The form of life. I. It is possible but not necessary

The Form of Life. I. It is Possible but Not Necessary

Yukio-Pegio Gunji

Department of Earth-Sciences

Faculty of Science

Kohe University

Nada, Kohe 657, Japan

ABSTRACT

However we deny naive realism, it is difficult to describe the form of life in a

Newtonian paradigm. Because any formal descriptions of biological systems in-

evitably include the concept of “function,” they entail the problem of self-refer-

ence. The description of function can be compared to that of meaning of a word,

and in philosophy we no longer understand that there exists meaning inherent in a

word of a language. As well as the problem of meaning, we cannot describe the

aspect resulting from the function, hecause we cannot identify the specific function

of a shape or morphology with the relationship between the shape itself and its

environment. Therefore, even if we construct a new formal language in which the

self-contradiction resulting from the self-reference is alleviated, that language is

just one of various possible ones. In this sense, the form of life in formal

description is possible but not necessary. The phrase “not necessary” just refers to

the outside of the formal description. In this paper reference to the outside of a

formal system is discussed in the context of time-space complementarity, and it is

strongly related to the origin of irreversible time. We have to introduce both

forward and backward time, as proposed previously by the author and coworkers,

even in this context.

1. INTRODUCTION

In this paper I discuss the form of life in the context of autonomous

systems. The word “autonomous” has b een used in various contexts. On the one hand, in classical system theory it means just a dynamical description including no explicit temporal term. On the other hand in naive biology it gives rise to various fuzzy concepts referring to organismic, vitalistic features and freedom from man’s control. I here use the word in the sense that an observer cannot control an autonomous system. There-

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fore, in the naive sense, if an autonomous system can be described, then it can be controlled. On the other hand, against this aspect, my aim is to formalize life: give it a formal description. That is a self-contradiction. It gives rise to an essential question: “Is it possible to describe the form of life self-consistently?”

The question mentioned above has been discussed by only a few authors. Rosen [l, 21 and Casti [3, 41 remarked that biological systems essentially consist of a metabolic map and a repair map which transforms the metabolic map. This means that the operand (or stable property of a system) and operator (transformational or evolutionary property of a system) cannot be conceptually separated. Hence they proposed a system which implicitly includes a tool to evaluate the structural stability of the system itself, and called it the M-R (metabolic-repair) system. This approach is very similar to the concept of autopoiesis proposed by Varela [5, S] in that he emphasizes that the state of “self’ coincides with the action of “self.”

M. Conrad [7, 81, K. Matsuno [9], and I [lo-121 proposed a similar but different idea, called unprogrummahility. In general, the mixture between operand and operator entails Godel’s theorem of incompleteness. Hence we cannot describe such systems in good (consistent and complete) language. M-R systems and Varela’s autopoietic systems can be regarded as newly constructed good languages in which self-contradiction of a language or discordance between syntax and semantics is alleviated. On the other hand, the concept of unprogrammability emphasizes that self- contradiction must be conserved in formal description, and that we cannot explicitly cure the self-contradiction in a positive sense. In order to overcome this contradictory situation where we have to describe what cannot be described, N. Konno and I proposed a model with autonomously emerging boundary, including backward time [12].

However, strictly speaking, the word unprogrammability is inadequate and possibly misleading in the sense required. Indeed, we have to discuss why biological systems are unprogrammable in detail, in order to compare this idea with the ideas of M-R system and autopoietic system. Hence I here examine the notion of unprogrammability and replace this word by “possible but not necessary” in the sense of Kripke [13]. I reexamine and reformulate the relationship between forward and backward dynamics proposed before [12], comparing it with a form which is possible but not necessary.

