BULLETIN OF MATHEMATICAL BIOLOGY

VOLUME39, 1977

OBSERVATION AND B I O L O G I C A L SYSTEMS-~

• ROBERT ROSEN D e p a r t m e n t of P h y s i o l o g y a n d Biophysics, D a l h o u s i e University, Halifax~ N o v a Scotia, C a n a d a B3H 4H7

A number of apparentlydifferent lines of inqmry into fundamental biological processes point to the central role played by the notion of observation in the theory of biological systems. Not only do we use the results of our own observations to obtain the system descriptions which are the starting-points for an understanding of biological processes, but it is a basic postulate of physics that the interactions between biological systems themselves can be regarded as observations. On this basis, it is clear that we cannot properly understand biological interactions unless the observables we employ for system description are the same as those involved in the interactions we are describing. To do this requires a general theory of observables and system description, establishing the relationship between different modes of description A sketch of such a theory is developed in the present paper, using only two postulates: (a) that all interactions are determined by the values of observables of a system evaluated on specific states, and (b) that real-valued observables suffice. As an application, a specihc test is proposed whereby it can be determined whether the observables employed to describe interacting systems are sufficient to specify the observables involved in the interaction itself.

I. Introduction. Most of the material to be developed in this paper is taken from a m o n o g r a p h concerned with the measurement problem, and its role in the representation of natural systems (Rosen, in press). We refer the reader to the monograph itself for most of the details; we can herein only sketch the major conclusions of this work, and the significance of some of its implications. Before proceeding to consider the work itself, it might be appropriate to develop the biological background or context in which it arose. To do this, it is helpful to begin with a review of some of my own earlier work. "~Presented at the Society for Mathematical Biology Meeting, University of Pennsylvama, Philadelphia, August 19-21, 1976 663

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1. Dynamical modelling. In one form or another, the preponderance of specific investigations into mathematical biology involve systems of rate equations, which may be written as dxi/dt=fi(xl, ..., x,),

i = 1 .... ,n.

(1)

Here the quantities xi represent state variables, whose numerical values at an instant of time determine the instantaneous state of the system being studied. The functions f~ represent, in some sense, the forces imposed on the system; the rate equations themselves express the manner in which instantaneous change of state depends both on the instantaneous state and on the imposed forces. Such equations are of a familiar form; if the state variables xi are interpreted as the positions and momenta of particles in space, then the rate equations become the equations of motion of a mechanical system; indeed, this is the context in which such rate equations originally arose. In general, the study of such equations involves not only the determination of specific solutions, but also their stability (i.e. their relation to nearby solutions) and their structural stability (the relation of the rate equations themselves to "nearby" equations). For the details of the mathematical theory of such systems, see for example Rosen (1971). In most studies of specific biological processes, the state variables xi represent measurable quantities associated with the system under study. For instance, in theories of cellular differentiation, the x, generally represent the concentrations of chemical species, and the functions f, correspondingly represent the specific chemical reactions occurring in the system, using the Law of Mass Action. In any case, the state variables represent specific observables of the system, which can be regarded as mappings from states to real numbers. In particle mechanics, any quantity associated with a mechanical system which can be measured (such as its total energy) can be thought of as an observable (or dynamical variable) in this sense. Conversely, it is traditionally assumed that any mapping which assigns numbers to states is an observable, in the sense that we could build a measuring instrument, or meter, which would assign numbers to states in the same way as the given mapping does. Thus, the set of all observables of a mechanical system becomes identified with the set of all real-valued mappings from states to numbers. In this context, a set of state variables xl .... ,x, is a family of observables with the following-additional properties: (a) If f i s any observable and s is any state, then we can write

f(s)

= f ( x I (s), ...,

Xn(S)) ;

i.e. the value of f on s is completely determined by the values of the x~ on s;

