Integral Biomathics: A Post-Newtonian View into the Logos of Bio

Integral Biomathics A Post-Newtonian View into the Logos of Bio Plamen L. Simeonov 28. February 2007 Integral Biomathics: A Post-Newtonian View into the Logos of Bio Plamen L. Simeonov Technische Universität Berlin [email protected] Abstract This work addresses the phenomena of emergence, adaptive dynamics and evolution of selfassembling, self-organizing, self-maintaining and self-replicating biosynthetic systems. We regard this research as an integral part of the emerging discipline of nature-inspired or natural computation i.e. computation inspired by or occurring in nature. Within this context, we are interested in studies which represent a significant departure from traditional theories about complex systems and selforganization, emergent phenomena and artificial biology. In particular, these include non-conventional approaches exploring (i) the aggregation, composition, growth and development of physical forms and structures along with their networks of production (autopoiesis), and (ii) the associated abstract information structures and computational processes. Our ultimate objective is to unify classical mathematical biology with biomathics (or biological mathematics) on the way to genuine biological system engineering. The convergence of these disciplines is going to be carried out both from the perspective of traditional (analytic) life and physical sciences, as well as from the one of engineering (synthetic) sciences. In this regard, our approach differs from most present day efforts of biomimetics in automata and computation design to develop autonomic systems by emulating a limited set of “organic” features using traditional mathematical methods and computational models which are suitable for physical sciences, rather than for life sciences. We call this new field integral biomathics. This paper presents a survey of approaches related to the above domain and defines a generalized epistemological model with the objective to set out an ecology for symbiotic research in life, physical and engineering sciences. ____________ Keywords: systems biology; synthetic biology; relational biology; autopoiesis; theoretical physics; evolving formal models; naturalistic computation; non-Turing and post-Newtonian computation; self* biosynthetic systems; artificial life. Integral Biomatics 28.02.2007 Integral Biomathics: A Post-Newtonian View into the Logos of Bio Plamen L. Simeonov Technische Universität Berlin [email protected] ”I’m not happy with all the because Nature isn’t classical… Can you do it with It is not a Turing analyses that go with just classical theory, How can we simulate the quantum mechanics?.. a new kind of computer - a quantum computer? machine, but a machine of a different kind.” Richard Feynman, Simulating physics with computers, 1981. 1. Introduction This work addresses the phenomena of emergence, adaptive dynamics and evolution of selfassembling, self-organizing, self-maintaining and self-replicating biosynthetic systems. We regard this research as an integral part of the emerging discipline of nature-inspired or natural computation i.e. computation inspired by or occurring in nature (Ballard, 1997; Shadboldt, 2004; MacLennan, 2005; Zomaya, 2006). Within this context, we are interested in studies which represent a significant departure from traditional theories about complex systems and self-organization, emergent phenomena and artificial biology. In particular, these include non-conventional approaches exploring (i) the aggregation, composition, growth and development of physical forms and structures along with their networks of production (autopoiesis), and (ii) the associated abstract information structures and computational processes. This study was motivated by previous research in system design and network engineering (Simeonov, 1998; Simeonov, 1999a/b/c; Simeonov, 2002a/b/c), where the limits of contemporary information technology and multimedia communication systems were identified and a novel approach towards autopoietic networking was proposed. The present work continues the above line of research towards deeper understanding of biological phenomena such as emergence and organisation in a holistic manner as seen in relational biology (Rashevsky, 1954 ff.; Rosen, 1958a/b ff.). Hence, our ultimate objective is to unify classical mathematical biology with biomathics 1 on the way to genuine biological system engineering. Therefore, this study is going to be carried out both from the perspective of traditional (analytic) life and physical sciences, as well as from the one of engineering (synthetic) sciences. In this regard, our approach differs from most present day efforts of biomimetics 2 in automata and computation design to develop autonomic systems by emulating a limited set of “organic” features using traditional mathematical methods and computational models which are suitable for physical sciences, rather than for life sciences. In addition, it is essential to note that also classical information theory (Shannon, 1948) should be developed along the same line of research in order to obtain an authentic picture of natural biological systems that will enable the creation of artificial ones. This viewpoint has certainly become an important issue in the design of complex networked systems deploying large numbers of distributed components with dynamic exchange of information in the presence of noise and under power and bandwidth constraints in the areas of telecommunications, transport control and industrial automation. 1 Biomahtics or biological mathematics is defined as the study of mathematics as it occurs in biological systems. In contrast, mathematical biology is concerned with the use of mathematics to describe or model biological systems, (Rashevsky, 1940). 2 Biomimetics is generally defined as the “concept of taking ideas from nature and implementing them in another technology such as engineering, design, computing, etc.”, cf. http://www.bath.ac.uk/mech-eng/biomimetics/about.htm. –1 – Integral Biomatics 28.02.2007 To address these critical issues, researchers pursue the amelioration and unification of classical theories such as those of control, thermodynamics and information. For instance, Allwein and colleagues propose the integration of Shanon’s general quantitative theory of communication flow with the Barwise-Seligman general qualitative theory of information flow to obtain a more powerful theoretical framework for qualitative and quantitative analysis of distributed information systems, (Allwein et al., 2004). Other authors are concerned with important theoretical issues such as the estimation of reliable noisy digital channel state (Matveev & Savkin, 2004) and the treatment of data density equilibrium analogous to thermodynamic equilibrium, (Kafri, 2006, 2007). Some works also introduce the physics of information in the context of biology and genomics, (Adami, 2004). However, what is important for the design of naturalistic systems is the perception of signalling and information content including their processing and distribution from the perspective of biological systems (Miller, 1978) and in correlation with autonomous regulation of power consumption and other life maintaining mechanisms. This topic has not been addressed sufficiently by present research in both natural and artificial systems. Therefore, it should become an integral part of the models and methods of our approach to naturalistic computation. This presentation is organized as follows. Our research develops along two planes 3 or conceptions of discourse, the physical or the ‘realization’ plane and the logical or ‘abstract’ one. The following two sections review previous research in naturalistic computation along these planes. Section four is devoted to non-classical computation models beyond the Turing machine model. Next, section five introduces the kernel part of this article, the integral approach to biological computation. Section six discusses the implications of the new field. Finally, section seven presents the conclusions with an outlook for research in integral biomathics. 2. The Physical Plane The physical plane lies within the domains of autonomous cellular automata (CA) and evolving complex systems such as autopoietic, autocatalytic and non-linear eco-networks. This area comprises classical systems and ‘discrete’ automata theories endorsed by artificial intelligence (AI) and artificial life (ALife) approaches such as evolutionary computation (Fogel et al., 1966), synthetic neural networks (Dyer, 1995) or adaptive autonomous agents (Maes, 1995) for transducing knowledge from biology and related life science disciplines into computer science and engineering. Such systems and automata are often referred to as bio-inspired or organic. The conceptual and theoretical foundations of these fields have been elaborated in previous works on self-replicating systems such as Boolean models of neural networks (Pitts, 1943; McCullough & Pitts, 1943), Moore’s artificial living plants (1956a), sequential machines (1956b) and other machine models (1962), von Neumann’s kynematic model and Universal Constructor (1966), Conway’s game of ‘Life’ (Gardner, 1970-71), Arbib’s self-replicating spacecraft (1974), Dyson’s self-replicating systems (1970, 1979) and Drexler’s bio-nanomolecular engines (1986, 1992). Some of these architectures such as the Codd’s and Morita’s simplified cellular automations have been realized in practice, (Codd, 1968; Winograd, 1970; Morita & Imai, 1995). However, discrete mathematics has its application limits in modelling and emulating biological systems. Richard Feynman came to a similar conclusion that classical computers are inappropriate for simulating quantum systems (1982, 1985). Apart from maintaining living functions, biological systems demonstrate complex computational mechanisms. Only those artificial or hybrid biosynthetic systems exhibiting a behaviour which is characteristic to the one of natural organic systems could truly reflect the essence of biological computing. 