Coherence My account of correspondence reconstructs how a physicalistic and a functional description of a technical artefact may be used in parallel without competition. In a two-sorted theory element there is, so to say, room for sets of physicalistic relations as well as of functional relations. This shows the consistency of two-model descriptions of technical artefacts. However, this is not yet a satisfying answer to the question of coherence of physicalistic and functional descriptions. It does not spell out the way in which the two models interact. In the introduction, I have pointed out that this interaction has to be reconstructed in terms of formal relations that hold between the elements of a body of knowledge. The body of knowledge that is in question here is the technical description as reconstructed in terms of a two-sorted theory element. What we are interested in now are the relations that hold between corresponding functional and physicalistic models. These may be of different kind. First, there might be classical inferential relations, as favoured by BonJour as mediators of coherence (1985). Due to the multiple realisability of functions and the heterogeneity of the functions that a function bearer may perform (see below, section 5), standard accounts of inference do not seem to be able to explain the coherence of two-sorted theory elements. The same holds true for the most important aspect of the approach of Thagard and co-workers, namely of explanatory and deductive relations (Thagard et al. 2002). Establishing relations of such kind would result in establishing a reduction relation between both sorts of models, which was shown not to hold in the cases considered here (cf. section I). But Thagard exposes another kind of relation that accounts for what he calls coherence by analogy, namely a mapping relation that holds between subsets of a body of knowledge (ibid.). I will concentrate on this mapping relation, which turns out to be a central mediator of coherence in the present case, without ruling out that other relations may hold that further increase coherence of the theory element. As I have mentioned in my exposition of the notions of structuralism, the core structure of a theory element defines the notions and relations that are available. But not every (if any) scientific theory or technical description is independent of any other theory. Concepts that are defined in other theories can be incorporated by so called intertheoretical links that hold between the two theories (Balzer et al. 1987). Being theoretical terms of other theories, such concepts can be applied in a theory element in question according to the rules that are specified within the theory they originate in. The links are additional constituents of the core structure of a theory element. My claim is that the coherence between a physicalistic and a functional description of a technical artefact is brought about by a concept that is defined in a more general theory and provided by a link: the concept of function. The link makes the notion of a function available as provided by a theory of function. Different such theories are discussed in the literature, resulting in different accounts of the concept of function. It does not matter for my present purpose which one is preferable, as long as it provides criteria for the functionality of processes or structures as described in a physicalistic way. In principle, this might be an etiological account like Millikan's (1984), based on the causal history of an entity, or one based on use plans like that of Houkes and Vermaas (2009). I have proposed an alternative explication that avoids shortcomings of other theories. It is based on the role that reference to design plays in accounts of the ontogeny or construction of an entity (Krohs 2004, chs. 4-5; Krohs 2009). "Design" includes intentional and natural design, the latter being the outcome of evolutionary processes. My approach therefore integrates biological and technical functions. However, I will not enter the discussion about the adequacy of different theories of Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/156 function here. For the reconstruction of two-sorted theory elements we merely need a workable concept of function that ascribes functions to components of technical (and biological) systems. Any such concept secures the interaction of physicalistic and functional descriptions. The role of the theory of function is the following: It provides the notion of function that can be applied to the objects Oi and the relations Rj of the physicalistic model. Only if elements of Rp can be shown to be functional according to the theory of function used, can it be regarded as justified to supplement the physicalistic model with a functional one: the concept of function becomes part of the physicalistic model, but it allows for the addition of a functional one. The theory of function that is applied maps the functional relations Rf onto the physicalistic relations Rp. This, obviously, need not be a one-to-one mapping. The structures of both models may differ greatly, as long as any relation out of Rf can be mapped onto Rp as a whole set (Fig. 1). Fig. 1: Visualisation of a two-sorted theory element. Right side of the figure: The physicalistic model consists of a set of objects, O1-O5 (only two of the five objects are labelled in the figure), and a set of two relations, Rp1 (solid lines) and Rp2 (dotted line). The functional model of the same aspect of the same entity consists, in this example, of the same objects plus an object Of1 that does not occur in the physicalistic model, and of a single relation, Rf (dashed line), that requires the additional object. The theory of function that is applied maps this relation onto the relations of the physicalistic model, as indicated in the left part of the picture. As is the case in the example, this need not be a 1:1 mapping. To sum up, coherence of models in a two-sorted theory element (given the identity of the intended application) is brought about by (i) an – at least partly – shared class of objects and (ii) by the status of some of the physicalistic relations that hold between the objects as being functional according to a link to the theory of function, therefore by a mapping of the functions to physicalistic relations. Such a mapping increases the coherence of a body of knowledge by elucidating, in Thagard's terminology, an analogy between its different subsets.6 However, since the mapping imposes relations on the components of the two-sorted theory element, it could in principle introduce not only an account of analogy, but also inconsistencies. This possible source Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/157 of inconsistencies is not covered by the argument about the structural independence of the pair of corresponding models (section 2), so it is crucial that it is observed by the theory of function. Not giving rise to inconsistencies in the mapping of functions to physicalistic relations might be regarded as a criterion that has to be met by any sound theory of function. 4. Application I now want to demonstrate the adequacy of my reconstruction using descriptions of the transistor as an example. A transistor is a semiconductor element that was developed as a device for amplification of electric current. From the literature, we can reconstruct a physicalistic model that describes the structure and electric characteristics of the transistor, and a functional model that describes how it works as an amplifier. In the descriptions given in the original U.S. Patents, both models are more or less blurred (Shockley 1950; Bardeen and Brattain 1950). For reconstruction, I therefore refer to the account of the transistor that Darlington gives on the basis of these first patents. In his patent of a transistor pair (Darlington 1953), he gives the following description of the structure of the transistor: Transistors comprise, in general, a body of semiconductive material and three connections, termed emitter, base and collector, to the body. ... Transistors ... may be classified further as to conductivity type; in an N-type junction transistor, the intermediate zone, i.e., the one with which the emitter and collector zones form junctions, is of N-conductivity type (Darlington 1953, p. 1). His description characterises the class of objects the model deals with: Elements of O are a piece of semiconductive material (usually silicon or germanium), atomic impurities, and three connections (made of metal). Free electrons may be added that did not need to be mentioned in the description because they are present in the used material anyway. Next, Darlington describes the physical relations that hold with respect to these objects: In the utilization of transistors, a number of what may be considered as circuit parameters or aspects are of prime moment. Among and illustrative of such parameters are the current multiplication factor, commonly designated alpha, and the emitter and collector resistances (ibid.). According to this description, elements of R are the mathematical functions stating the current multiplication factor, and the emitter and collector resistances. In the structuralist reconstruction, the mentioned parameters belong to the class Rp of relations of the physicalistic model. In addition, so-called "holes" or defect electrons as defined by the model of semiconductivity (Wilson 1931) will be elements of this class.7 Darlington further characterises the physicalistic model by describing how the parameters just mentioned are influenced by structural variations: [The current multiplication factor] is dependent upon a number of controllable factors, such as the width of the intermediate zone ... . Similarly, the emitter and collector resistances are amenable to control by design (ibid.). So, if the required data were provided, the physicalistic model could be stated in a completely quantified way. What about the