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(b) If f is any observable, and sl, s2 are any states, then

f(sl)~f(sz)

implies

xi(&).Pxi(&)

for at least one of the state variables x~ (i.e., if two states can be distinguished by any observable, they can already be distinguished by at least one of the state variables). There is thus nothing unique about a family of state variables; they are simply a family satisfying the above property. The basic postulate of particle mechanics, embodied in Newton's Laws, is precisely that the displacements and m o m e n t a of the particles of a system comprise such a set of state variables. It should also be noticed that the functions f appearing in the rate equations defining a system's dynamics are assumed to be observables in this sense, although this is a highly non-trivial assertion (cf. Rosen, 1972). N o w let ~b= q~(xl,..., x,) be an observable. T h e n 4} inherits a dynamics from the rate equations governing the state variables xv This equation for d4(dt will not in general be very instructive or significant. IfO is another observable, it too will inherit a dyn&mics in this way. However, it m a y happen that we can write dqS/dt =q)(qS, O)

dO/dt = tP({b, ~)

{2)

for appropriately chosen functions (I), ~ . It will be noted that these equations themselves comprise a set of rate equations. Furthermore, q~ and ~, are observables of the system; hence by definition they are in principle directly measurable quantities. N o w let us notice that it is entirely accidental that we come equipped to readily measure the observables xi, and not the observables q~, tk. If on the other h a n d we had been equipped to measure the observables ~band 0, and not the x~, we would not "see" our original system (1) at all: but rather the system (2). Indeed, it is possible to show that there exist systems of the form (1) which are universal, in the sense that, given any set of rate equations whatsoever, there exist observables of (1) which inherit precisely these rate equations from the dynamics governing the state variables. (Rosen, 1968). N o w what does all this m e a n ? Let us note that an observable represents by definition the capability of a state of our system to move a meter. It thus represents a capability of a given system to interact with another. We characterize a system in a particular way because we come equipped to interact conveniently with certain of its observables. But there is no reason to expect that every other system will interact with it t h r o u g h these same observables. The above argument thus implies that the consequences of such interactions can be quite different from those we would expect; as we showed in (op. cit.), we might observe a gravitating particle, and another system, interacting with

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different observables, might see a harmonic oscillator. Since all observables are equally "real", in any physical sense, it follows that (a) we cannot ascribe to our description (1) a greater measure of"physical reality" than could be ascribed to any other description of the form (2); (b) on the contrary, we would have to admit that the same physical system admits many modes of dynamical description, all equally valid, and each appropriate for the study of a class of interactions of the given system with another. Such considerations lead to a basic re-assessment of the nature of physics itself. For they imply that the basis for the physical interactions between systems is far wider than has been recognized heretofore. Moreover, since the basic analytical units for the study of physical systems are constructed out of those measurable quantities which we most directly perceive, a widening of the class of observables forces a corresponding widening of the analytical possibilities for decomposing a physical system into subsystems. Now physics has been dominated by the idea that there is ultimately but a single correct mode of analysis, involving a resolution of a system into "elementary particles". Our considerations, on the other hand, suggest that there exist many other modes of analysis of physical systems, all equally "real" in any physical sense, and often far more appropriate for the understanding of any particular interaction. Let us give a couple of simple examples. The first of these is the well-known three-body problem of particle mechanics. It is well known that the three-body problem cannot be solved in closed form, so that in particular we cannot answer asymptotic questions regarding the stability of such systems. We might think to decompose such a three-body system into its particulate subsystems; two-body and one-body systems. Such a decomposition can be performed physically; and furthermore, such subsystems are completely tractable analytically. But we cannot thereby solve the three-body problem, primarily because the physical process of isolating the subsystems irreversibly destroys the original dynamics in which we were interested. To solve the three-body problem means therefore that we must discover a new set of analytical units; a new set of subsystems defined by new observables, from which the dynamics of the three-body system can be effectively reconstructed. Such subsystems will look strange to us, because we are accustomed to think in terms of particulate subsystems, and because the new subsystems will not correspond to any physical decomposition of the system. The second example is the familiar solution of a system of linear equations of the form (1). We solve these equations by (formally) introducing a new set of observables, which reduce the system to a canonical form which is maximally uncoupled. These new observables, and not the original state variables, are the appropriate units of analysis for such a system. Thus, we have seen that we are led naturally to a consideration of (a) a wider

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set of observables than the ones which we find it convenient to measure, and (b) a correspondingly wider class of analytical units, or subsystems, with which to study the interactions of a given system with another, it should be emphasized that there is nothing "unphysical" about our arguments; on the contrary, they are all developed entirely within the traditional physical fi°amework; their only novelty is to take seriously, so to speak, the full richness available wi|hin that flamework.