3 We refer here to the first two spheres of Penrose's “Three Worlds Model“ (Penrose 1995, p. 414; Penrose, 2004, p. 18), the physical world of phenomena and the Platonic mathematical world. –2 – Integral Biomatics 28.02.2007 Therefore, we argue that this area of research needs to be placed on broader foundations which more adequately reflect the emergence and organisation of artefacts and processes in nature than modern discrete automata and computation approaches. In this work, we are willing to expand and enforce frontier research on the emergence and organisation of living forms (Thompson, 1917; Bertalanffy, 1928; Franck, 1949; Rashevsky, 1954; Miller, 1978; Thom, 1989; Rosen, 1991) and their networks of production within a broader discourse and beyond the classical state automata theory and traditional models in life and physical sciences. There has been a number of artificial life techniques using computational models and algorithms adopted from life science disciplines such as genetics, immunology and neurology, as well as evolutionary and molecular biology (Schuster 1995; Ray 1995). Among the most prominent examples for artificial life systems are Tierra (Ray 91) and Avida (Adami 98). Other well known references are the SCL model (McMullin 1997a), the self-assembling cells (Ono & Ikegami 1999, 2000) and the selfassembling lipid bilayers (Rasmussen et al., 2001). Recently, artificial chemistry approaches have been obtaining much attention (di Feroni & Dittrich, 2002; Hutton, 2002; Matsumaru et al., 2006). In particular, experimentation on design and implementation of living systems from scratch is becoming now an intensive research area (Bedau 2005; Bedau 2006). A good intermediate report of the field is given in (McMullin, 2004). As yet, the disciplines of artificial life and chemistry still hide many open issues (Bedau et al., 2000; McMullin, 2000a) including such as the controversial matter about the possibility to engineer molecular and self-replicating assembler (chemistry with/-out mechanics or the Smalley vs. Drexler debate) in nanotechnology (Baum, 2003; Freitas & Merkle, 2004). One of them refers to a basic question and probably the most known effort to explain life in general, the autopoietic theory (Varela et al., 1974; Maturana & Varela, 1980) which has influenced various sciences including biology and sociology for the past 30 years (Kneer & Nassehi, 1993). Autopoiesis provides a good model for understanding the organization and evolution of natural living systems. The question in artificial life research is, however, whether this model can serve as a base for creating synthetic organisms that mimic real ones. To this moment, autopoiesis was used as the base for a number of computational models, simulations, engineering and architecture solutions (Zeleny & Pierre, 1975; Zeleny, 1978; McMullin & Varela 1997; Cardon & Lesage 1998; Ruiz-del-Solar & Köppen, 1999; McMullin & Groß 2001; Simeonov, 2002; Kawamoto, 2003; Keane, 2005). However, autopoietic theory has obviously failed to provide a consistent view on the spontaneous organization of living systems in a quite early stage of its development (Zeleny, 1980). Since then, it has been subject of various critics and controversial discussions. Thus, a key point in understanding the mechanisms of self-assembly and self-organisation in living systems is the notion of organisational closure (McMullin 2000b). Acording to Maturana and Varela, an autopoietic system is not only one that is (a) clearly separated from its environment by a boundary, but also one that has (b) an internal organisation capable of dynamically sustaining itself (including its boundary). It is yet not clear how stringent this definition should be taken. Today, autopoiesis appears to be still the kind of “theory-at-work” which is very general and undifferentiated both in terms of mathematical formalization and technical implementation. The following two sections illustrate this evidence. 2.1 The Formalization Gap In his book “Life itself” (Rosen, 1991), Robert Rosen presents a category theoretical framework for formalization of living systems he studied over three decades, (Rosen, 1958 ff.). Rosen places what we call the Fundamental Question of Artificial Life. According to his conclusion, living systems, which are essentially metabolism/repair (M, R) systems, are not realizable in computational universes. –3 – Integral Biomatics 28.02.2007 If Rosen was right, his conclusion could mean that Artificial Life cannot exist at all, or at least in computational spaces as we know them now. It could be the case that the entire ALife research is going in the wrong direction. Letelier et al. (2004) analysed (M, R) systems from the viewpoint of autopoietic theory. They provided an algebraic example on defining metabolic closure while suggesting a relationship to autopoiesis. In a series of works (1997, 2002, 2006), reflecting the contributions of Rosen (1972, 1991), Luhman (Kneer & Naseli, 1993) and Kawamoto (1995, 2000), Nomura reviewed the formal roots of autopoiesis in the light of category theory (Mitchell, 1965) and proposed a mathematical model of autopoiesis based on Rosen's definition (1997). After having examined some of the central postulates of the autopoietic theory, Nomura comes to the conclusion that previous research has not delivered an unambiguous description of the phenomenon (♣ 4 ) and proposes a more general and strict formal definition. Another category theoretical argument and revision of Rosen’s theorem was provided by Chu and Ho (Chu & Ho, 2006). The authors review the essence of Rosen's ideas leading up to his rejection of the possibility to simulate real artificial life in computing systems. They argue that the class of machines Rosen distinguished from closed systems is not equivalent with realistic formalization of ALife and conclude that Rosen's central proof, stating that living systems are not mechanisms, is wrong. As a result, Rosen's claim remains an open issue. The conclusion that some of Rosen's central notions were probably ill defined provides some interesting theoretical concerns which deserve further investigation. Yet, Rosen himself warned in his book: “there is no property of an organism that cannot at the same time be manifested by inanimate systems” (Life itself, page 19). Thus, having him apparently failing in his own proof does not change the matter at all. From the viewpoint of contemporary logic, we cannot take for granted that organisms are mechanisms and that they can be constructed in the way we used to build machines. In a recent paper (2007), Nomura analyses the possibilities of algebraic description of living systems and clarifies the differences between the aspects of closedness required in (M, R) systems and autopoiesis. He discovers two essential differences between autopoietic and (M, R) systems. The first one is the difference on forms of their closedness under entailment of the components and categories required for the description of closedness. The second one is the distinction between organization and structure. Nomura points out that the first difference depends on the assumption that completely closed systems, modelled as an isomorphism from the space of operands to the space of operators, are necessary conditions of autopoiesis. However, this requirement has not been yet proved in a mathematically strict way. Furthermore, the definition of autopoiesis itself deserves a special attention. There were differences in the interpretation of autopoiesis between Maturana and Varela. When Varela collaborated with McMullin on computational models of autopoiesis, the original algorithm was revised within the same year (McMullin, 1997b; McMullin & Varela, 1997). We assume that Varela may not have considered the implications of autopoietic theory on the axiomatization of discrete mathematics for modelling biological processes. Summarizing all these facts, we made the following two conclusions: (i) we need a more precise and formal definition of autopoiesis and its relationship to (M, R) systems, as outlined by (Nomura, 2007); (ii) we need novel mathematical techniques and tools which adequately describe and simulate biological processes, (Rashevsky, 1954, 1960, 1961). In other words, in order to make progress in this area, we need to either provide further means for formalization, or invent new formal approaches that suite best the original definition, or redefine (refine, extrapolate) the theory itself. Rosen tried to distinguish life systems from machines with his original definition. Although his proof was incomplete, Chu and Ho identified the importance of Rosen's idea itself. Nomura also agrees with them in this point (2007). Therefore, in order to provide the engineering base for artificial organisms and systems in the physical space, we need to further investigate the necessary conditions for modelling characteristics of living systems and provide more stringent definitions of life systems and machines based on Rosen's attempt. 4 With the symbol ♣ we denote the anchor arguments of our conclusion about integral biomathics henceforth. –4 – Integral Biomatics 28.02.2007 2.2 The Implementation Gap In the C5 database product presentation, Stalbaum makes the point that we should differentiate between the implementation of computational autopoiesis as proof of concept (‘computational autopoiesis is possible') and the potential practical applications of such a system. He reckons that Varela's early work makes a strong case for the former (McMullin & Varela, 1997), but that it were a substantially different problem to design autopoietic automata implementing computing applications such as database (self-)management systems. Stalbaum states that the challenge in finding or engineering congruency between autopoietic systems and problems that yield solutions were enormous. Indeed, the demonstration that autopoiesis can be used for computational and communication processes using a minimal implementation, e. g. of an artificial chemistry model, is a relatively simple However, the truthfulness of computational autopoiesis does not necessarily imply that autopoiesis can be effectively implemented to perform work. This is because the internal purpose of an autopoietic system is restricted to the ongoing maintenance of its own organization. Yet, this goal becomes a problem for anyone with the intention to use autopoiesis for the purpose of computation as we know it, i.e. to deliver an output result from a given number of inputs within a limited number of steps. In fact, computation might be a by-product of ongoing structural coupling (a posteriori) between a collection of autopoietic elements such as neurons and their environment, but it cannot be deterministically defined as a purposeful task for the solution of a specific problem or class of problems in the way we are used to expect from today's computational and engineered systems. In other words, we can not count on the natural drift inherent in living systems to directly solve problems that do not primarily serve for conservation, adaptation and maintenance of those systems. In addition, the multiple orders of structural coupling in autopoiesis define even a more complex picture of the interacting units. In this respect, autopoietic computing is analogous 5 to both associative computing (Wichert, 2000) and quantum computing (Feynmann, 1982-1985). It demonstrates an overlayered potential multiplicity of results which becomes apparent at the very moment of system interrogation. The following example illustrates the problem of implementing artificial biology using conventional computing