functional model? Darlington, like Shockley and Bardeen & Brattain, speaks about amplification of a signal. "Amplification" and "being a signal", in contrast to the already Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/158 mentioned physicalistic concepts of a current multiplication factor and of a current, are functional classifications of the transistor's electric characteristics and of the change or time course of an electric current, respectively. This view is supported by the fact that there is not one single correlate to the signal. It occurs in various realisations, as it may in general be the case with functions. A voltage change of different currents may be the very same signal. Since signal processes are functional, we are dealing with an element of the class Rf of functional relations that are part of the functional model. Since the structure of the transistor as referred to in the functional model does not differ from the description given in the physicalistic model, domain O is identical for both models. (However, only two connections instead of three may be required in some other functional descriptions.) The "sandwich-structure" of the two-sorted theory element is therefore made up of two classes of relations, Rp and Rf, and a common domain O. Both models, the physicalistic and the functional one, share the intended application Ii, which is the behaviour of the transistor in an electric circuit of a certain type. Therefore, all three requirements for correspondence as posed in section 2 are met: sort difference, identity of the described entity, and identity of the modelled aspect. With respect to the two models, (corr) holds, and the technical description given in the patents has the structure of a two-sorted theory element. To account for the coherence of a two-sorted theory element, we need to have a look at the mapping of the relations of both models. Above, I have introduced this mapping as a matter of the application of the concept of function. We have seen that the functionalities of amplification and of being a signal are mapped on the physicalistic roles of a current multiplication factor and of fractions of different currents. We do not rely on any particular explication of the concept of function here, so no restriction applies in this regard. It is clear from Darlington's description that such a mapping holds, mediated by a concept of function that is not further specified. So, coherence by analogy can in fact be found in the transistor example. 5. Accounting for functional ambiguity We have not yet addressed the fact that the twinning of functional with physicalistic models may be ambiguous. Most technical artefacts can serve different functions, and in general, functions can be realised differently. This is known as heterogeneity of functionality and as multiplicity of realisation, respectively (Carrier 2000). It has to be asked on the one hand, whether the reconstruction presented can account for both, and on the other hand, whether an account of ambiguity brings in any unwanted arbitrariness. Sticking to the example of the transistor, I will first have a look at the heterogeneity of its possible functions. Above, I was concerned with the functional model of the transistor as used as an amplifier. However, it may be used in different ways. Some of these are mentioned in the quoted patents, others not.8 Most prominent might be the function of the transistor as a switch. The switch model refers to offand on-states of the transistor. They are characterised by saturation of collector current in the first state, and by zero collector-emitter-voltage in the second state. The characteristic of the transistor in the range in between, which is most important for the amplifier-model of the transistor, almost does not matter for the switch model. Nevertheless, the physicalistic model that corresponds to the functional switch-model is the same one that corresponds to the functional amplification model. Each pair of corresponding models has a common intended application (transistor in electric circuit of a certain type). So we have two different two-sorted theory elements describing the transistor. Functional heterogeneity of technical artefacts (see the two left columns of table 1) can easily be accounted for. Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/159 Table 1: Multiplicity and heterogeneity. Function Artefact Amplification Transistor Tube Multiple realisability of amplification Switching Transistor Relay Multiple realisability of switching Functional heterogeneity of transistor use Multiple realisability of a function can be accounted for in a similar way. Let us regard the function of signal amplification. This function can be realised by a transistor and described by the two-sorted theory element already discussed. Alternatively, it may be realised by a vacuum tube, which was functionally the predecessor of the transistor. Similarly, there are alternative realisations of electrical switches. The switching function may be realised, e.g., by a relay instead of a transistor (see right column of table 1). Let us regard the two realisations of an amplifying element only, a transistor and a tube. Physically, they are completely different. So the physicalistic models are different as well. However, the functional model will differ only with respect to the elements of the domain that refer to the physical entity: to the structure of the vacuum tube and of the transistor, respectively. But the two functional models will define almost the same functional relations over these domains. So again, we have two different two-sorted theory elements, in this case describing two different technical artefacts in similar functional contexts. This accounts for multiple realisability of functions. At first sight, my approach seems to be fairly liberal with respect to the functions that might be assigned to an artefact. This might be judged as being problematic. Therefore, I want to add some considerations about this issue. I was dealing with descriptions of technical artefacts. Doesn't this mean that I have to accept any function as relevant for a reconstruction in a two-sorted theory element that anybody likes to describe, no matter whether it has any relation to the actual or intended use of the artefact, the goals that anybody might pursue by using it, or the norms that are set on its construction? First, it seems quite clear that functions of technical artefacts are in fact context dependent in some way. Herbert Simon conceives an artefact as the interface of an internal structure and a (physical) environment. In his view, exactly this interface position seems to bring about the association of a goal with the artefact (Simon 1996, pp. 6-7). Peter Kroes has adjusted this conception and made clear that the relevant environment is not the physical environment but human action (Kroes 2002). This results in the view of the dual nature of a technical artefact as embedded in a context of human action. My account seems to go another step further and change the environment of human action into an environment of human description. This, I agree, would be a step too far. However, the impression of complete dependency on the description is false. There are two reasons for my claim. (i) I admit that my conception of correspondence allows for a wide range of combinations of physicalistic and functional models. Nevertheless, there is an additional demand for coherence: the theory of function itself has to be applicable in the framework of the physicalistic model. This guarantees that not any effect or relation that can be ascribed to a component of a complex entity may be considered in the functional model, but only functional relations that can be isolated from physicalistic models. So the theory of function has to be specific enough to prevent arbitrary function ascriptions. Reconstruction of theory structure only has to show that the theory of function is crucial in this respect; it does not have to develop this theory. (ii) The validity of the functional part of a two sorted theory element has to be proven by standards as strong as those that are used for the physicalistic part. A functional model that is not in accordance with observations of the structure, dynamics, and possible use of an artefact is not a good model and Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/160 must not be regarded as a valid functional description. (Such an evaluation would mostly fall within the scope of the special sciences.) But discrimination of good and bad models cannot be a matter of reconstruction. Reconstruction deals with theories only and has no grip on the modelled phenomena. Moreover, it has to be possible to reconstruct even implausible combinations of physicalistic and functional descriptions for the sake of analysis. My claim is only that the reconstruction method has a grip on "mixed" models, not that it, purely as a method, sorts out the best model. 6. Conclusion Descriptions of technical artefacts consist of physicalistic and functional elements, which can be ascribed to different models. The description as a whole was reconstructed as a two-sorted theory element, which, in the simplest case, combines a pair of corresponding physicalistic and functional models in a sandwich-like structure. This accounts for the consistency of a mixed physicalistic-functional description of a technical artefact. The same theory structure can be found in mixed physicalistic-functional descriptions in biology. It was shown how the concept of function, as provided by an adequate theory of function, maps the relations of the functional model onto relations of the physicalistic one. The mapping relation brings the coherence of a twosorted theory element about, in terms of what Thagard has dubbed coherence by analogy. References Agazzi, E., ed. 1991. The Problem of Reductionism in Science. Dordrecht: Kluwer. Balzer, W., C.U. Moulines and J.D. Sneed. 