2. Quamum gene~ics. In earlier research (Rosen, [960) the following question was considered: what, if any, are the implications of the hypothesis that primary genetic processes involve microphysical (i.e. quantum) events? Such a question was naturally suggested by the revolutionary developments occurring at that time in the fields of molecular biology and molecular genetics~ and by the corresponding implication of molecular interaction as the vehicle for the transmission of genetic information. One can approach such a problem in a straightforward fashion in terms of the q u a n t u m -theoretic formalism of yon Neumann (1955). In this formalism, a measurable quantity, or observable~ is represented by tt self-adjoint operator acting on a function space; the values which it can assume are identified with the spectrum of the operator. Measurements are limited by uncertainty, so ~hat precise determ£nation of a spectral value is paid for by uncertainty in conjugate observables; simultaneous measurability is equivalent to the commutativity of the corresponding operators. Moreover, basic to the formalism is the idea that all physical events are ultimately reducible to observables evaluated on states; hence the only way in which systems can physically interact is, so to speak, to evaluate observables on each other. A microphysical system, then, is simply a collection of observables satisfying a number of mathematical properties. However, in order to faithfully transmit genetic information under repeated observations, we must avoid uncertainty effects. Thus, a microphysical system appropriate for the transmission of genetic information must be a very special collection of observables, which can be characterized. Now the physical theory of such systems is dominated by the idea of the Hamiltonian. However, I was unable to formulate a Hamiltonian for the kind of microphysical system which was appropriate for transmitting genetic information. Hence, the results of the analysis were the isolation of valid microphysical systems which (a) contained no Hamiltonian and hence (b) did not correspond to particles. Such systems could be embedded (in many ways) into particulate systems (i.e. could be regarded as subsystems of particulate systems) but they could not be recovered from such a particle by a physical fractionation procedure analogous to the way in which particulate subsystems could be separated from each other. We may met~tion that such a picture is highly reminiscent of, say, the active site of an enzyme:-this too is a subsystem

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which can be embedded into a particle in many ways, but which cannot be separated from the particle by any physical procedure. Thus here again, starting from a conventional physical viewpoint, we are led to the concept of a wider class of subsystems than are ordinarily recognized, directly related to a specific functional activity (i.e. a specific interaction) in which we are interested. And again, we are led to this class by taking seriously, so to speak, the richness available in the formalism of quantum theory, which rests ultimately on the availability of observables in microphysical systems. We may note that similar conclusions have been reached by Howard Pattee (1970) and by numerous others.

3. Relational biology. The first definitive studies in relational biology, as well as the term itself, are due to N. Rashevsky (1954). It was his insight to recognize that what we are really interested in, as far as biological systems are concerned, is primarily their function and behavior, not their structure. Indeed, he pointed out that the same behavior and function can be manifested by organisms of the most diverse structure. On the other hand, it is structure which we can most readily study quantitatively. In reductionistic terms, the two are related by the view that "structure implies function". However, as Rashevsky perceived, even if the reductionistic hypotheses were correct in principle (which, as we shall see, there is ample reason to doubt) this might be the hard way to approach biology. Rashevsky therefore proposed an alternative approach to the theory of the organism; one which was based from the outset on functional considerations. He attempted to construct a mathematical framework in which organizational and behavioral considerations provided the basic analytical units, quite apart from any particular structural basis. Rashevsky's point of departure lay in two observations: (a) that the same functions or behaviors are manifested throughout the biological world, and (b) that these behaviors stand to each other in the same relations throughout the biological world. He pointed out that these relations could be expressed for each particular organism in terms of an oriented graph, whose detailed structure would vary fiom organism to organism; but the graphs of different organisms could be related to each other through a homomorphism which maps corresponding functions on each other while preserving the basic relations between them. This was the essence of Rashevsky's Theory of Biotopological Mapping. He also showed that such a class of homomorphic graphs could be generated from a single graph, which he called the primordial, by means of a simple set of rules, which he called transformation rules. The specification of a primordial, and a set of transformation rules, comprised what he called an abstract biology. He proved a number of theorems valid for all such abstract biologies; for instance, he showed that the primordial, and hence the entire biology, could be determined from one (or at most a few) graphs chosen at