techniques more vividly. A bottom-up synthesis approach, Substrate Catalyst Link (SCL), build on the concept of evolving autopoietic artificial agents (McMullin & Gross, 2001) was integrated within a top-down analytical system, Cell Assembly Kit (CellAK), for developing models and simulations of cells and other biological structures (Webb & White, 2004). The original top-down design of the CellAK system was based on the object-oriented (OO) paradigm with the Unified Modelling Language (UML) and the Real-Time Object-Oriented Methodology (ROOM) formalisms, with models consisting of a hierarchy of containers (ex: cytosol), active objects with behaviour (ex: enzymes, lipid bilayers, transport proteins), and passive small molecules (ex: glucose, pyruvate). Thus, the enhanced CellAK architecture comprised a network of active objects (polymers), the behaviour of which causally depended partly on their own fine-grained structure (monomers), where this structure was constantly changing through interaction with other active objects. In this way, active objects influence other active objects by having an effect on their constituent monomers. The enhanced tool was validated quantitatively vs. GEPASI (Mendes, 1993) and demonstrated its capability for simulating bottom-up synthesis using the cell bilayer active object. The authors claim that this result clearly confirms the value of agent-based modelling techniques reported in (Kahn et al., 2003). However, there is a major difficulty in implementing this method for more complex organic structures than lipid bilayers such as enzymes and proteins (Rosen, 1978). This is because amino acids that compose proteins are coded in the DNA; their order to form a folding 3D shape is of crucial importance. Therefore, the behaviour of a protein is an extremely complex function of its fine-grained structure. This turns quite easily the design and validation of artificial biological structures, such as medicaments, using conventional computing techniques into a problem of polynomial complexity. Causal relationships may not provide the unique base for investigating biological processes. We do not believe that significant progress can be made in this area without a paradigm change. 5 meaning proportional (Latin: rational), or following the principle of mediation in the Pythagorean sense, (Guthrie, 1987). –5 – Integral Biomatics 28.02.2007 Taking into account the above arguments about formalization and implementation, we expose the need for a new broader and unifying automata theory for studying ‘natural’ and artificial living systems, as well as the combination of both, the cybernetic organism (cyborg) or the hybrid between an animal/plant and a machine. Furthermore, we need a new kind of unifying and real Artificial Intelligence, the expected ‘quantum leap for AI’ (Hirsch, 1999), that goes beyond the pioneer days heuristics and hypothesis-driven modelling by placing itself much closer to the essence of living systems and closer to the nature of the underlying processes in both organic and inorganic system complexes. 3. The Logical Plane The second plane for research we are interested in is the abstract or logical plane in the Pythagorean sense of the word 6 , i.e. one that implies analogy or proportion, relation that is generalized as a law, habit, principle or basic pattern of system organization. The essential distinction of this conception from Hilbert’s pure syntactic definition is the inclusion of semantics or context-related information which allows the multiplicity of interpretations, a characteristics typical for biological, social and ecosystems. The logical plane is concerned with the development of new integral computational paradigms that reflect the emergence and organization of information for living systems in a more adequate way than traditional formal approaches based on binary logic and the Church-Turing thesis, (Church, 1932-41; Turing, 1936-51). The latter represents an idealized, but not necessarily unique, model for computation, where representations are formal, finite and definite (MacLennan, 2004). Exposing these characteristics itself, the Church-Turing theory of discrete states excludes per definition alternative approaches to computation. Even worse, as it has been successful for the past 70 years, the Church-Turing model evokes the conviction in the majority of our contemporaries that it is universal, as demonstrated in (Shannon, 1956), and that there is no other option for computation at all. Thus, binary logic remains a good working, and of course, economic model for our everyday discourse, just as the Newtonian celestial mechanics was useful for global navigation until the arrival of Einstein’s theory of relativity which made extraterrestrial journeys possible. Therefore, we expose the need for new theories of computation in order to understand tough issues in science such as the emergence and evolution of brain and thought. In his books “The emperor's new mind” (Penrose, 1989) and “Shadows of the mind” (Penrose, 1994), Roger Penrose’s main theme was the understanding of mind from the perspective of contemporary physics and biology. While discussing the non-computational physics of mind, he placed indirectly what we call the Fundamental Question of Strong Artificial Intelligence 7 . Penrose claimed to prove that Gödel's incompleteness theorem 8 implies that human thought cannot be mechanized. In fact, Penrose does not actually use Gödel's theorem, but rather an easier result inspired by Gödel, namely, Turing's theorem that the halting problem is unsolvable. Penrose's key idea was essentially the same as that of the philosopher J. R. Lucas (Lucas, 1961) who argued as follows. Gödel's incompleteness theorem shows that, given any formal system, there is a true sentence which the formal system cannot prove to be true. But since the truth of this unprovable sentence is proved as part of the incompleteness theorem, humans can prove the sentence in question. Hence, human abilities cannot be captured by formal systems made by humans. This corresponds to the eye’s blind spot paradox 9 which in our view should be regarded rather as a principle in biological systems. 6 Guthrie, 1987 “...according to strong AI, the computer is not merely a tool in the study of the mind; rather, the appropriately programmed computer really is a mind", (Searle, 1980). 8 stating that Number Theory is more complex than any of its formalizations (Gödel, 1931-34) 9 The blind spot in an eye’s vision field is caused by the lack of light-detecting photoreceptor cells on the optic disc of the retina where the optic nerve passes through it. Usually, this ‘defect’ is not perceived, since the brain compensates the missing details of the image with information from the other eye. 7 –6 – Integral Biomatics 28.02.2007 The question here is again, as in Rosen’s thesis about Artificial Life (Rosen, 1991), s. Section 2.1, whether this proof is correct or not. If Penrose and Lucas are right, their conclusions might leads to the interesting result that Strong Artificial Intelligence cannot be realized at all or at least in computational spaces as we know them now. Lucas’s standpoint was already criticized by Benacerraf (1967). Some of Penrose's technical errors were pointed out in (Boolos et al., 1990) and (Putnam, 1995). LaForte and colleagues review Penrose’s arguments and demonstrate, following Benaceraf’s line of thought, that they depend crucially on ambiguities between precise and imprecise definitions of key terms (♣♣), (LaForte et al., 1998). The authors show that these ambiguities cause the Gödel/Turing diagonalization argument to lead from apparently intuitive claims about human abilities to paradoxical or highly idiosyncratic conclusions, and conclude that any similar argument will also fail in the same ways. This is a similar situation as with the contra-proof of Rosen’s thesis discussed above, (Nomura, 2006; Chu et al., 2006). In fact, the arguments of Lucas and Penrose have the same foundation as Rosen’s discussion about the entailment of formal systems (Rosen, 1991, pp. 47-49), stating that “from the standpoint of the formalism [assuming here Gödel's thesis], anything, that happens outside is accordingly unentailed.“ However, Aristotle’s fourth category about the Final Cause 10 , which appears to violate the unidirectional (temporal) flow from axioms to theorems, places (some) human abilities inside the formal system. Rosen finds a solution to this contradiction by postulating that final causation requires modes of entailment that are not generally present in formalisms. Following Rashevsky’s concept of relational biology (Rashevsky, 1960), he also suggests the possibility to separate finality from teleology by retaining the former while e. g. discarding the latter. A similar approach could be taken here to investigate the Lucas/Penrose arguments more precisely. Another solution may be placed entirely or partly on quantum mechanical foundations. The final word about strong artificial intelligence has not been said yet. However, whatever result the polemic around computation and human intelligence may deliver, as in the case with artificial life, we cannot take for granted that human thought is mechanistic, i.e. formal, finite and definite, and that it can be constructed in the way we used to build computers so far. The following example illustrates the above viewpoint. Although being purely subjective, it does not require a formal proof to be apprehended by the reader. It is the question of human intuition as a creative act, a process which cannot be expressed by formal means. In our case, a two-dimensional concentric multi-ring topology of a logical peer-to-peer overlay communications network with the best robust resource discovery and routing algorithm known yet (Wepiwé & Simeonov, 2006) came into being as an analogy, i.e. involving semantics, of a natural system model, the electromagnetic field gradient structure of a metal sphere. It simply popped up intuitively during a discussion with the persuasion that this is the optimal system architecture for the domain in question (Wepiwé & Simeonov, 2005). This happened a priory to proving that very fact. There are numerous examples in the scientific literature that confirm such common experiences and the Gödelian implicit truthwithout-proof, so that the process of its discovery, or disentanglement of semantics into the axiomatic frame of a specific formalism, can be taken as objective reality and principle in living systems (Penrose, 1994, 2004). Hence, there is no need of hypothetical theories about time travels and precognition; we simply face a different kind of awareness and computation from those that contemporary technology delivers to us. Ultimately, from the standpoint of Aristotelian epistemology, it is not the question what and how was something proved, but rather why did this happen. A reasonable answer may contain the assumption that the solution of a problem is delivered by the human system itself, as a useful answer to the stimulus of a purposeful (re-)search through autopoietic response (natural drift) which belongs to the nature of the phenomenon itself. 