1987. An Architectonic for Science: The Structuralist Program. Dordrecht: Reidel. Bardeen, J. and W.H. Brattain. 1950. Three-electrode Circuit Element Utilizing Semiconductive Materials. U.S. Patent No. 2,524,035, Oct. 3, 1950. Beckermann, A., H. Flohr and J. Kim, eds. 1992. Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism. Berlin: De Gruyter. BonJour, L. 1985. The Structure of Empirical Knowledge. Cambridge, MA: Harvard University Press. Carrier, M. 2000. "Multiplicity and Heterogeneity: On the Relations between Functions and their Realizations," Studies in the History and Philosophy of Biology and Biomedical Science, 31: 179-191. Darlington, S. 1953. Semiconductor Signal Translating Device. U.S. Patent No. 2,663,806, Dec. 22, 1953. Fodor, J. 1974. "Special Sciences (Or: The Disunity of Science as a Working Hypothesis)," Synthese, 28: 97-115. Houkes, W. and P.E. Vermaas. 2009. "Produced to Use: Combining Two Key Intuitions on the Nature of Artefacts," Techné, 13 (2): 123-136. Kroes, P. 2002. "Design Methodology and the Nature of Technical Artefacts," Design Studies, 32: 287-302. Kroes, P. 2006. "Coherence of Structural and Functional Descriptions of Technical Artefacts". Studies in History and Philosophy of Science, 37: 137-151. Krohs, U. 2004. Eine Theorie biologischer Theorien: Status und Gehalt von Funktionsaussagen und informationstheoretischen Modellen. Berlin: Springer. Krohs, U. 2008. "Conservative and Non-Conservative Models in Molecular Biology and the Concept of Genetic Information," presented at the 21st Biennial Meeting of the Philosophy of Science Association, Pittsburgh. Krohs, U. 2009. "Functions As Based on a Concept of General Design," Synthese, 166: 69-89. Techné 13:2 Spring 2009 Krohs, Structure and Coherence of Two-Model-Descriptions/161 Krohs, U. (forthcoming). "Dysmalet non: l'autre côté de la fonctionnalité," in: J. Gayon & M. Mossio, eds., Épistémologie de la catégorie de fonction: des sciences de la vie à la technologie. Paris: Presses Universitaires de France. Millikan, R.G. 1984. Language, Thought and Other Biological Categories: New Foundations for Realism. Cambridge, MA.: MIT Press. Putnam, H. 1964. "Psychological predicates," in: W. Capitan and D. Merrill, eds., Art, Mind, and Religion. Pittsburgh: University of Pittsburgh Press: 37-48. Scheibe, E. 1993. "A new theory of reduction in physics," in: J. Earman, A.I. Janis, G.J. Massey and N. Rescher, eds., Philosophical Problems of the Internal and External Worlds: Essays on the Philosophy of Adolf Grünbaum. Konstanz: Universitätsverlag Konstanz / Pittsburgh: University of Pittsburgh Press: 249-271. Shockley, W. 1950. Semiconductor amplifier. U.S. Patent No. 2,502,488, Apr. 4, 1950. Simon, H.A. 1996. The Sciences of the Artificial. 3rd edition, Cambridge, MA.: MIT Press (1st ed. 1969). Sneed, J.D. 1971. The Logical Structure of Mathematical Physics. Dordrecht: Reidel. 2nd ed. 1979. Sober, E. 1999. "The Multiple Realizability Argument against Reductionism," Philosophy of Science, 66: 542-564. Stegmüller, W. 1986. Theorie und Erfahrung, dritter Teilband: Die Entwicklung des neuen Strukturalismus seit 1973. Berlin: Springer. Thagard, P., C. Eliasmith, P. Rusnock and C.P. Shelley. 2002. "Knowledge and Coherence," in: R. Elio, ed., Common Sense, Reasoning, and Rationality, Vol. 11, Oxford: Oxford University Press: 104-131. van Fraassen, B. 1972. "A Formal Approach to the Philosophy of Science," in: R.G. Colodny, ed., Paradigms and Paradoxes. The Philosophical Challenge of the Quantum Domain, Pittsburgh: University of Pittsburgh Press: 303-366. Van Regenmortel, M.H. and D.L. Hull, eds. 2002. Promises and Limits of Reductionism in the Biomedical Sciences. Chichester: Wiley. Wilson, A.H. 1931. "Theory of Electronic Semi-Conductors," Proceedings of the Royal Society London, A 133: 458-491. Endnotes 1 The case of partial reduction is all but an exception. Even for the mentioned standard example for reduction we cannot obtain more than this (Scheibe, op. cit.). 2 For a discussion of reductionism, see, e.g., the contributions in Agazzi (1991), Beckermann et al. (1992), and Van Regenmortel and Hull (2002). 3 For a dissenting view, see Sober (1999). 4 The concept of function has even more normative aspects than the generally acknowledged case of dysfunctionality (Krohs, forthcoming). 5 Specification of some further classes would be required to fully characterise the theory core; only one of them needs to be introduced below. 6 In addition, practical inferences may play an important role, as Peter Kroes has recently pointed out (Kroes 2006). 7 The concepts of Wilson's model are again provided via a link between his theory and the physicalistic model of the transistor that makes use of it. 8 Bardeen and Brattain (1950, p. 1) generously write about "a novel method of and means for translating electrical variations for such purposes as amplification, wave generation, and the like."


What is Radical Recursion?

Philosophy

Recursion or self-reference is a key feature of contemporary research and writing in semiotics. The paper begins by focusing on the role of recursion in poststructuralism. It is suggested that much of what passes for recursion in this field is in fact not recursive all the way down. After the paradoxical meaning of radical recursion is adumbrated, topology is employed to provide some examples. The properties of the Moebius strip prove helpful in bringing out the dialectical nature of radical recursion. The Moebius is employed to explore the recursive interplay of terms that are classically regarded as binary opposites: identity and difference, object and subject, continuity and discontinuity, etc. To realize radical recursion in an even more concrete manner, a higher-dimensional counterpart of the Moebius strip is utilized, namely, the Klein bottle. The presentation concludes by enlisting phenomenological philosopher Maurice Merleau-Pontyâs concept of depth to interpret the Klein bottleâs extra dimension.

What is Radical Recursion? Steven M. Rosen Departments of Psychology and Philosophy (Emeritus) College of Staten Island/City University of New York email@example.com [This paper is published in S.E.E.D. Journal, 2004, 4 (1), 38–57.] ABSTRACT Recursion or self-reference is a key feature of contemporary research and writing in semiotics. The paper commences by focusing on the role of recursion in poststructuralism. It is suggested that much of what passes for recursion in this field is in fact not recursive all the way down. After the paradoxical meaning of radical recursion is adumbrated, topology is employed to provide some examples. The properties of the Moebius strip prove helpful in bringing out the dialectical nature of radical recursion. The Moebius is employed to explore the recursive interplay of terms that are classically regarded as binary opposites: identity and difference, object and subject, continuity and discontinuity, etc. To realize radical recursion in an even more concrete manner, a higher-dimensional counterpart of the Moebius strip is utilized, namely, the Klein bottle. The presentation concludes by enlisting phenomenological philosopher Maurice Merleau-Ponty's concept of depth to interpret the Klein bottle's extra dimension. 1 SEMIOTICS, POSTSTRUCTURALISM, AND RECURSION In classical signification, the stability of the relationship between the signifier and what it signifies is maintained by preserving the anonymity of the former. Attention is fixed solely on the meanings that are signified, not on the act of signification itself. With the advent of semiotics this changes. Semiotics is the discipline that studies the process of signification. Here the sign becomes recursive; instead of focusing exclusively on signified meanings, it comes to focus on itself. The signifier, which had played a predominantly tacit role in classical semiosis, is now itself explicitly signified. Despite this role reversal inherent in the very existence of the discipline of semiotics, structuralist semioticians like Saussure still sought to preserve the invariance of the link between the given signifier and what it signifies. The problem is that, once classical sig2 nification is surpassed by signifying the signifier, the door is opened to an infinite regress. For now, it seems that no signifier is exempted from mutation into that which is signified. A new signifier is presumably needed to signify what had been the signifier, but this new signifier is subject to signification by a still newer signifier, and so on ad infinitum. And each time the tacit operation of the signifier is undermined by being explicitly signified, the functioning of what had been signified by that signifier is also affected. Ultimately then, we have in this "hall of mirrors" neither signifier nor signified in any stable, abidingly meaningful form. Poststructuralist writing exemplifies the recursive "sliding" or "slippage" of the signifier. The approach of psychoanalyst Jacques Lacan is a prime illustration. For Lacan, language "is constituted by a set of signifiers" that involves what "I call the Other" (1966/1970: 193). The "otherness" of language results from the fact that, in its "chain of signifiers" (194), every act of self-reference, rather than affirming the identity of the self or subject that is referred to, always slips away into what is other, into a new and anonymous signifier. As Lacan puts it: All that is language is lent from this otherness and this is why the subject is always a fading thing that runs under the chain of signifiers. For the definition of a signifier is that it represents a subject not for another subject but for another signifier. This is the only definition possible of the signifier as different from the sign. The sign is something that represents something for somebody, but the signifier is something that represents a subject for another signifier. The consequence is that the subject disappears...." (1966/1970: 194) In this way, the sign-which had constituted for earlier semioticians a fixed relationship between a signifier and its signified meaning, with the subject operating stably behind the scenes (the "somebody" to which Lacan alludes)-now dissolves into an evanescent flux of differences wherein the subject loses its substance, becoming a "nobody," a ghost-like quasi-presence. Much the same process of dissolution is reflected in the deconstructionist writings of Jacques Derrida. In the "primary writing" (1976: 7) of which he speaks, "[s]ign will always lead to sign, one substituting the other...as signifier and signified in turn" (Spivak 1976: xix). In Derrida's own words, language must be understood as a field "of freeplay, that is to say, a field of infinite substitutions" (cited by Spivak 1976: xix) in which identity fragments into sheer difference (différance). The specific way this takes place is by the process of self-referential mirroring in which, time and again, the signifier is displaced by being made into what is signified by a newly implicit signifier. 