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random from the biology. This entire approach was characterized by emphasizing those characteristics of organisms which made them appear alike (function and relation) rather than those which made them appear different (structure). My own independent approach to relational biology arose from an attempt to characterize the most general kinds of organizations which we would want to call cellular. In brief, I argued (Rosen 1958a, 1958b, 1959) that any system we would want to call a cell must involve at least two inter-related activities: (a) a metabolic activity, characterized by the processing of materials ultimately derived from the environment, and (b) a repair aspect, involving the continual synthesis of the components responsible for metabolism. It turned out that systems organized in this fashion, and which were accordingly called (M, R)systems, possessed many interesting properties. One of them was that, under precisely definable conditions, there was intrinsic in the formalism a mechanism for the replication of the repair components. Another of them was that the distinction between metabolic, repair and replication activity was not an absolute distinction, but a contingent one; it was dependent on the choice of a particular way of observing the system. That is, a given (M, R)-system could in principle be extended in such a way that the original metabolic components became repair components, and the original repair components became replication components. The entire relational approach to biological systems has two striking characteristics. The first is its emphasis on commonalities between structurally dissimilar systems. The extraction of such a commonality can only be done in abstract mathematical terms, and is in fact analogous to the characterization of an abstract group from a set of explicit representations of it. It thus again suggests that there are ways of analyzing physical systems distinct from the conventional structural decompositions. Further, the fact that such relational analyses are contingent rather than absolute is exactly analogous to the situation we found earlier in dynamical system analysis, and in microphysics; that different choices of observables lead to the isolation of different subsystems, and hence different modes of analysis, of the same system. Moreover, it must be emphasized in the strongest terms that there is nothing speculative about relational arguments; our perceptions of relational aspects of organisms is just as real as our perception of their structures, and, in a precise sense, just as much a part of their physics. Now all of the examples mentioned above fall within the framework of the general theory of observation and measurement, but they all lead outside the traditional framework of the application of this theory as it is found in physics itself. They lead to the recognition of new classes of analytic units for interactions in physical systems, chosen so as to be most appropriate for the treatment of any particular mode of interaction. Therefore, it is suggested that

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we reconsider the m a n n e r in which we employ the results of measurement to represent systems in general, so that we can put the conclusions reached from our special examples into a more general framework, and see what else is implied by these conclusions. We therefore now turn to a study of these questions.

II. The Basic Theory.

The material presented in the preceding section suggests that any system may admit m a n y non-equivalent descriptions, depending on which observables of the system are involved in the description. Moreover, each of these descriptions is an equally valid representation of the corresponding system, in the sense that each captures some aspect of the system's interactive capabilities. The question thus arises: how do these different descriptions fit together to generate more comprehensive descriptions? What, if anything, do we learn about a system when we pass from one mode of description to another? It is with such questions that the present section will be concerned. We shall develop our discussion within the context of physical measurement, but as we shall see later, it has a much wider range of applicability. F r o m the examples of the preceding section, we can see that the basic ingredient in system description is what we have called an observable, or dynamical variable. The domain of definition of an observable is a set of states of a system. If we assume that a system must be in some state at every instant of time, then an observable provides information regarding such an instantaneous state. As we noted earlier, physics makes two explicit assumptions about the relation between observables and states: (1) that every physical event can be represented in terms of observables evaluated on states, and (2) realvalued observables suffice, We shall be concerned with exploring the implications of these two hypotheses. Let us consider the prototypic situation, in which we have a set S of states, and a real-valued function f : S ~ R which represents an observable. The image f ( S ) _ : R will be called the spectrum of J. We observe that the observable J induces an equivalence relation Ry in S, defined as follows:

s1Rfs 2 i f a n d o n l y i f

f(Sx)=J (S2)