10 a proposition that requires something of its own effect. –7 – Integral Biomatics 28.02.2007 Therefore, we claim that computation which occurs in nature always involves semantics and cannot be expressed within formalisms in purely syntactic terms. Whereas this argument is relatively weak in physical systems, an adequate picture of biological phenomena requires semantics. In our view, new opportunistic theories of computation should basically regard computation as major property of living matter and be able to develop their own principles within their specific domains of application. In this respect, our approach to the logical plane is conform with the one of MacLennan in his definition of computation as a physical process for the abstract manipulation of abstract objects, the nature of which could be discrete, continuous or hybrid (MacLennan, 2004, 2006). Besides, the introduction of broader and integral concepts for computing is supported by such arguments as the fact that a computer’s functioning is based on the state superposition principle can be realised both with classical and quantum elements (Oraevsky, 2000). Basically, MacLennan’s investigation into these alternative models of information processing led to similar conclusion for computation as the ones of Feynman for quantum systems (1982, 1985) and Rosen for artificial biology (1991, 1999). He argues that conventional digital computers are inadequate for realizing the full potential of natural computation, and therefore that alternative and more brain-like technologies should be developed. In his view, which we share, the Turing Machine model is unsuited to address the class of questions about natural computing (MacLennan 2003a). Furthermore, MacLennan argues that natural computation which occurs in natural neural networks requires continuous and non-Turing models of computation (MacLennan 2003b), although these processes cannot be regarded as ’computation’ in terms of the Turing machine definition (Turing, 1937). Yet, Turing’s thesis is not the ultimate verdict about computing. We only need to refer to Euclid’s geometry in a historical context. There are numerous examples about how new generalizations in mathematics and physics emerged out of apparent axioms and postulates. When shifting the perspective, the latter turned out to be conceptual deadlocks as they were not able to deal with new ideas or factual observations. When the new theories were finally reconciled with the established world order, the apparent ‘paradoxes’ turned out to be new, more general facts and the old frame of thought became a special case within the new one (Rosen, 1991). It is a well known fact 11 that science requires sometimes a few iterations of denial and rediscovery over the centuries until a new idea or paradigm is accepted by the dominating majority (Sacks, 1995). We have a similar situation with the dominant computing paradigm today, the Turing Machine (TM) model. This concept is leading now to a major crisis 12 in computer science, analogous to the one of irrational numbers in mathematics a few centuries ago. A major paradox with adopting the TM model for natural computing was pointed out in (MacLennan, 2004). Accordingly, formal logic considers a function computable if for any input data the corresponding output would be produced after finitely many steps, i. e. a proof can be of any finite length. For the sake of completeness and consistency, the TM model imposes no bounds on the length of the individual steps and on the size of formulas, so long as they are finite. The concept of time assumed in the TM model is not discrete time in the familiar sense that each time interval has the same duration. Therefore, it is more accurate to call TM time a sequential time (MacLennan, 2006). Since the individual steps have no definite duration, there is no use to count the number of steps to translate that count into real time. Consequently, the only reasonable way to compare the time required by computational processes is in terms of their asymptotic behaviour. Therefore, once we have chosen to ignore the speed of the individual steps, all we have as complexity measure is the size of the formulas produced during the computation or the growth degree of the number of steps with the size of the input. 11 Max Planck: "A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it." 12 Indeed there is nothing wrong with the TM model, as long as it is exploited within its frame of relevance which deals with questions about formal derivability and the limits of effective calculability.(MacLennan, 2006). –8 – Integral Biomatics 28.02.2007 In fact, Turingian sequential time is reasonable in a model of formal derivability or effective calculability, since the time duration required for individual operations was irrelevant to the research questions of formal mathematics. However, this perspective leads to the result that any polynomialtime algorithm is “tractable“ and “fast” (e.g. matrix multiplication ~ O(N3)) and that exponential-time algorithms are “intractable“ (e.g. Traveling Salesman), whereas problems which are polynomial-time reducible are virtually identical. This step duration independent situation is indeed peculiar for natural systems. MacLennon ironically describes it as one where “an algorithm that takes N100 years is fast, but one that takes 2N nanoseconds is intractable.” We cannot really afford to ignore the duration of the steps in natural computing, where instant response has the value of survival for a living organism. „In nature, asymptotic complexity is generally irrelevant... Whether the algorithm is linear, quadratic, or exponential is not so important as whether it can deliver useful results in required real-time bounds for the inputs that actually occur. The same applies to other computational resources. … it is not so important whether the number of neurons required varies linearly or with the square of the number of inputs to the net; what matters is the absolute number of neurons required for the numbers of inputs there actually are, and how well the system will perform with the number of inputs and neurons it actually has.“ (MacLennan, 2006). In addition, adaptation and reaction to unpredicted stimuli, large scale cluster optimizations and continuity of input and output values in space and time, such as those occurring in neural networks, immune systems and insect swarms are other characteristics of the natural system that lie outside the scope of the TM model and cannot be addressed appropriately with traditional analytical approaches. Finally, evolvability and robustness, (Wagner, 2005), i.e. the continuous development and the effective (not necessarily the correct!) operation in the presence of noise, uncertainty, imprecision, error and damage complete the fragmentary set of characteristics of living systems, (Rashevsky, 1954, 1960, 1965; Miller, 1978). The following section discusses some alternative approaches to the TM model in more detail. 4. About Non-classical Computation Models The term non-classical computation denotes computation beyond and outside the classical Turing machine model such as extra-Turing, non-Turing and post-Newtonian computation (Stannet, 1991). Representative approaches include Super-Turing and hypercomputation, as well as nano-, quantum, analog and field computation. Super-Turing computation (Siegelmann, 1995, 1996a, 2003) is a synonym for any computation that cannot be carried out by a Turing Machine, as well as any (algorithmic) computation carried out by a Turing Machine. Furthermore, Super-Turing computers are any computational devices capable to perform Super-Turing computation, e. g. non-Turing computable operations such as integrations on real-valued functions that provide exact rather than approximate results. In fact, Turing himself proposed a larger class of “non-Turing” computing architectures including oracle machines (o-machines), choice machines (c-machines), and unorganized machines (umachines). He did not anticipate that his original I/O model will dominate computer science over five decades. Other approaches include pi-calculus (Milner, 1991 & 2004), $-calculus (Eberbach, 20002001), Evolutionary Turing Machines 13 (Eberbach & Wegner, 2003; Eberbach, 2005) and (recurrent) neural networks 14 (Garzon & Franklin, 1989; Siegelmann & Sontag, 1992; Siegelmann, 1993). 13 14 more complete model for evolutionary computing than common Turing Machines, recursive algorithms or Markov chains some of them with real numbers as weights –9 – Integral Biomatics 28.02.2007 Furthermore, there are Interaction Machines 15 (Wegner, 1997-98), Persistent Turing Machines 16 (Goldin & Wegner, 1999; Goldin, 2000), Site and Internet Machines (van Leeuwen & Wiedermann, 2000a/b) and finally, self-replicating cellular automata (von Neumann, 1966). The latter we regard as a special class generic self-realizable computing architecture that we assigned to the physical plane for the purpose of our dyadic model of an evolving intelligent “computational organism” discussed in the next section. In fact, CA and some other Super-Turing architectures belong to both planes since they can be both an abstract theoretical concept and its physical realization at the same time. Hypercomputation (Copeland, 2004) studies models of computation that expand the concept of computation beyond the Church-Turing thesis and perform better than the Turing machine. It refers to various proposed methods for the computation of non-Turing computable functions such as the general halting problem. A good survey report in this area is given in (Ord, 2002). Here the author introduces ten different types of hypermachines 17 and compares their capabilities while explaining how such non-classical models fit into the classical theory of computation. His central argument is that the Church-Turing thesis is commonly misunderstood. Ord claims that the negative results of Gödel and Turing depend mainly on the nature of physics 18 (♣♣♣). Other authors also approve this view (Kieu, 2002). We also share Ord's thesis that the Turing machine model is based on concepts conform to the Newtonian physics and on inadequate (from biological viewpoint) mathematical abstractions such as negligible power consumption and unlimited memory. On the other hand, quantum computation has already demonstrated 19 that the feasibility of algorithms depends on the nature of the physical laws themselves. Thus, extrapolating new physical laws would automatically mean new computation approaches. This holds also in the case of