3 It seems clear that the slippage of the signifier results from the indefinite repetition of recursion. This is the nature of semiosis, we are told; it is a "mirror game" in which self-identity is perpetually subverted and we continually slide into otherness and difference. While I certainly agree with the general consensus among semioticians that symbolic operations are inherently recursive, I propose that the infinite regress to which poststructuralism is prone actually derives from its failure fully to achieve recursion. Again, when the signifier "slips," it becomes something that is now itself signified. Yet does this really constitute a concrete instance of self-signification, or does it merely entail the close juxtaposition of two semiotic acts neither of which are recursive in themselves? Initially, X signifies Y. Then there is the reversal of this in which X itself becomes signified by a new signifier, Z. This is obviously not to say that X signifies X. Poststructuralist "self-signification" then does not truly involve a signifier's reference to itself within the same actual occasion, to use Whitehead's (1978) term for a fundamental concrete event; it entails only a switching of roles between signifier and signified from one occasion to another. (No doubt occasions may follow each other in close succession and may be broadly construed as belonging to the "same" occurrence; but, on a more concrete level, the occasions of poststructuralism constitute distinct semiotic acts.) Therefore, poststructuralist signification is not radically recursive, not recursive all the way down into the roots of semiosis, for the signifier of occasion 2 does not signify itself but only that which was the signifier on occasion 1. If poststructuralism fails to meet the challenge of radical self-reference because its signification within the actual occasion is strictly a reference to what is other, does radical recursion involve reference to the self in the sense of simple self-identity (X≡X)? It surely cannot. For without the aspect of the other, of difference, meaning is trivialized and collapses. Radical recursion therefore entails neither external reference nor selfidentity. What it constitutes, I suggest, is the dialectical interplay of these. I propose that, in radical recursion, though the self that is signified is not simply the same self that does the signifying; though the very act of reflecting upon the self turns it into what is other; this other flows right back into the source from which it arises, rather than appearing merely as an other cast before a new self. The semiotic act I am intimating thus would give us neither self nor other, in the categorically opposed sense of these terms. We would realize instead their paradoxical interpenetration. I suggest that this dialectic is what we require to supersede the supremacy of linear signification in a meaningful way. Signifier and signified would be more than reciprocally interdependent in such a self-signification. They would be identical, utterly one. Yet they also would be two. By virtue of the latter aspect, meaningful signification would continue; by virtue of the for4 mer, recursion would go all the way down; it would be realized concretely in the heart of the actual occasion. To be sure, this construal of radical recursion requires further explication. 2 RADICAL RECURSION IN TOPOLOGY Psychoanalyst Jacques Lacan had turned to the science of linguistics in order to clarify the language of the psyche. What we see precisely in the slippage of the signifier through which the subject is "a fading thing" (1966/1970: 194) is the functioning of the unconscious. But Lacan was not content to stop with a merely linguistic clarification of psychic process. In an effort to achieve an even higher level of precision, he appealed to mathematics, and, in particular, to topology, the qualitative sub-discipline that deals with the properties of surfaces. By way of elucidating the signifying activity that constitutes the unconscious discourse of the human subject, Lacan presented a diagram of a Moebius strip: This diagram can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. (Lacan 1966/1970: 192–193) Comparing the sphere and the Moebius strip, we can say that both are recursive, insofar as they both turn back upon themselves. But unlike "that old symbol for totality," the Moebius possesses a "fundamental cut"; a knot, twist or gap. In Lacan's view, this cut represents the division inherent in the subject that prevents it from realizing the selfidentity symbolized by the sphere. The cut functions like a crack in a mirror, leading the Moebius to signify itself in such a way that it distorts or displaces itself. Or, speaking diachronically, we can say that Moebius recursion is interrupted by the cut and we cut to a new occasion. In moving into the Moebius's twist, the signifier we started with is twisted into that which is signified by a newly implicit signifier that is a "mirror image" of the original yet out of step with it. On Lacan's reading then, the Moebius strip embodies the 5 self-alienating kind of recursion that falls short of what I have called radical recursion. I suggest, however, that there is a different way of reading the Moebius. To see how Moebius recursion can be grasped in the radical sense adumbrated above, let us look more closely at this curious topological structure. We may bring out most effectively the dialectical character of the Moebius strip by comparing it to a non-dialectical structure more similar to it than is the sphere: the cylindrical ring (see Rosen 1994, 2004a). Figure 1. Cylindrical ring (a) and Moebius strip (b) A cylindrical ring (Fig. 1a) is constructed by cutting out a narrow strip of paper and joining the ends. The surface of Moebius (Fig. 1b) is produced by giving one end of such a strip a half twist (through an angle of 180°) before linking it with the other. The cylindrical ring possesses the familiar property of two-sidedness: at any point along its surface, two distinct sides can be identified. Commencing on either side, rotation about the ring traces out a circle of simple self-return like that found on the sphere. The two-sidedness of the cylinder of course precludes continuous passage from one side to the other. Such a transition is inevitably cut short at the surface's edge; the singularity we encounter there tells us that we cannot reach the far side without a break in contact, a cut to a new occasion. We are therefore able to say that, whereas rotation about a single side of the twosided ring signifies the simply continuous affirmation of self-identity, passage between sides expresses a simply discontinuous cut to what is other. Now, in the case of the Moebius strip, it is true that if you place your index finger anywhere on the surface, you will be able to put your thumb on a corresponding point on the opposite side. The Moebius strip does have two sides, like the cylinder. But this only 6 holds for the local cross-section of the strip defined by thumb and forefinger. Taking the full length of the strip into account, we discover that points on opposite sides are intimately connected-they can be thought of as twisting or dissolving into each other continuously, as being bound up internally. Accordingly, mathematicians define such pairs of points as single points, and the two sides of the Moebius strip as but one side. I want to emphasize that the Moebius surface is not one-sided in the homogeneous sense of a single side of the cylindrical ring. It is one-sided in the paradoxical sense, onesided and also two-sided, for the local distinction between sides is not just negated with expansion to the Moebius as a whole. In coming to interpenetrate each other, the sides do not merely lose their distinct identities. And yet, though the sides remain different, they also become one and the same. Thus, if the cylindrical ring embodies the dualism of identity and difference, of continuity and discontinuity, the Moebius strip signifies their dialectical entwinement. We can say as well that while the cylinder dualistically expresses both trivial recursion (through movement on a single side) and non-recursion (through passage to the other side), the Moebius models radical recursion. Let us focus on the unique recursive action of the Moebius. With 360° of rotation about this surface, we appear to return to our point of origin. But this return is in fact also a departure, since, instead of remaining on the same side of the strip as in the case of cylindrical rotation, we are carried to the opposite side. So Moebius recursion incorporates an element of discontinuity not evident in its cylindrical counterpart. It is this distinctive feature that Lacan picked up on in contrasting the Moebius with the sphere. What Lacan apparently missed is that the discontinuity of the Moebius, its twist or cut-unlike the cut required in passing from one side of the cylinder to the other-is also