Obviously the quotient set S/Rz, the set of equivalence classes in S under the relation R:, is in 1-1 correspondence with the spectrum f(S). In general, the observable f c o n v e y s limited information about the set S on which it is defined, because by definition it cannot distinguish between states lying in the same equivalence class. If our only access to the set S were through the observable f, then for us the state set of our system would appear to us to be S/R:. N o w this set is in 1-1 correspondence with f(S), which is a set of numbers

O B S E R V A T I O N A N D B I O L O G I C A L SYSTEMS

671

serving to label the "states" we see. Thus we may as well call f (S) "the state space" of our system. As a set of numbers, f ( S ) possesses a variety of structures. Most important, it possesses a metric topology, which allows us to decide when two of its elements are "close" to each other. We use these structures in f (S) to impute structure to S itself; we would say that two states s~,s2~S are close if their images f(s~ ), J (s2) are close. It cannot be too strongly stressed that such structure is not intrinsic to S; it is imputed to S through the vehicle of a particular observable J, and the topological properties of f (S). Indeed, as we have seen, an observable f does not allow us to deal directly with S at all, but rather with a partition of S into equivalence classes. N o w let us suppose that we are given another observable g: S ~ R . It too partitions S into a set of equivalence classes by virtue of the equivalence relation Ro, and defines another "state space" g (S) = S/R o. The question is: how do the descriptions of S obtained w i t h f a n d with g compare with each other? H o w can we combine these descriptions to obtain a more comprehensive description of S? Given an observable f, suppose that sl, s2 are states in S such that J (Sl), J (sz) are close in f(S). This means that, as far as the observable f is concerned, the two states s~, s2 approximate each other; we may substitute s2 for s~ and make only a small change in observed values. Let us ask the following question: given that sa, s2 approximate each other under f, when will they also approximate each other under g ? If it is always true that g (s 1 ), g(s2) are close whenever f (s2), f (s2) are close, then in a certain sense the structures imputed to S by f and by g are the same; we can then say that g approximates f, and hence can be substituted for J, with no change in imputed structure. Alternatively, we could say that g "conveys the same information as f " , so we learn nothing essentially new a b o u t S by employing g rather than f M o r e precisely, let s~S. We shall say that s'~S is J-close to s if If (s) - 1 (s') I is small. We shall say that s is a stable point for g if every state J:close to s is also gclose to s. If s is not a stable point, then every neighborhood of s contains a state s' which is f-close to s but not g-close. In that case, we shall say that s is a biJurcation point of g with respect to fi Interchanging f and g, we obtain dual concepts of stable point and bifurcation point of J with respect to g. These are in general not the same as those obtained from g with respect to f It is easy to see that the sets of stable points give rise to open subsets in f ( S ), g(S); the complements of these open subsets are the respective sets of bifurcation points. The bifurcation points represent states at which one of the observables J, g does not approximate the other. Thus, these are the states at which essentially new information is obtained from the employment of the second observable.

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ROBERT ROSEN

O u r use of the term "bifurcation" is perfectly consistent with (and in fact is a generalization of) its conventional usage in the context of dynamical system theory. Let us now introduce a related but equally important notion, that of the linkage of observables at a state. As we have seen, an observable f partitions a state set S into a family of equivalence classes. Given an f-equivalence class, we m a y ask how this class is "split" by another observable g; i.e. for the manner in which states indistinguishable to f can be discriminated by g. The linkage of g to f at a state s is reciprocally related to this "splitting"; g is totally linked to J at s if the g-class of s contains the f-class; unlinked to f at s if every g-class intersects the f-class of s. Thus, if g is totally linked to f at s, g (s) is determined by f (s). If g is unlinked to f at s, f ( s ) gives no information at all about g(s); g(s) may be anything in g(S). Partial linkages of g to f at s are typically expressed in the form of constraints, or, in q u a n t u m mechanics, as selection rules. Interchanging the roles of f and g in the above discussion, we obtain a dual concept of the linkage of f to g at a state s. Once again, the two concepts of linkage are not in general equivalent. The linkage concept represents another way to compare the information provided about a state set S by a pair of observables. Total linkage means that the two observables convey essentially the same information; unlinked observables convey information which is, in a precise sense, "orthogonal". However, it should be noted that linkage is not a topological concept; though it is related to the notions of stability and bifurcation, it is not equivalent to them. Let us now see how the descriptions of a state set S provided by two observables f, g may be incorporated into a larger description. Formally, given the two equivalence relations RI, R0, we can obtain a new relation Rio by writing