biology. In fact we face an epistemological problem. The more we know about the natural phenomena, the more we expand our models. Therefore, to deal with the rising complexity of abstractions, we need an evolving networked model for living systems (Capra, 1997) that addresses a new, inclusive theory of computation and automation based on the principles about emergence and self-organisation from general system theory (Bertalanffy, 1950 ff.), living systems theory (Miller, 1978) and systems biology (Kitano, 2002). Mathematical models for hypercomputers include: • • • • a Turing machine that can complete infinitely many steps, (Shagrir & Pitowsky, 2003); an idealized analogous computer (MacLennan, 1990; Siegelmann, 1996b), a real (numbers) computer that could perform hypercomputation if physics allows in some way the computation with general real variables, i.e. not only computable real numbers; a relativistic digital computer working in a Malament-Hogarth space-time which can perform an infinite number of operations while remaining in the past light cone of a particular spacetime, (Etesi &. Németi, 2002); a quantum mechanical system, but not an ordinary qubit quantum computer, which uses (e. g.) an infinite superposition of states to compute non-computable functions, (Feynmann, 1982, 1985). Currently all these devices are only theoretical concepts, but they may move some day to the physical (realization) plane and become our everyday reality in the same way as von Neumann’s cellular automata did. 15 while Turing machines cannot accept external input during computation, interaction machines extend the TM model by input and output actions that support dynamic interaction with an external environment. 16 multitape machines with a persistent worktape preserved between successive interactions; they represent minimal extensions of TM that express interactive behaviour characterized by input-output streams. 17 infinite state TM, probabilistic TM, error prone TM, accelerated TM, infinite time TM, fair non-deterministic TM, coupled TM, TM with initial inscriptions, asynchronous networks of TM, O-machines (also classified as Super-Turing computation). 18 This result is comparable with the ones in the discussions on the mechanization of life (♣) and thought (♣♣). 19 e.g. through such effects as quantum superposition, quantum entanglement or wave function collapse. – 10 – Integral Biomatics 28.02.2007 Nanocomputation (MacLennan, 2006) involves computational processes with nano-devices and information structures which are not fixed, but in constant flux and temporary equilibria. It includes sub-atomic and molecular modes of computation such as quantum computation (Nielsen & Chuang, 2000) and DNA computation (Amos, 2005). A fundamental characteristic of nanocomputation is the microscopic reversibility in the device and information structures. This means that chemical reactions always have a non-zero probability of backwards flow path. Therefore, molecular computation systems must be designed so that they accomplish their purposes in spite of such reversals. Furthermore, computation proceeds asynchronously in continuous-time parallelism and superposition. Also, operations cannot be assumed to proceed correctly and the probability of error is always non-negligible. Therefore, errors should be built into nanocomputational models from the very beginning. Due to thermal noise and quantum effects errors, defects and instability are unavoidable and must be taken as given. Examples of nanocomputing devices include quantum logic gates and DNA chips. Analog / continuous computation uses physical phenomena (mechanical, electrical, etc.) to model the problem being solved, by using uninterrupted varying values of one kind of physical parameter (e.g. water/air pressure, electrical voltage, magnetic field intensity, etc.) to obtain, measure and represent other as a goal function. A major characteristic is the operation on signals without conversion (sampling and integration) and in their natural continuous state. Analog computers has been used since ancient times 20 in agriculture, construction and navigation (Bromley, 1990). When in 1941 Shannon proposed the first General Purpose Analog Computer (Shannon, 1941) as a mathematical model of an analog device, the Differential Analyser (Bush, 1931), this invention announced the age of electronic analog computers (Briant et al., 1960). From the very beginning, they stepped in competition with their digital brothers and initially outperformed them by optimally deploying electronic components (capacitors, inductors, potentiometers, and operation amplifiers). Analog computers have three major advantages over digital ones: i) instantaneous response, ii) inherent parallelism, and iii) time-continuity (no numerical instabilities or time steps). They are well suited to simulating highly complex and dynamic systems in real time and accelerated rates such as aircraft operation, industrial chemical processes and nuclear power plants. Until 1975 analog computers were considered to be unbeatable in solving scientific and engineering problems defined by systems of ordinary differential equations. However, their major disadvantage, the limited precision of results (3 to 5 digits), their size and price made them unsuitable for future applications with the advent of the transistor and the growing performance of integrated circuits in digital computers. Nevertheless, the interest of the scientific community in continuous computation arises now from several different perspectives, (Graca, 2004). Recent research in computing theory of stochastic analog networks (Siegelmann, 1999), neural networks and automata (Siegelmann, 1997, 2002) challenged the longstanding assumption that digital computers are more powerful than analog ones. The analog formulation of Turing’s computability thesis suggests now that no possible abstract analog device can have more computational capabilities than neural networks, (Siegelmann & Fishman, 1998; Siegelmann et al., 1999; Natschläger & Maass, 1999). Field computation (MacLennan, 1990, 1999, 2000) can be considered as a special case of neural computation which operates on data which is represented in fields. The latter are either spatially continuous arrays of continuous value, or discrete arrays of data (e.g. visual images) that are sufficiently large that they may be treated mathematically as though they are spatially continuous. A Fourier transform e.g. of visual images is an example of field computation. 20 e. g. the Antikythera mechanism, the earliest known mechanical analog computer (dated 150-100 BC), designed to calculate astronomical positions, (François, 2006). – 11 – Integral Biomatics 28.02.2007 This approach provides a good base for naturalistic computation. It is a model for information processing inside of cortical maps in the mammalian brain. Field computers can operate in discrete time, like conventional digital computers, or in continuous time like analog computers. Other realizations of field computation could be analog matrices of field programmable gate arrays (FPGAs) and grids thereof for image and signal processing problems. Further examples for field computation include optical computing (Stocker & Douglas, 1999; Woods, 2005), as well as Kirchhoff-Lukasiewicz machines 21 and very dense cellular automata (MacLennan, 2006). Finally, quantum computation which was traditionally concerned with subatomic discrete systems such as qubits has realized that many quantum variables with continuous character, such as the position and momentum of electromagnetic fields, can be quite useful. Noise is a difficult problem for quantum computation and continuous variables are more susceptible to noise than discrete ones. Thus, quantum computation over continuous variables becomes interesting option towards a robust and fault-tolerant quantum computation and the simulation of continuous quantum systems such as quantum field theories. In (Lloyd & Braunstein, 1999), the authors provide the necessary and sufficient conditions for the construction of a universal quantum computer capable to perform "quantum floating point" computations for the amplitudes of the electromagnetic field. The above list of alternative models to classical computation is not exhaustive, but it provides a good starting base for transition to the next section which introduces the main part of this contribution. 5. Integral Biomathics The review of the diverse non-classical computation approaches beyond the Turing frame in the previous section leads to the conclusion that these models hypothesize different post-Newtonian era laws of physics as a precondition for their implementation. Although modern computer science has not really entered the relativistic sub-nuclear age of modern physics yet, the abundance of computational ideas and approaches indicates that researchers are well aware of the fact about the limitations of the Turing computation model. They are certainly going to use every discovery and invention in physics to realize their concepts. Perhaps the most significant aspect of this finding in historical perspective is the comeback of analog computing now based not on mechanical components and electronic circuits, but on artificial neural networks and continuous quantum computation. This is a very interesting fact which shows that computation is now closer to biology than to physics. Indeed, computer science and biology maintain today a similar relationship like 19th and 20th century mathematics and physics. The progress in the one filed will influence the progress in the other and vice versa, for we cannot avoid analogy and semantics (in terms of formal logic as we know it) and limit ourselves to a priori established conventions (based on past facts) about what is general and what is special in theoretic research. In this respect we are lead here by the ideas presented in the introductory chapters of Rosen’s Life itself (1991). It becomes evident from the discussion in the previous two sections that the two fields or planes of research are closely interdependent through synergy and correlation and by addressing a number of parallelized phenomena, paradoxes and questions, thus representing a congruent pair, a dyad, of knowledge in complex system design and automation that deserves a special attention by the scientific community. It is therefore our intention to get beyond the limits of autopoietic theory and Turing computation and explore new computational models in complex autonomous systems. Our focus represents the research in natural automation and computation, including models of information emergence, genetic encoding and transformations into molecular and organic structures. This area is the joint domain of life sciences, physical sciences and cybernetics. 