continuous. It is in glossing over the paradox of Moebius recursion that Lacan stopped short of radical recursion. Lacan was apparently unable to recognize that the Moebius signifier does not merely short-circuit its reference to itself by prematurely cutting away from itself to an alterself operative on a new occasion. Rather, the Moebius signification of self as other (and other as self) transpires within the same concrete occasion thereby surpassing the dualism of self and other. The application of Moebius topology has been taken up by thinkers with diverse orientations and disciplinary backgrounds. From a feminist perspective emphasizing embodiment, theorist Elizabeth Grosz expands on Lacan's use of the Moebius by portraying it as expressing "the inflection of mind into body and body into mind" (1994: xii). Anthropologist Peter Harries-Jones (2002) suggests that the paradoxical link between culture and environment as understood by Bateson is best depicted in the form of a Moebius strip. Communications philosopher Brian Massumi (carrying forward Deleuze and Guat7 tari's call for a "topology of multiplicities" [1987: 483]) demonstrates the need to reconceive human transactions via a "strange one-sided topology" that recursively surmounts the old dichotomies by working at a "paradoxically creative edge" (1998). Semiotician Floyd Merrell (1998) uses the paradox of the Moebius to model C. S. Peirce's concept of abduction. And philosopher Yair Neuman-in this issue of SEED-applies the Moebius to the structure of boundary events in semiotic systems. (My own work with the Moebius dates back to the 1970s; see Rosen 1994.) In these writings, sustained emphasis on paradox allows the authors to surmount Lacanian "slippage" and employ Moebius topology to question effectively "the binary oppositions...[of] mind/body, nature/culture, subject/object and interior/exterior" (Grosz 1994: 164). It is all too easy, however, to lose one's paradoxical edge. This is evidenced in Grosz and Massumi when-after using topology to successfully challenge binary opposition on one level of analysis, they appear to fall prey to it on another. Thus, in the case of certain root philosophical oppositions that implicitly structure their thinking-such as the one and the many, identity and difference, being and becoming-they wind up privileging "the fields of difference, the trajectories of becoming" (Grosz 1994: 210). The "onesidedness" of such a reaction to the totalizing propensities of structuralism (and classicomodernism in general) is certainly not of the Moebius kind. Instead of genuinely questioning the categorial purity of the old approach by consistently applying topological paradox to the most basic philosophical dichotomies, there is a slippage into a sort of "reverse purism" (Rosen, 2004b). Pure identity (totality, unity, being, continuity, etc.) is supplanted by a mode of difference every bit as pure: Derridean différance. It is in the process of unambiguously affirming one member of the philosophical binary over the other that the Moebius edge is lost. So what I am proposing is that the application of topological paradox needs to be implemented in a consistent and thoroughgoing manner all the way down. From my own experience, I know how difficult this is to achieve and it would not surprise me to learn that I myself lose my edge in places in this very text. As a dweller in a "glass house," I must be careful then about the "stones" I hurl. We are all challenged to avoid limiting our applications of topological paradox to the surface of our discourse while allowing our deepest assumptions and forms of expression to remain tacitly governed by the system of binary logic that has controlled our thinking for so many centuries. To keep our topological edge all the way down and thus achieve radical recursion, we must consistently exceed mere "logics of presence or position" and employ "qualitative topologics," as Massumi (1998) so well puts it. 8 3 THE KLEIN BOTTLE I must now acknowledge a limitation in the Moebius expression of radical recursion. The Moebius does effectively signify the dialectic of continuity and discontinuity. In traversing the twist, we depart from the circle of self-identity associated with continuous rotation about a single side of the cylindrical ring, and we make the transition to the other side of the surface, which, when enacted on the cylinder, brings simple discontinuity. Yet, in the Moebius case, the departure from cylindrical continuity happens continuously. However, the discontinuous aspect of the Moebius dialectic is in fact something of an abstraction. While the effect of discontinuity is surely created in passing through the twist to the far side, there is never any true cut or break, as occurs when actually crossing an edge. The Moebius therefore signifies the continuity-discontinuity dialectic in a continuous way; the discontinuous element is symbolized but not concretely embodied. I suggest that, for a full-fledged realization of the radically recursive dialectic, we require a topological structure in which continuity and discontinuity are interwoven not merely in effect but in actual fact. There exists a higher-dimensional counterpart of the Moebius surface. By way of introduction, consider an interesting attribute of the Moebius: its asymmetry. Unlike the cylindrical ring, the Moebius has a definite orientation in space; it can be produced either in a leftor right-handed form (depending on the direction in which it is twisted). If both a leftand right-oriented Moebius surface were constructed and then "glued together," superimposed on one another point for point, a Klein bottle would result (Lacan's passing allusion to this topological structure is cited above). Figure 2. The Klein bottle The Klein bottle (Fig. 2) has the same property of asymmetric one-sidedness as the twodimensional Moebius surface but incorporates an added dimension (Rosen 1994). Note, however, that we cannot actually produce a continuous model of this curious container, for leftand right-facing Moebius bands cannot be superimposed on each other in three9 dimensional space without tearing the surfaces. Therefore, while each Moebius enantiomorph is continuous within itself, joining these mirror twins to form a Klein bottle brings discontinuity. The feature of Kleinian discontinuity can be illustrated by means of a different but mathematically equivalent way of making the bottle. Once again a comparison is called for. Figure 3. Construction of torus (upper row) and Klein bottle (lower row) Both rows of Figure 3 depict the progressive closing of a tubular surface that initially is open. In the upper row, the end circles of the tube are joined in the conventional way, brought together through the three-dimensional space outside the body of the tube to produce a doughnut-shaped form technically known as a torus (a higher-order analogue of the cylindrical ring). By contrast, the end circles in the lower row are superimposed from inside the body of the tube, an operation requiring the tube to pass through itself. This results in the formation of the Klein bottle. Indeed, if the structure so produced were cut in half, the halves would be Moebius bands of opposite handedness. But in threedimensional space, no structure can penetrate itself without cutting a hole in its surface. So, from a second standpoint, we see that the continuous construction of a Klein bottle cannot be carried out in the three dimensions available to us. The Klein bottle thus seems to possess the element of concrete discontinuity missing from its lower-dimensional Moebius counterpart. Whereas the twist in the Moebius mediates the transition from one side of the surface to the other in a continuous fashion, the Kleinian passage from inside to outside requires a hole. Of course, a simply discontinuous structure will serve us no better than a simply continuous one if we are seeking to express the dialectic of continuity and discontinuity. What is needed is a structure that embodies the paradoxical inter10 weaving of continuity and discontinuity. And, in fact, the Klein bottle does just that, provided that we approach it in a truly dialectical way. How does modernist mathematics approach the Klein bottle? Mathematicians certainly do not just accept the discontinuity of the Klein bottle. Instead they rely on the idea that a form that penetrates itself in a given number of dimensions can be produced without cutting a hole by invoking an added dimension. The point is nicely illustrated by the mathematician Rudolph Rucker (1977). He asks us to imagine a species of "flatlanders" attempting to assemble a Moebius strip. Rucker shows that, since the space inhabited by these creatures would be limited to two dimensions, when they would try to make an actual model of the Moebius, they would be forced to cut a hole in it. Of course, no such problem arises for us human beings, who have full access to three dimensions. It is the continuous construction of the Klein bottle that seems problematic for us, since this would appear to require a fourth dimension, but, try as we might, we find no fourth dimension in which to execute the operation. For modernist mathematics, however, there is actually no problem. Although dimensions higher than the third may be unavailable to concrete experience, mathematicians feel free to proceed abstractly, calling forth as many extra dimensions as they wish. Added dimensions are summoned into being by extrapolation from the known three-dimensionality of the physical world. This theoretical procedure of dimensional proliferation presupposes that the nature of dimensionality itself is left unchanged. In the case of the Klein bottle, the "fourth dimension" required to complete its formation remains an extensive continuum as is three-dimensional space, though the "higher" space is taken as "imaginary"; the Klein bottle, for its part, is regarded as an "imaginary object" embedded in this space. Enclosed as