R fg = R f ~ R o. This means that s Riq s' if and only if f (s) - - f (s') and g (s) = g (s'). Note that we do not assume the existence of an observable h such that R h = R f o . We now assert that there always exists an injective mapping

re: S/RIg-* S/R I x S / R g = f (S) x g(S). That is, we can always map the states which we can discern through the application of both f and g into the cartesian product of the state spaces we obtain through employing J and g separately. The image of S/RIg in f ( S ) x g(S) can also be used as a "state space" for S, of higher dimensionality than the original spaces f (S), g (S). The fact that the mapping rc is generally into is an

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expression of the linkage between f and g: if f and g are totally linked at every state, this image is one-dimensional (essentially the diagonal map); if f and g are unlinked at every state, the mapping is onto all of f (S) x g (S). The above concepts provide the basic ingredients for investigating the interrelationships of system descriptions. Mathematically, nothing more sophisticated than the algebra of equivalence relations on a set, and a small amount of metric topology is involved. Nevertheless, many surprising results are obtained; for instance, the adding of a new observable to a previous description may cause the description to collapse to a much lower dimension (i.e. be expressible with a smaller set of state variables than previously) and conversely. For a fuller discussion of these matters, we refer to the forthcoming m o n o g r a p h cited previously. So far, the discussion has been entirely static. We now enlarge the discussion to include dynamical considerations. We do this by first restricting attention to the dynamics of the special systems called meters through which our observables are defined. We shall then see that general dynamics, involving change of state in arbitrary systems through the interaction of their states, is in fact a corollary of the properties of meters and our basic hypothesis that all physical events can be expresssed in terms of observables evaluated on states. Heuristically, a meter is itself a system in which dynamics can be induced through interaction with the states of a system S. We assume only that the meter is initially in some "reference state", or "zero-state" too. Upon interaction of m0 with a state s s S, a change of state (i.e. a dynamics) is induced in the meter, which carries it asymptotically to a state re(s). These asymptotic states are themselves parameterized or labeled by real numbers; the number corresponding to the state re(s) is then assigned to s as the value of the observable defined by the meter. The basic proposition underlying our analysis of dynamics is that every dynamical interaction between systems can be locally represented in terms of the dynamics induced on meters. For any change of state is a physical event, hence by our basic hypothesis, must be representable in terms of the evaluation of observables on states. But the evaluation of observables on states can be represented in terms of change of state of meters; therefore an arbitrary change of state is, locally, essentially a meter. The paradigm of arbitrary dynamics then becomes: two meters looking at each other. In general, the vehicle for state transitions in a state set S is an automorphism T: S ~ S. A dynamics is an assignment of such an automorphism to each instant of time;i.e, it is a one-parameter group { Tt} of such automorphisms. The nature of these automorphisms in general is determined locally through the identification of local dynamics with meter dynamics. In this way, a general representation of dynamics, in terms of the observables defined locally by the corresponding meters, can be obtained.