21 J. Mills, http://www.cs.indiana.edu/~jwmills/ANALOG.NOTEBOOK/klm/klm.html. – 12 – Integral Biomatics 28.02.2007 In this respect, we are interested to investigate natural systems which are robust, fault-tolerant and share the characteristics of both living organisms and machines which can be implemented and maintained as autonomous organisations on molecular and atomic scale without being planned. Therefore, we are going to use concepts, models and methods from such disciplines as evolutionary biology, synthetic microbiology, molecular nanotechnology, neuroscience, quantum information processing and field theory. We are going to enhance them with other techniques and formalisms from classical and non-classical computation, network and information theory to address specific challenges in the pervasive cyberzoic era of integral research beyond nano-robotics and nano-computation into molecular self-assembly, synthetic morphogenesis, evolutionary computation, swarm intelligence, evolvable morphware, adaptive behaviour and co-evolution of biosynthetic formations (Zomaya, 2006). What we mean by this is ultimately the convergence and transformation of biology, mathematics and informatics into a new naturalistic discipline that we call integral biomathics. This new field is founded on principles of mathematical biophysics (Rashevsky, 1948; Rosen, 1958), systems biology (Wolkenhauer, 2001; Alon, 2007) and information theory (Shannon, 1948) endorsed by biological communication and consciousness studies such as those described in (Sheldrake, 1981; Crick, 1994; Penrose, 1996; Hameroff, 1998; Edelman & Tononi, 2000). Here the term ‘integral’ implies also ‘relational’ and denotes the associative and comparative character of the field from the perspective of cybernetics (Bateson 1972) and systemics (François, 1999). Hence, the goal of integral biomathics is the integration of the numerous system-theoretic and pragmatic approaches to artificial life and natural computation and communication within a common research framework. The latter pursues the creation of a stimulating ecology of disciplines for studies in life sciences and computation oriented towards naturalistic system engineering. Our approach does not antagonize old theories and results. It does not defeat recent or new ones either. In this way, we answer the rising appeal of systems biologists to develop integrative and reconciling philosophies to the diverse approaches to genetics, molecular and evolutionary biology, (O’Maley & Dupré, 2005). Thus, following the line of research set up by the pioneer works of Rashevsky’s (1940 ff.), Rosen (1958 ff.), Bateson (1972), Miller (1978), Maturana and Varela (1980), integral biomathics is going to address questions arising in the widened relational theory of natural and artificial systems. It is concerned with the evolutionary dynamics of living systems (Nowak, 2006) in a unified manner while accentuating the higher-layer dynamic relationship, interplay and cross-fertilization among the constituting research areas. In particular, integral biomathics is dedicated to the construction of general theoretical formalisms related to all aspects of emergence, self-assembly, self-organisation and selfregulation of neural, molecular and atomic structures and processes of living organisms, as well as on the implementation of these concepts within specific experimental systems such as in silico architectures, embryonic cell cultures, wetware components (artificial organic brains, neurocomputers) and biosynthetic tissues, materials and nano-organisms (cyborgs). Therefore, the research methods of integral biomathics include not only the traditional ones of pragmatic and theoretical biology, involving such disciplines as molecular biology and functional genomics, but also the dynamic inclusion of novel computational analysis and synthesis techniques which are characteristic for the corresponding frame of relevance, FoR (MacLennan, 2004) and beyond the existing taxonomic framework for modelling schemes (Finkelstein et al., 2004). This corresponds to a qualitatively new development stage in systems biology and engineering. The algorithmization of sciences not only placed biology closer to the traditional ‘hard’ sciences such as physics and chemistry, but also provided the base for a paradigm shift in the role distribution between biology and mathematics, (Easton, 2006). Therefore, the goal of this new field, – biologically driven mathematics and informatics (biomathics), and biological information theory, – is the elaboration of naturalistic foundations for synthetic biology, systems bioengineering, biocomputation and biocommunication which are based on understanding the patterns and mechanisms for emergence and development of living formations. – 13 – Integral Biomatics 28.02.2007 Our major objection, however, to previous efforts for unification in this field is the one about the roles of causation and entailment in the process of creating and organising life forms. These processes are qualitatively different from the system models we know in physical sciences. Therefore, integral biomathics aims at: (i) removing the restrictive reductionist hypotheses of contemporary physics, (ii) adopting the appropriate mathematical formalisms and models, and (iii) deriving new formalisms out of the biological reality that face the complexity of living systems more adequately. Such arguments are in line with pioneer research in mathematical biophysics (Rashevsky, 1948, 1954) as well as with recent results in systems biology (Westerhoff et al., 2004; Mesarovic et al., 2004). Therefore, we provide an extended model about the interdependence between the various disciplines in this field based on Rosen’s category-theoretical definition (Rosen, 1991, p. 60). Figure 1 illustrates Rosen’s relational model of science 22 based on the concept of Natural Law (Whitehead, 1929, 1933). The arrows 1 and 3 represent the recursive entailment structures within the corresponding domains, – causation in the natural (physical) world and inference in the formal (abstract) world, – whereas the arrows 2 and 4 express the possibility of consistent use of syntactic and semantic truth through encoding/measurement and decoding/realization. In particular, arrow 2 depicts the flow path of abstraction and generating hypotheses about the natural world in physical (analytical) sciences. Arrow 4, in turn, represents the path of de-abstraction and creating forecasts about natural world events using formal models and theories. These forecasts are then used to prove the truth of scientific theories, inferred from hypotheses in the formal world, by observation and measurement in the “internal” loop 1 of the physical world 23 . At the same time, arrow 4 shows the path of invention and engineering of artificial systems in synthetic sciences (e.g. computer science) that emerge out of mathematical models and theorems within the “internal” loop 3 in the formal world. Thus, arrow 4 can be regarded as a double one. In engineering, when we start with inference systems in the formal world (e.g. our minds) that are intended to realize physical world systems systematically, i.e. “by natural law”, we face the so-called realization problem, according to Rosen, which involves “modes of entailment falling completely outside contemporary science” 24 . Figure 1: Rosen’s modelling relation for science and natural law (Rosen, 1991, p. 60) 22 associated with physical systems. There are two separate paths of cognition in (physical) science: 1 and 2 + 3 + 4. If both of them deliver the same replicable results, the theory derived from the hypothesis in the formal world becomes a model of the natural world, (Rosen, 1991). 24 e.g. knowledge through intuition mentioned in section 3. 23 – 14 – Integral Biomatics 28.02.2007 Indeed, both arrows 2 and 4 of encoding and decoding remain unentailed 25 according to Rosen’s model. These arrows are not part of the natural world, nor of its environment; they do not belong to the formal world either. They appear like mappings, but they are not such in any formal sense. This finding is consistent with Gödel’s incompleteness theorems 26 that state there is always a true statement within a formal system that cannot be proved within that system and requires a higher level formalism. Therefore, in a next step, we propose four substantial amendments to Rosen’s modelling relation in Fig. 1 defining the major distinctions of biological systems when compared to physical systems. Figure 2 depicts these changes which extrapolate and endorse the interchange circle model between synthetic biology and systems biology presented in (Barret et al., 2006). Firstly, we adopt Rosen’s postulate about the generalization of biology over physics (Rosen, 1991, 1999). In other words we regard physical systems as a subset of the more complex biological systems and not the reverse way. Secondly, we adopt and develop some concepts related to the inductive force field in morphogenesis (Thompson, 1917; Franck, 1949) and the formative causation or ‘morphic’ resonance hypothesis 27 (Sheldrake, 1981) as basic organization principles in biological systems, analogous to those in quantum physics 28 . Thirdly, we replace the human-centric concept of ‘natural law’ by the biological one of natural habit 29 which is synonymous with ‘natural pattern’. Finally, we apply the principle of ‘lateral’ induction 30 in the formal world for creating new formalisms about natural systems through pattern recognition and analogy, based on observations, experiments and associations with other formalisms. Lateral induction corresponds to morphic resonance in the natural world and addresses the larger set of formal systems that reflect the behaviour of biological systems from the standpoint of integral biomathics. In this way, biology is generalized and physics becomes its special case with causation and inference being entailments of resonance and induction respectively. Figure 2: The revised Rosen’s modelling relation with generalization of biology and natural habit 25 There is no mechanism within the formal world to change an axiom or a production rule and there is no such mechanism within the physical world to change the flow of causation. 26 in particular, the second incompleteness theorem: “For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.” 27 cf. section 6 for more details. 28 "The universe -- being composed of an enormous number of these vibrating strings --is akin to a cosmic symphony. " (Greene, 1999). 