it is in the hypothesized fourdimensional continuum, the imaginary Klein bottle itself is presumed simply continuous. Like the Moebius strip of three-dimensional space, it is thought to possess nary a hole. Now, in his phenomenological study of topology, the mathematician Stephen Barr advised that we should not be intimidated by the "higher mathematician....We must not be put off because he is interested only in the higher abstractions: we have an equal right to be interested in the tangible" (1964: 20). The tangible fact about the Klein bottle that is glossed over in the higher abstractions of modernist mathematics is its hole. Because the standard approach has always presupposed extensive continuity, it cannot come to terms with the inherent discontinuity of the Klein bottle created by its self-intersection. Therefore, all too quickly, "higher" mathematics circumvents this hole by an act of abstraction in which the Klein bottle is treated as a closed object embedded in a hyper-dimensional continuum. To be sure, an "added dimension" is needed if the Klein bottle is not to be regarded as merely discontinuous. When limited to the three dimensions of ordinary 11 space, the Klein bottle cannot give expression to the dialectic of continuity and discontinuity. But the "added dimension," rather than being a continuum, must itself blend continuity and discontinuity. In a continuum, all interactions occur between fixed terms that are externally related. The point elements of which the continuum is composed are themselves related in this manner. As philosopher Milič Čapek put it in his reflection on classical space, "no matter how minute a spatial interval may be, it must always be an interval separating two points, each of which is external to the other" (1961: 19). In the words of Martin Heidegger, the continuum is essentially constituted by the "'outside-of-one-another' of the multiplicity of points" (1927/1962: 481). Given the fundamental exteriority of classical space, relations among objects and events contained within it must also be external. In the continuum, systems "interact through forces that do not bring about any changes in their essential natures...[they interact] only through some kind of external contact" (Bohm 1980: 173). Generally speaking then, the notion of the continuum implies that all boundaries are external in nature. This includes the point elements that bound space; the boundaries between and among interacting objects, systems, and events in space; and the figure-ground boundary that distinguishes an entity from its spatial context. One other kind of exterior boundary is implicit in the classico-modernist approach: the one that separates the object being observed from the subject that observes or analyzes it. Whereas objects are embedded in the extensive continuum, the subject entails discontinuity. In the language of Descartes, the object is res extensa and the subject res cogitans, thus unextended, not manifested in space. However, this distinction is complicated by the subtlety of the continuum idea. The continuum actually possesses its own aspect of discontinuity. Even though the points composing space are packed densely together, because these points are related to one another externally, the continuum is infinitely divisible; it can be indefinitely partitioned into ever smaller segments (the mathematician Charles Muses was thus prompted to describe the continuum as actually constituting an "infinite discontinuum" [1968: 37]). It naturally follows that the objects embedded in this medium are themselves partible; they can be rendered discontinuous. The discontinuity associated with the subject, on the other hand, signifies its transcendence of the continuum. So, whereas the breach one may produce in an object in fact reflects a property of the continuum, the subject constitutes a break with that continuum. Although it may rightly be said that classico-modernism favors continuity over discontinuity, what we are seeing is that there is indeed a place for discontinuity in the conventional paradigm, albeit a tacit or negative one. At the deepest level, it is the division of continuity and discontinuity that classico-modernism upholds. 12 4 RADICAL RECURSION AND THE DIMENSION OF DEPTH The classical concept of dimension has prevailed from the time of Descartes and Kant to the physics and mathematics of today. In mainstream science and philosophy, the exteriority of relations among objects in space, and between object and subject, has not been questioned in a fundamental way. Yet countercurrents do exist. We find evidence of these in the works of process-oriented thinkers such as Heidegger (1962/1972), Gendlin and Lemke (1983), and Bateson (see Neuman's topological interpretation of Bateson in this issue of SEED, and Harries-Jones's [1995] account of Bateson's recursive vision); in each case, internal dynamics are given precedence over the static externality of the spatial continuum (see also Rosen 1994, 2004a). One of the most explicit formulations of process dimensionality is found in the notion of depth advanced by the phenomenological philosopher Maurice Merleau-Ponty (1964). This idea provides us with an insight into dimension that permits us to surpass the limits of classico-modernism and arrive at a radically recursive understanding of space that is well suited for expressing the Kleinian dialectic of continuity and discontinuity. By way of introducing Merleau-Ponty's depth dimension, let us consider in greater detail the traditional dichotomy between the objects contained in space and their spatial container, or, as Plato put it, between "that which becomes [and] that in which it becomes" (1965: 69). A visible form "becomes," whereas that "in which it becomes" is "invisible and formless" (1965: 70). Whatever changes may transpire in the objects that "become," however they may be transformed, the containing space itself does not change. Indeed, for there to be change, there must be difference, contrast, dialectical opposition of some kind. But the point-elements that make up the classical continuum, rather than entailing opposition, involve mere juxtaposition. Unextended and thus devoid of inner structure, the elements of space possess no gradations of depth; no shading, texture, or nuance; no contrasts or distinctions of any sort. Instead of expressing the dialectical interplay of shadow and light, space itself is all light, as it were. A condition of "total exposure" prevails for the point-elements of the continuum, since these elements, having no interior recesses, must be said to exist solely "on the outside." All that can be said of the relations among such eviscerated beings is what Heidegger said: the points of classical space are "'outside-of-one-another'" (1927/1962: 481). So, rather than actively engaging each other as the beings that are contained in space seem to do, the densely packed elements of the classical container sit inertly side by side, like identical beads on a string. 13 In fact, even though the beings that dwell in such a space can be described as "actively engaged," we have seen that the quality of their interaction is affected by the context in which they are embedded: since the continuum is constituted by sheer externality, the relations among its inhabitants must also be external. Classical dynamics are essentially mechanistic; instead of involving a full-fledged dialectic of opposition and identity wherein beings influence each other from core to core, influence is exerted in a more superficial fashion, "only through some kind of external contact" (Bohm 1980: 173). We may say then that classical space contains dialectical process in such a way that it externalizes it, divesting it of its depth and vitality. It is the classico-modernist view of space that Merleau-Ponty calls into question. What he demonstrates is that the spatial continuum appearing to contain dialectical process actually originates from it. In his essay "Eye and Mind," Merleau-Ponty emphasizes the "absolute positivity" of traditional Cartesian space (1964: 173). For Descartes, space simply is there; possessing no folds or nuances, it is the utterly explicit openness, the sheer positive extension that constitutes the field of strictly external relations wherein unambiguous measurements can be made. Merleau-Ponty speaks of this space without hiding places which in each of its points is only what it is....Space is initself; rather, it is the in-itself par excellence. Its definition is to be in itself. Every point of space is and is thought to be right where it is-one here, another there; space is the evidence of the "where." Orientation, polarity, envelopment are, in space, derived phenomena inextricably bound to my presence [thus "merely subjective"]. Space remains absolutely in itself, everywhere equal to itself, homogeneous; its dimensions, for example, are interchangeable. (1964: 173) Merleau-Ponty concludes that, for Descartes, space is a purely "positive being, outside all points of view, beyond all latency and all depth, having no true thickness" (1964: 174). Challenging the Cartesian view, Merleau-Ponty insists that the dialectical features of perceptual experience ("[o]rientation, polarity, [and] envelopment") are not merely secondary to a space that itself is devoid of such features. He begins his own account of spatiality by exploring the paradoxical interplay of the visible and invisible, of identity and difference, that is characteristic of true depth: The enigma consists in the fact that I see things, each one in its place, precisely because they eclipse one another, and that they are rivals before my sight precisely because each one is in its own place. Their exteriority is known in their envelopment and their mutual dependence 14 in their autonomy. Once depth is understood in this way, we can no longer call it a third dimension. In the first place, if it were a dimension, it would be the first one; there are forms and