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I~rom this procedure for representing arbitrary dynamics we can obtain insight into the appropriate modes for the qnalysis of dynamics. This involves the inter-relation of dynamical processes induced by particular observables with system descriptions obtained through the employment of, in general, other observables. To deal with such questions, we must first study the effect of a single automorphism T : S ~ S on a system description; how does the automorphism appear when viewed through an observable f ? The appropriate notion here is the compal ibiIity of the automorphism with the equivalence relation RI; Tis compatible with R I if and only ifs Rj s' implies Ts R ! Ts'; i.e. T does not split the .y-classes. If Tis not compatible, T does not even induce a mapping on S/R~; its effect is apparently random in S/R I, and can only be expressed through concepts like transition probabilities (a concept closely related to linkage, or constraint). If Tis compatible, it may appear to induce a mapping of S/R I into itself 0.e. a map without an inverse, which will appear as an irreversible process), or a mapping of S/R I onto itself, which may be the identity (i.e. the action of Tis invisible to f). From such considerations we may pass to the study of one-parameter groups of such automorphisms (i.e. dynamics), and the effect of such dynamics on a system description through observables like f It is apparent (a) that not every set of observables J will allow us to see a dynamics at all; (b) that the sets of observables f which will allow us to see dynamics in S/Rj are sharply circumscribed by the automorphisms in {Tt}, and hence by the observables involved in generating the dynamics itself. What we would like to do in the analysis of dynamics is to decompose a system description of that dynamics into subdescriptions, on each of which a dynamics is induced, and from which the original dynamical description can be reconstructed. The type of formulation we have been developing allows us to characterize when this is possible, in terms of relationships between the observables generating the dynamics and those employed for a representation or description of the dynamics. Let us conclude this section by mentioning a few ramifications of the concepts developed above. One of these is the connection between dynamics and linkage. It is easy to see that dynamics can establish linkage relations between observables. For instance, the very essence of a meter is to establish a linkage between observables defined on a system S and observables defined on the meter. Conversely, dynamics can break linkages established by anterior dynamical interactions. Thus, dynamics can be viewed as the vehicle by which linkage relations are shifted within and between systems. Such puzzling phenomena as emergence can be understood in a coherent way as a consequence of the establishment and breaking of linkage relations through dynamical interactions. Another important concept in physical interactions is that of symmetry. In its

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most basic form, a symmetry operation is one which is visible in one description of the system but invisible in another. Now a set of symmetry operations forms a group, which can be regarded as a subgroup of the group of automorphisms of a set S. Thus we can discuss symmetries in exactly the same terms as we discussed dynamics; indeed, symmetry provides an alternate language for the development of many of the concepts we have employed above. Once again, we refer to the monograph previously cited for fuller details.

III. An Application: Biological Observables. Let us conclude our brief review of the theory of observation and description with a sketch of a specific application of the techniques developed above. Now our entire thesis has been that all system descriptions arise through the choice of observables, and that such choices in general lead to inequivalent descriptions. Indeed, we have argued elsewhere (Rosen, 1977) that the number of different (inequivalent) descriptions of a system available to us underlies our notion of the complexity of a system (thus making complexity also a contingent concept, dependent on the particular observables available to us). The problem is then one of interrelating inequivalent descriptions; this is of a quite different character from that traditionally found in physics, where it is assumed that all observers, and hence their descriptions, are related to one another by simple coordinate transformations. In physics, a "law of nature" is something on which all observers can agree: as we have seen, the study of complex systems requires essentially inequivalent observers; this may be the reason there are so few "laws" in biology. We have already mentioned some previous work (Rosen, 1960) on quantum theory in genetics. In that work, we suggested that the observables on which the transfer of "primary genetic information" depended might not be the same as the ones with which the physicist traditionally deals. That is, we suggested that the vehicle through which system interactions take place need not be the same as the vehicle through which we, through our meters, interact with these systems. At first sight, this is hard to reconcile with quantum physics, which is dominated by the notion of a Hamiltonian; on the other hand, the upshot of our work in this area was that there can exist microphysical systems for which no Hamiltonian could be directly defined. Hence there is ample room for perfectly physical observables (i.e. involved in the interactions between physical systems) which were different from the ones with which the physicist has heretofore been mainly concerned. Such observables would be characterized by the fact that the), induce dynamics which can make discrimination~ between states which appear identical to the traditional observables of physics. How would we go about demonstrating the existence of such observables ? Let us suppose that we have a system S, which we can observe directly through meters which define a set F = {fl,..., f,} of observables of S. As we have seen,

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this will give rise to a description of the form S/RF, where R F = R f l ( h R f z ( - h ... ~ R f .