29 30 coined by Sheldrake; the term “law” is too strong for biology, whereas habits are less restrictive towards changes. lateral induction is analogous to de Bono’s concept of ‘lateral thinking’ (de Bono, 1967); cf. section 6 for more details. – 15 – Integral Biomatics 28.02.2007 We still do not know how does an idea emerge from, relates to, coincides/reacts with or is influenced by other ideas in human’s minds. Thoughts exhibit such properties as quantum entanglement (Aspect et al., 1982). Yet, we do not know the idiosyncratic nature of such mechanisms as (spontaneous) inversion, correlation, fusion, fission of concepts. This is an interesting research area along the pathways of arrows 3 and 4 on figure 2. Furthermore, in the bidirectional relation of natural habit between the two worlds on figure 2, we identify two epistemological processes denoted by arrows 2 and 4. First, it is the process of pattern recognition in the natural world and the phenomenon of memory viewed by the self or the formal world (arrow 2). Second, it is the case of relational science in the formal world based on formal theories of inference in the first concentric circle of mathematical models, but also on metaphors, analogies and non-local relations or induction in the second incorporating circle (arrow 4). The latter is associated with the next level entailments in the formal world and with resonance in the natural world related to such phenomena as life, thought and consciousness. Thus, arrows 2 and 4 represent an open infinite spiral of science development rather than a closed stuttering loop. Figure 3 represents the evolution of formalizations and realizations with the shifted perspective of higher layer entailments along the time axis. It suggests that a vertical view on the slices of shifted flat world layers along the temporal axis may deliver new insights into the cross-layer links between the entities in the diagram and about the real dimensions of the interplay within the formal and natural worlds. Similar models of the epistemological ontogenesis are present in Rashevsky’s topological overlays of allocated organic functions (Rashevsky, 1961) and in the amino-acid interrelationships within the 3D folding structure of DNA (Crick, 1988). This layered interrelationship within the developmental spiral of the world “versions” results from paradigm changes and inflection points in the formal world. Complexity in the natural world is manifested through implicate and explicate order which corresponds to semantic enfolding and unfolding in the formal world (Bohm, 1980). The later is associated in our enhanced model with recursive pattern generation (including failures) followed by reflexive processes of evaluation, change and adaptation in response to external stimuli in the presence of noise and disturbance. Essentially, we propose an integral evolving model of a layered dynamic interdependence between the logical formal meta-computational and reasoning system world and the natural autonomous biophysical system world. The two worlds represent a dyad in perpetual development which ultimately embeds the unentailed relationships 2 and 4 of Rosen’s original modelling relation in figure 1. Figure 3: The evolving Rosen’s modelling relation for science – 16 – Integral Biomatics 28.02.2007 The above amendments in Rosen’s modelling relation are necessary because in a historical perspective mathematics has been derived from and developed for descriptions of physical phenomena. Mathematics was proved to be a suitable tool for statics, celestial mechanics, thermodynamics, electromagnetism and relativistic theory. However, most of its formalisms might be insufficient to handle biology in a straightforward manner. We may need to develop new mathematical foundations for modelling different types of biological systems just as von Neumann and Dirac proposed their own formalisms to explain quantum mechanics. Indeed, quantum theory and theoretical biology are closely related to each other (Rashevsky, 1961), which means that biomathics could successfully adopt and develop the formal toolset of quantum theory. The basic characteristic of physical systems is that they are closed and that everything that could not be included within a closed system is neglected. However, in biology we face an inverted world to the one of physics, because systems are basically open and because the second law of thermodynamics about the system equilibrium is defined in terms of order (negentropy 31 ) instead of chaos (entropy). Therefore, we need now a different, kind of mathematics, a new kind of science (Wolfram, 2002) that is devoted to the discovery of recursive patterns of organization in biological systems (Miller, 1978), such as those of neural activity where neural systems could be understood in terms of pattern computation and abstract communication systems theory, (Andras, 2005). This science should be able to deal with the complexity of biological systems 32 by restructuring its ontology base to correct its models whenever appropriate. 6. Discussion The presented approach to integral biomathics in the previous section appears related to the classical dialectical scheme of thesis-antithesis-synthesis, (Hegel, 1807). Yet, we have a different motivation for explaining the philosophy of this new discipline that we derived from the analysis of the latest developments in the participating scientific fields discussed in sections 1-4. In particular, we identified the following categories that play special roles in integral biomatics: Syntax vs. Semantics. Formal concepts are either purely syntactic in the Hilbert’s sense or they are created through recursive extrapolation from other formal concepts. In the second case, they contain a semantic component of truth that links them to the context of those previous concepts they were derived from. Rashevsky pointed out that there are different ways in mathematics to approach the relational problem in biology using such means as set theory, topology or group theory (1961). Each one of these approaches is able to represent different aspects of relations in a system within different contexts. Thus, context (semantics) and relation (syntax) can be interchanged depending on the purpose of the description. Therefore, the author regards even pure syntactic models in the formal world as semantic inclusions of axiomatic truths which are obtained in an empirical way from the physical world through the associative link of natural habit. In order to change the historical discrimination between syntax and semantics in the formal world imposed by the different perspective of our present day understanding of the natural world, we need to step back and generalize the description of the domain of discourse. This can be realized by taking out the restrictions placed by the TM model 33 to a degree that allows to reconcile traditional Turing based (incl. Super-Turing) and non-Turing based approaches in computation (Rosen, 1991; Hogarth, 1994). The above considerations demand a systematic study of the traditional and the opportunistic approaches to computation and automation along with their relations and frames of relevance. 31 Schrödinger, 1944. At a certain level of complexity, human beings and machines cannot recognise patterns anymore. 33 derived from the principles of Newtonian mechanics 32 – 17 – Integral Biomatics 28.02.2007 Resonance vs. Causation. Sheldrake’s hypothesis of formative causation or morphic resonance states that morphogenetic fields shape and organize systems at all levels of complexity (atoms, molecules, crystals, cells, tissues, organs, organisms, societies, ecosystems, planetary systems, galaxies, etc.). Accordingly, “morphogenetic fields play a causal role in the development and maintenance of the forms of systems“, (Sheldrake, 1981, p. 71); they contain an inherent memory given by the processes of morphic resonance in the past, where each entity has access to a collective memory. Our approach differs from Sheldrake’s original definition in two points. Firstly, we clearly distinguish between causation and resonance as universal organization principles of the Natural World (Fig. 2). Whereas in causation we can semantically identify multiple linear cause–implication chains about events in the domain of discourse, we understand resonance as non-linear, non-local and (sometimes) semantically ‘hidden’ spatio-temporal relationships between entities in the broad sense. The entailment of causation within resonance is however allowed in our model analogously to the scales of interactions in physics. Secondly, resonance is for us a dyad consisting of (i) energetic resonance as we know it in physics (wave mechanics, electromagnetism, quantum mechanics, string theory, etc.), and (ii) information resonance corresponding to both classical communication theory (Shannon, 1948) and to Shelldrake’s morphogenetic field theory. Induction vs. Inference. With the term ‘induction’ in integral biomatics we do not mean the classical mathematical induction used for formal proofs. It is the counterpart of resonance within the formal world which entails not only the classical formal reasoning theories in mathematics, but also (yet) unknown structures and pathways of logic based on complex, semantically enfolded relationships within and between the formalisms. We can imagine induction as a the self-organized process of generating and evolving formalisms from a set of basic axioms and theorems that can mutate depending on the results of cognition. Synchronicity and Chance vs. Determinism. When we observe the above categories as an evolving dynamic set of cross-interacting patterns of structures and processes for organization and exchange between the natural and the formal worlds, we can realize the third epistemological dimension about the multi-relational helix Gestalt of scientific exploration along the temporal axis (Fig. 3). We believe that the usage and verification of this macro-model can deliver new insights into the natural phenomena of synchronicity and chance in the context of emergence, differentiation, organization and development of biological structures and processes. Following the line of thought from the previous paragraphs, we can hypothesize that determinism is entailed within synchronicity and chance. 