definite planes only if it is stipulated how far from me their different parts are. But a first dimension that contains all the others is no longer a dimension, at least in the ordinary sense of a certain relationship according to which we make measurements. Depth thus understood is, rather, the experience of the reversibility of dimensions, of a global "locality"- everything in the same place at the same time, a locality from which height, width, and depth [the classical dimensions] are abstracted. (1964: 180) Speaking in the same vein, Merleau-Ponty characterizes depth as "a single dimensionality, a polymorphous Being," from which the Cartesian dimensions of linear extension derive, and "which justifies all [Cartesian dimensions] without being fully expressed by any" (1964: 174). The dimension of depth is "both natal space and matrix of every other existing space" (1964: 176). Merleau-Ponty goes on to observe that primal dimensionality must be understood as self-containing. This is illustrated through a discussion of contemporary art, and, in particular, the work of Paul Cézanne: "Cézanne knows already what cubism will repeat: that the external form, the envelope, is secondary and derived, that it is not that which causes a thing to take form, that this shell of space must be shattered, this fruit bowl broken" (1964: 180). In breaking the "shell," one disrupts the classical representation of objects in space. Merleau-Ponty asks: [W]hat is there to paint, then? Cubes, spheres, and cones...? Pure forms which have the solidity of what could be defined by an internal law of construction...? Cézanne made an experiment of this kind in his middle period. He opted for the solid, for space-and came to find that inside this space, a box or container too large for them, the things began to move, color against color; they began to modulate in instability. Thus we must seek space and its content as together. (1964: 180) The work of Cézanne is Merleau-Ponty's primary example of the exploration of depth as originary dimension. The foregoing passage describes Cézanne's discovery that primal dimensionality is not space taken in abstraction from its content, but is the unbroken flow from container to content. It is in this sense of the internal mediation of container and content that Cézanne's depth dimension is self-containing. Merleau-Ponty also makes it clear that the primal dimension engages embodied subjectivity: the dimension of depth "goes toward things from, as starting point, this body to which I myself am fastened" (1964: 173). In commenting that, "there are forms and defi15 nite planes only if it is stipulated how far from me their different parts are" (180; italics mine), Merleau-Ponty is conveying the same idea. A little later, Merleau-Ponty goes further: The painter's vision is not a view upon the outside, a merely "physical-optical" relation with the world. The world no longer stands before him through representation; rather, it is the painter to whom the things of the world give birth by a sort of concentration or coming-toitself of the visible. Ultimately the painting relates to nothing at all among experienced things unless it is first of all "autofigurative."....The spectacle is first of all a spectacle of itself before it is a spectacle of something outside of it. (1964: 181) In this passage, the painting of which Merleau-Ponty speaks, in drawing upon the originary dimension of depth, recursively draws in upon itself. Painting of this kind is not merely a signification of what is other, but a concrete self-signification that undercuts the external boundary between signifier and signified. In sum, the phenomenological dimension of depth as described by Merleau-Ponty is (1) the "first" dimension, inasmuch as it is the source of the Cartesian dimensions, which are idealizations of it; it is (2) a self-containing dimension, not merely a container for contents that are taken as separate from it; and it is (3) a dimension that blends subject and object concretely, rather than serving as a static staging platform for the objectifications of a detached subject. In realizing depth, we surpass the concept of space as but an inert container and come to understand it as an aspect of an indivisible cycle of action in which container, contained, and "uncontained"-space, object, and subject-are integrally incorporated. The work of Merleau-Ponty provides us with an insight into the nature of the "added dimension" that is required for the Kleinian signification of radical recursion. It would not be enough to say that the Klein bottle makes use of the dimension of depth to realize its dialectic of continuity and discontinuity-not if "makes use" connotes the operation of a model employing a containing medium to signify a meaning external to itself. It is perhaps more accurate to say that the Klein bottle is the depth dimension. For, rather than being a model contained as object-in-space, the Klein bottle-grasped in terms of depth-is the inseparability of object, space, and subject, the unbroken circulation of these intimated by Merleau-Ponty. It is the unique hole in the Klein bottle that plays the pivotal role. This loss in continuity is necessary. One certainly could make a hole in the torus, or in any other object in three-dimensional space, but such discontinuities would not be necessary inasmuch as 16 these objects could be fully assembled in space without rupturing them. It is clear that whether an object like the torus is cut open or left intact, the closure of the space containing that object will not be brought into question; in rendering such an object discontinuous, we do not affect the assumption that the space in which it is embedded is a continuum. Indeed, we have seen that the divisibility of an ordinary object derives from the infinite divisibility of the continuum itself. With the Klein bottle it is different. Its discontinuity does challenge the continuity of three-dimensional space as such, for the necessity of the hole in the bottle indicates that space is unable to contain the bottle the way ordinary objects appear containable. We know that for the Kleinian "object" to be brought to completion, assembled without a hole, an "added dimension" is required, and I am proposing that the dimension to be engaged is that of Merleau-Pontean depth (assuming we do not wish merely to skip over the hole by a continuity-maintaining act of abstraction, as in the standard mathematical stratagem for dealing with the Klein bottle). In the depth dimension, while the hole in the Klein bottle is no mere breach in an object in space, neither is it simply a rupture in space per se that corresponds to the subject. Rather, the Kleinian "hole" is in fact a dialectical (w)hole resulting from an act of selfintersection wherein the purported object does the "impossible": it passes unbrokenly through itself, and, in so doing, flows backward into its own subjective ground (in Merleau-Ponty's terms, it is "autofigurative"). Elsewhere, I noted the resemblance of the Klein bottle to the hermetic vessel of old alchemy (Rosen 1995). The design of the enigmatic vessel is essentially that of the uroboros, the serpent that consumes itself by swallowing its own tail. To contain itself, the serpent must intersect itself, an operation requiring a hole (corresponding to the opening that is its mouth). The hole in the Klein bottle is of this sort. It is neither solely a hole in a container, nor a hole in that which it contains, but the hole produced by the recursive act of self-containment that integrates the container with its contents in this way giving (w)holeness. The Kleinian process of self-containment enacted through the dimension of depth is surely no trivial recursion, no regression of meaning to simple self-identity. The Klein bottle refers to itself, but it also makes reference to what is other and a boundary is crossed. Of course, the boundary in question is not of the exterior sort so familiar to us; instead it is paradoxical, a boundary that is not a boundary (see Neuman, this issue of SEED, and Rosen 1997, 2004a). In passing through Kleinian depths from self to other (subject to object, the discontinuous to the continuum), we cross over the boundary to the "far side," yet at once remain on the "near side." In this way, while the self-other distinction is not just abrogated, the supremacy of this distinction is overcome and we realize a harmony of self and other so intimate that the prior meanings of these terms are trans17 muted. The erstwhile categorical purity of self and other is supplanted by an odd uroboric hybrid, a "hermaphroditic" fusion wherein self and other, though assuredly different, are one and the same. This profoundly paradoxical manner of self-reference is what I mean by radical recursion. 5 SEMIOTIC POSTSCRIPT The subtlety of the notion of radical recursion has not been exhausted by what I have written above. I will note another layer of meaning before I conclude. Consider the words "Klein bottle." Although these signifiers point to the depthdimensional structure that embodies the paradox of radical self-signification, the signifiers themselves-"K-l-e-i-n" and "b-o-t-t-l-e"-are but arbitrarily devised, conventionally agreed upon tokens that refer to their content in a merely external manner. These onedimensional typographic marks appearing on the two-dimensional surface of this page obviously fall short of tangibly delivering the three-dimensional Kleinian depth they signify. To the extent that "unmotivated" conventional marks constitute the primary mode of signification for this text, the old division between signifier and signified will be upheld and the meaning of the Klein bottle will remain an abstraction. In seeking to close the gap between signifier and signified, it might be feasible to place greater emphasis on our two-dimensional images of the Klein bottle (Figs. 2 and 3), or, better still, to work with a full three-dimensional model of this paradoxical structure (a model can be constructed with a flexible length of tubing such as that illustrated in Fig. 3). However, it should be clear by now that for the Klein bottle to be fully dimensioned, our model cannot be limited to an object in three-dimensional space. The Klein bottle must be realized in Merleau-Pontian depth. To this end, rather than regarding the Klein bottle as but an object appearing before us, something "out there" in space that we see and can handle, we must resist this compulsion of naïve realism and take the bottle as something that we read. What I am suggesting is that, while the Klein bottle cannot be actualized in the abstract universal medium of words alone, neither can it be brought to fruition as but a particular concrete thing; to be realized in depth, it must be realized as a hybrid of word and thing (a "general thing"; Merleau-Ponty 1968: 139), as a "tangible word" or iconic text. In the capacity of iconic sign, the Klein bottle can serve to "motivate" the process of signification by raising it from one-dimensional arbitrariness to a fully committed three-dimensional intercourse of signifier and signified. 