Now let us suppose that the states of S can interact with the states of another system S', through some observable 4~. We suppose further that we can observe S' directly, through a family of observables G = {gl, ..., gin}- We thus have a diagram of the form S

s,,R~

4~

~S'

S'/R~

where the dotted arrow represents a dynamics, and a corresponding asymptotic state, in S', induced through interaction with an element of the quotient set S/R e . N o w the dotted path ~2qb can be regarded as representing a meter f o r the observable ~) oj S. That is, the dynamical interaction between S and S', coupled with our own capability to observe S' directly through G, provides us with a new description of S. This description can be compared with the description of S which we obtain directly with F. In particular, we can ask whether the ne'~ description of S will bifurcate with respect to the original description. As we pointed out earlier, a bifurcation represents a situation in which essentially new information about a system is being conveyed by a description, with respect to some reference description. In the present context, a bifurcation would mean precisely that the observable ~b, involved in the dynamical interaction between S and S', is not one of the observables in F. It would thus mean precisely that the observable q~ through which this interaction takes place is not one which we can measure with our meters. We would thus have found an observable of the kind suggested in our previous work. Such a conclusion would have profound implications for the prevailing ideas of reductionism, which asserts that all biological interactions can be understood in terms of physical ones; o r more precisely, that they can be understood in terms of the specific observables with which contemporary physics is concerned. The identification of an observable like ~b, crucially involved in a biological interaction, but distinct from these "physical" observables, would not only be an explicit indication that this narrow interpretation of

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reductionism is untenable, but would also be of the greatest significance for physics itself. In particular, it would indicate that, far from biology being swallowed up by physics, physics may rather be enormously extended by biology, to the same extent that it was extended by the discovery of the observable now called spin. It is worth pointing out that the procedure we have sketched here has been explicitly applied by S. Comorosan in a study of the observables involved in enzyme-substrate interactions (cf. Comorosan, 1977). His experimental work indicates that simple enzyme substrates may exist in a class of states which appear physically indistinguishable to our meters, but which can be split by enzymes; the enzymic discrimination appears as a small but significant modification of enzyme rate. Such a result, if experimentally confirmed, would serve precisely to specify an observable of the kind we have suggested; in turn, the formalism we have described above provides the theoretical framework in terms of which such results may be understood. We would suggest that other kinds of bio@namic interactions can be studied in a similar way, according to the scheme sketched in this section; the results should be most interesting and most profitable. It may also be remarked that the formalism we have developed applies to any situation involving discrimination, classification or recognition in terms of numerical (or in fact, any other) kinds of invariants. There is no essential distinction between the evaluation of an observable on a state and, say, the extraction of a feature from a pattern; thus our formalism for measurement and observation goes over into a theory of pattern recognition. There is likewise no essential distinction between assigning a number to a state by a meter and the assigning of a n u m b e r to an object through a process of computation; thus our formalism applies equally well to such diverse situations as the classification of topological spaces through topological invariants in mathematics and the taxonomic classification of organisms and societies. The essential conceptual unity of all these areas, and their embodiment in a single formalism, seems also to be of some value.

LITERATURE Comorosan, S. 1977. "Biological Observables." Progress m Theoretical Biology, Vol. 4. AcademicPress. Pattee, H. H. 1970. "Can Life Explain Quantum Mechanics?" Quantum Theory and Beyond, Ed. E. W. Bastm pp. 307- 319. Cambridge University Press. Rashevsky,N. 1954. "Topology and Life." Bull. Math. Bml., 16,317--348. Rosen, R. 1958a. "A Relational Theory of Biological Systems."Bull. Math. Biol., 20, 245-260 --. 1958b."'The Representation of BiologicalSystemsfrom the Standpoint of the Theory of Categories." ibid., 20,317 342. --. 1959. "A Relational Theory of Biological Systems II." ibid., 21,109- 128.

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--. 1960. "A Quantum-Theoretic Approach to Genetic Problems." ~bld., 22,227 256. • 1968. "On Analogous Systems." ibid., 30,481--492. .1971. Dynamical System Theory in B~ology. Wiley, New York. .1972. "Are the Dynamics of a System Operationally Determinable ?'" J. Theor. Btol., 36~ 635- 640. - - - . 1977. "Complexity as a Systems Property." Int. J. Gen. S~stems, 3, 227 232. • 1977. The Measurement and Representation oJ Natural Systems. Elsevier. in press. yon Neumann, J. 1955. Mathematical Foundatiolzs of Quantum Mechamcs. (R. T. Beyer, trans.) Princeton University Press.