7. Conclusions Recent efforts in autonomic computing (IBM, 2001; Kephart & Chess, 2003) and autonomic communications (Smirnov, 2005; Dobson et al., 2006) is interested in studying how such systems can automatically adjust their performance and behaviour in response to the changing conditions of their work environments. The goal of this research involving the whole scale of contemporary computer science from automata theory to artificial intelligence is to improve and enhance the complex design of modern computing and communications architectures with capabilities that occur in living systems such as self-configuration, self-optimization, self-repair and self-protection. Expected is the development of autonomic algorithms, protocols, services and architectures for the next generation pervasive Internet that "evolve and adapt to the surrounding environments like living organisms evolve by natural selection" (Miorandi, 2006). Formality in contemporary computing means that information processing is both abstract and syntactic and that the operation of a calculus depends only on the form, i.e. on the organisation of the representations which has been considered as deterministic. Yet, the more computation pervades into natural environments, the more it faces critical phenomena of incompatibility, (Koestler, 1967). – 18 – Integral Biomatics 28.02.2007 One of them is the oversimplification of state-based computing models discussed in the previous sections. Although we are equipped today with a whole range of theories and tools for dealing with complexity in nature (Boccara, M. 2004; Sornette, 2004), we miss the real essence of natural computation in such areas as geology, meteorology and stock exchange. The diverse models and tools for solving differential and recurrence equations, for modelling stochastic processes and power law distributions using cellular automata and networks are operating on Turing machine computer architectures based on the concept of state. The latter is central for Newtonian mechanics which describes reality as sets of discrete interactions between stable elements. Yet, physics had already a paradigm shift towards quantum mechanics in the past century and natural systems are now seen at their utmost detail to match the wave function equation which describes reality as a mesh of possibilities where only an undifferentiated potential describes what might be observed a priori to measurement. It was shown in this paper that biological systems are even more complex than physical ones and that we need a different paradigm embedded in the underlying computation architecture in order to achieve the vision of truly autonomic computing and communications. Networks are seen as the general organizing principle of living matter. A recent IBM report in Life Sciences (Burbeck & Jordan, 2004) referred to a Science article which made this conclusion (Oltvai & Barabasi, 2002). Indeed, this notion originates from Rashevsky’s biological topology (1954) and Rosen’s metabolism-repair model (1958) which became the foundations of relational and theoretical biology. The network concept in life was then reinforced 20 years later by the autopoietic theory of Maturana and Varela (1974) and by Miller’s living systems (1978). Almost a decade before carrying out the first computer simulation experiments of autopoiesis, (MacMullin & Varela, 1997), Varela and Letelier tested Sheldrake's theory of morphic resonance in silicon chips using a microcomputer simulation, (Varela & Letelier, 1988). Briefly, they let a crystal grow in silico. It was expected that if the morphic fields theory were correct, the synthesis of the same crystal pattern will be accelerated after millions of iterations. Yet, Varela and Letelier found that there was no detectable acceleration of the growing process at all. They concluded that either Sheldrake's hypothesis is falsified or that it does not apply to silicon chips. However, there might be other explanations of this result. One of them could be that conventional Turing machines were used to simulate the above experiments. In this case, new computation models beyond the TM concept are needed to address such issues as autopoiesis. Several issues arise in the investigation of non-Turing computation: (i) What is computation in the broad sense? (ii) What frames of relevance are appropriate to alternative conceptions of computation (such as natural computation and nanocomputation), and what sorts of models do we need for them? (iii) How can we fundamentally incorporate error, uncertainty, imperfection, and reversibility into computational models? (iv) How can we systematically exploit new physical processes (molecular, biological, optical, quantum) for computation? (MacLennan, 2006). Integral biomatics is a new approach towards answering these questions and towards shifting the computation paradigm closer to the domains of quantum physics and biology (Baianu, 1980, 1983, 2004) with the ultimate objective of creating artificial life systems that evolve harmoniously with natural ones.. This new discipline is particularly interested in four essential HOW 34 questions: • how life and life-like properties and structures emerge from inorganic components. • how abstract ontological categories and semantics entailments emerge in living systems. • how cognitive processes emerge and evolve in natural systems. • how life related information is transferred in space and time. 34 of course, these questions imply also the Aristotelian WHY. – 19 – Integral Biomatics 28.02.2007 A starting point in this quest is the definition of the theoretical framework where the understanding autopoisis plays a central role. Kawamoto's extensive definition of autopoiesis can be described as follows 35 (Kawamoto, 2000): "An autopoietic system is organized as a network of processes of production of elements. Then, (i) the elements of the system become the components only when they re-activate the network that produces these elements, (ii) when the sequence of the components construct a closed domain, it constitutes the system as a distinguishable unity in the domain in which they exist." The main difference of Kawamoto’s definition of autopoiesis from the original one by Maturana and Varela lies in the second part of the definition (ii). It distinguishes between the living system itself 36 (German: "sich") constituted as the network of productions from the self 37 (German: "Selbst") as the closed domain in the space. According to Kawamoto, this extension makes it possible to represent the aspect that the entity Selbst (self, syntax) changes while sich (itself, semantics), which ultimately represents its self-awareness, is maintained in such way as Schizophrenia 38 . Kawamoto argues that this distinction is ambiguous in Maturana’s and Varela's original definition and that it causes misunderstanding of autopoiesis. Nomura’s formal model of autopoiesis in (Nomura, 2006) is actually affected by the above aspect. The organization is closed and maintained in a specific category and the structure is open and dynamic in a state space. Indeed, the first part of Kawamoto’s definition of autopoiesis is more precise than the original one of Maturana and Varela. It provides an initial condition which can be regarded as the "birth" of the living system. The second part of the definition also identifies a more distinct characteristic than the original one (Maturana & Varela, 1980). To be more formal, we could add here also the gerund form of the verb, because the construction of a the closed system takes place permanently, i. e. in every single moment, so that the system does not die and then revives again and again. Thus, the distinction from the environment is always present and includes the processes of metabolism and repair which maintain the development of the living entity and its equiibrium/exchange with the environment (homeostasis). Living systems are open in physical spaces. But autopoiesis requires closedness of organization in living systems. This implies that openness – closedness, (enfolding – unfolding or syntax – semantics) lie within observers' physical perspective level and another level beyond it, as described in Nomura’s model of two levels illustrated with the relations on figure 3. Indeed, the layering of perspectives has also the dimensions of Maturana, Varela and Luhman who defined the categories of first, second and third order autopoiesis starting from molecules and cells, and moving up through organic systems and individual beings to species and social organisations. Here we could ask ourselves whether Nomura’s model represents an orthogonal view to the classical model of autopoiesis while containing subsets or overlays of the three orders of autopoiesis defined there. The above problem is not explicitly dealt with in Nomura’s paper (Nomura, 2006). However, the existence of an isomorphism between operands and operators, the necessary condition of completely closed systems, is implied from the orthogonal view mentioned in (Soto-Andrade & Varela, 1984). This perspective appears also when we re-consider Rosen's idea (Rosen, 1991). Nevertheless, some hardliner philosophers including Kawamoto argue that since the view of the relation between inputs and outputs in the system is the one of the external observer, it does not clarify the organisation or the operation of the productions in the system itself, (Nomura, 2007). 35 36 37 38 personal correspondence with Tatsuya Nomura, November 2006. or the “self”-part of the process definition or the physical entity of the system as epistemological distinction or even awareness (e.g. of a social group) Kawamoto developed this extension from psychiatric perspective. – 20 – Integral Biomatics 28.02.2007 Consequently, any description of this level is impossible with the current mathematics at hand. Nomura reckons 39 that this impossibility implies the difference between the perspectives of quantum physics and autopoiesis regarding the role of the observer (Toschek & Wunderlich, 2001). Finally, each perspective has its own rules and frame of relevance as MacLennan states, so that we should rather ask ourselves at which level do we define autopoiesis in the classical way. At the cellular level that could be a good model, but at molecular level we begin to encounter quantum effects such as nonlocality or entanglement (Aspect et al. 1982). Since Rosen claimed that a material system is an organism if and only if it is closed to efficient causation, the above evidence leaves open the question at which level a system can be defined as open or closed and if there can be provided a strict separation of concepts. The whole circle of questions around these definitions is not complete at the current state and their formulation and answer will require further studies. The most interesting result of the research surveys presented in this paper are the parallels in the discussions about the formalizations of life (♣), thought (♣♣) and computation (♣♣♣) which we regard as the fundamental questions of a new scientific discipline. Integral biomathics is envisioned to provide a generalized epistemological framework and ecology for symbiotic research in life, physical and engineering sciences. It is going to be another challenging mountaineering experience in intellectual development. Yet, we remain optimistic, for history of science knows also other unusual discovery pathways which proved to be successful in the long run (Crick, 1988). _________________ Acknowledgements: The author deeply appreciates the valuable help of Prof. Tatsuya Nomura from Ryukoku University (Japan) for discussing questions on the formalization of autopoiesis using category theoretical models. 8. References Adami, C. (1998). Introduction to artificial life. 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