18 In reading our iconic Kleinian text, we must of course read its hole. Instead of interpreting the hole as a gap in an ordinary object contained in space, we are to read it as an opening to a "higher dimension," and read that dimension "autofiguratively," grasping it as the prereflective source of our very own reading. The hole then becomes a (w)hole that we actualize in depth as we pass unbrokenly from text back to subtext in a uroboric act of radical recursion. Here the Klein bottle signifies radical recursion by signifying itself. Or in Peircian terms, we may say with semiotician Paul Ryan that the Klein bottle is a "sign of itself" (1993: 345–347). REFERENCES Barr, Stephen. 1964. Experiments in Topology. New York: Dover. Bohm, David. 1980. Wholeness and the Implicate Order. London: Routledge and Kegan Paul. Čapek, Milič. 1961. Philosophical Impact of Contemporary Physics. New York: Van Nostrand. Deleuze, Gilles, and Felix Guattari. 1987. A Thousand Plateaus: Capitalism and Schizophrenia. Minneapolis: University of Minnesota Press. Derrida, Jacques. 1976. Of Grammatology. Trans. Gayatri C. Spivak. Baltimore: The Johns Hopkins University Press. Gendlin, Eugene T., and Jay L. Lemke. 1983. A Critique of Relativity and Localization. International Journal of Mathematical Modelling, 4: 61–72. Grosz, Elizabeth. 1994. Volatile Bodies. Bloomington, Ind.: Indiana University Press. Harries-Jones, Peter. 1995. A Recursive Vision: Ecological Understanding and Gregory Bateson. Toronto: University of Toronto Press. ---. 2002. Where Bonds Become Binds: The Necessity for Bateson's Interactive Perspective on Biosemiotics. Sign Systems Studies, 30 (1): 163–181. Heidegger, Martin. 1927/1962. Being and Time. Trans. J. Macquarrie and E. Robinson. New York: Harper & Row. ---. 1962/1972. Time and Being, in J. Stambaugh, trans., On Time and Being, pp. 1–24. New York: Harper & Row. 19 Lacan, Jacques. 1966/1970. Of Structure as an Inmixing of an Otherness Prerequisite to Any Subject Whatever, in R. Macksey and E. Donato, eds., The Languages of Criticism and the Sciences of Man: The Structuralist Controversy, pp. 186–200. Baltimore: The Johns Hopkins University Press. Massumi, Brian. 1998. Strange Horizon: Buildings, Biograms, and the Body Topologic. Retrieved 15 February 2003, from the Indiana University Web Site: http://www.indiana.edu/~thinkmat/strange.doc. Merleau-Ponty, Maurice. 1964. Eye and Mind, in J. M. Edie, ed., The Primacy of Perception, pp. 159–190. Evanston, Ill: Northwestern University Press. ---. 1968. The Visible and the Invisible. Evanston, Ill.: Northwestern University Press. Merrell, Floyd. 1998. Simplicity and Complexity. Ann Arbor, Mich.: University of Michigan Press. Muses, Charles. 1968. Hypernumber and Metadimension Theory. Journal of Consciousness Studies, 1: 29–48. Neuman, Yair. 2003. Moebius and Paradox: On the Abstract Structure of Boundary Events in Semiotic Systems. SEED Journal, this issue. Plato. 1965. Timaeus and Critias. Trans. D. Lee. New York: Penguin. Rosen, Steven M. 1994. Science, Paradox, and the Moebius Principle. Albany, N.Y.: State University of New York Press. ---. 1995. Pouring Old Wine into a New Bottle, in M. Stein, ed., The Interactive Field in Analysis, pp. 121–141. Wilmette, Ill.: Chiron. ---. 1997. Wholeness as the Body of Paradox. Journal of Mind and Behavior, 18 (4): 391–424. ---. 2004a. Dimensions of Apeiron: A Topological Phenomenology of Space, Time, and Individuation. Amsterdam and New York: Editions Rodopi, in press. ---. 2004b. Topologies of the Flesh: A Multidimensional Exploration of the Lifeworld. Unpublished manuscript. Rucker, Rudolph. 1977. Geometry, Relativity, and the Fourth Dimension. New York: Dover. Ryan, Paul. 1993. Video Mind/Earth Mind: Art, Communications, and Ecology. New York: Peter Lang. 20 Spivak, Gayatri C. 1976. Translator's Preface, in J. Derrida, Of Grammatology, pp. ixlxxxvii. Baltimore: The Johns Hopkins University Press. Whitehead, Alfred North. 1978. Process and Reality. New York: Free Press.


E. J. Lowe's Metaphysics and Philosophical/Analytic Theology. Special Issue.

Philosophy

The essays in this special issue focus on connecting the relevant aspects of Loweâs metaphysics to issues in philosophical theology. In this regard, the essays focus on Trinity, divine causal agency, atonement, embodied existence, physicalism vs. dualism, natural science, and theological claims.

TheoLogica An International Journal for Philosophy of Religion and Philosophical Theology CALL FOR PAPERS E. J. LOWE'S METAPHYSICS AND ANALYTIC THEOLOGY Guest editors Mihretu P. Guta Biola University, Addis Ababa University & Azusa Pacific University Eric LaRock Oakland University & University of Michigan, Center for Consciousness Science Edward Jonathan Lowe was one of the most distinguished metaphysicians of the last 50 plus years. He made immense contributions to analytic philosophy in as diverse areas as metaphysics, philosophy of mind, philosophy of language, philosophical logic, modern philosophy (especial on John Locke) and philosophy of religion. Lowe was a realist metaphysician. Like Aristotle, he thought that, with sustained reflection and responsible engagement with empirical research, the nature of a mind independent reality can be discovered. In all of this works, Lowe consistently maintained that our common-sense pre-philosophical convictions about reality should not be ignored unless there is a good reason to do so. Even in such cases, Lowe firmly believed that common-sense should rather be corrected and further enriched in light of relevant empirical discoveries. But Lowe never accepted the idea that, in light of the advancement of science, somehow we should entirely stop our reliance on common-sense in our inquiry into the nature of reality. Partly in defence of this very view, Lowe developed his most influential and highly original work: the four-category ontology. The gist of this work concerns metaphysics as an inquiry into the structure of ultimate reality (taken in general), provides a foundation for natural science. Lowe strongly believed that it is metaphysics not science that can set the terms for what is possible and not possible. Lowe believed that figuring out what actually exists in the natural world falls within the purview of science. On Lowe's view, metaphysics and science can and should work in synergy, each playing its distinctive role in enhancing our knowledge of a mind independent reality. Lowe extended his realist view of reality to causation, laws of nature, modality, personal identity, logic, language, God's existence, time and space, human ontology, properties and many other issues. Lowe's views on ontological issues also have direct implications for issues in philosophical theology as well as philosophy of religion such as incarnation, trinity and divine attributes. One of the things that makes Lowe's work uniquely suitable to apply to various issues in either philosophical theology or philosophy of religion has to do with its systematic nature. Lowe built an extremely sophisticated ontological system as shown in his the Four-Category Ontology. In so many ways, Lowe's highly original ontological system will prove relevant to address questions that arise in philosophical theology. Many contemporary metaphysicians influenced by Lowe's system also have an interest both in philosophical theology and philosophy of religion, and have integrated elements of Lowe's metaphysics in their treatment of these questions. Yet, to this date, no attempt has been made to take a general look at how Lowe's metaphysics relates to various issues in the philosophy of religion. This is the first attempt to take concrete steps to fill in the existing gap in this regard. To this effect, we would like to invite paper contributions that connect any relevant aspect of Lowe's work to any issue in philosophical theology or philosophy of religion, especially incarnation, trinity, divine attributes, human agency and divine sovereignty, unified experience and the existence of God, divine causation, divine temporality or atemporality et cetera. Deadline for submissions: June 30th, 2020 Full papers should be submitted via our website: https://ojs.uclouvain.be/index.php/theologica/index or sent to: firstname.lastname@example.com. In order to contribute equally to scientific international discussions held in several languages, articles written in English, French, German, Italian, and Spanish are accepted. Visit the TheoLogica homepage for a description of the journal and instructions to authors. For a brief biography on Lowe's life and work, click on the link below: https://www.jstor.org/stable/43047040?seq=1#metadata_info_tab_contents For an extended discussion on Lowe's work, click the link below: https://www.iep.utm.edu/lowe-ej/ Yours sincerely, Mihretu P. Guta & Eric LaRock