This paper is organized as follows. First I examine the notion of explanation, and show that explanation is possible. Though most biologists often misunderstand the notion, explanation properly has nothing to do with the infinite regression of explanation. Second, I argue that the

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explanation of biological systems is possible but not necessary. Here I examine how to describe the dynamic hierarchical structure observed in biology, and show that any attempt to describe them entails the problem of self-reference. With respect to how to deal with this problem, I show that the ideas on autonomous or biological system can be classified into a Newtonian paradigm, a recursive form which is merely possible, and a form which is possible but not necessary. Third, I reformulate the relationship between virtual forward- and backward-time dynamics [12] in the context of a form which is possible but not necessary.

2. WHAT IS EXPLANATION?

What we explain (the object of explanation) is not a natural system but a formal system. Here natural and formal system are terms of Rosen’s [2]; the former denotes real substances or phenomena, and the latter denotes symbol sequences in a formal description or language. Strictly speaking, a natural system by no means represents raw material or “reality” in Wittgenstein’s sense [14]. It means a system which consists of a word used in natural language and our behaviors related to that word. Real substances (the meanings of words) do not exist, but are imagined or consti- tuted as a result of language games. Of course, a language game consists not only of what we call words, but also of any performative behaviors or actions by us. Here we should call all tokens of our actions words, according to Wittgenstein. The reason why we can use words or languages, despite the fact that there are no meanings, is that words are performative in language games.

We here summarize the relationship between formal and natural systems in language games as shown in Figure 1. On the one hand, a formal system consists of both syntax (axioms and well-defined propositions) and semantics (the meaning of propositions), and on the other hand, a natural system consists of just words and/or names. However, this does not mean that only formal language has semantics instead of performativeness. Note that the semantics of formal language is not meaning (however, for the sake of convenience, we call the meaning of a formal language its meaning value) in Wittgenstein’s sense, and that we can use a formal language not because there exists a unique rule defined by the axioms, but because we do not need to doubt a rule (though it is possible to doubt whether it is possible to say that we follow the rule). Hence the reason why we can use a formal language also results from its performativeness in language games.

A formal system is surrounded by a network of words but is formally separated from that network, because words in a natural system have no

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(A)

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(Natural System)

FK:. 1, (a) A natural system consists of just words. In fact, therr is no axiom which

defines how to use a word. IIence words do not correspond to syntax in a formal system. (II)

A formal system consists of syntax and semantics. (c) A language game is articubed into

formal and natural systems. What we call the meaning does not exist, and we can use a

language without meaning. The meaning results from the performance of the language game, which is the behavior originating in the network of wvrds (repwsentcd by hoken lines).

meaning values, and we cannot compare words in the natural system with the symbols of the formal system by the criteria of meaning values (there is just a topological relation to compare with propositions in the formal system). The network itself constitutes the property of performativeness [Figure l(c)]. Therefore there cannot exist mappings or relations between formal and natural systems in principle. Of course, symbols of the formal system may have reference in the form of a word in the natural system; however, it is mere reference, and we cannot talk about a formal relation.

These ideas are often misunderstood among scientists, and many problems in theoretical sciences result from this misunderstanding. Because there is no meaning of words in principle, we cannot understand symbols in a formal system as an abstraction on words (what we call real phenomena). In view of this, the coding-decoding relation between formal and natural systems proposed by Rosen [2] cannot be formally realized; however his insights are valuable.

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Rosen and Casti took care to avoid naive misunderstanding related to the terms “natural” and “real,” and then introduced the concept of the “observable.” An observable is defined as a character which can be represented by real values. They emphasize that we can explain only observables. In this sense, the observables are not related to natural reality but are just symbol sequences. However, as long as we use the terms natural system and observables, we mistakenly think as if we could abstract some properties of natural reality. By examining the very delicate concept of unprogrammability, we here avoid any problems originating from naive realism. Hence the observable in Rosen’s sense is part of a formal system in this paper.

Now meaning of words is replaced by performativeness. Note that the formal system is also included in language games. In the strict sense there is no meaning of formal language. In fact, the meaning of formal language is just the definitive relation between symbols and/or propositions in syntax. Semantics just gives rise to the definition of topological space. We confirm that the meaning of the formal system in Wittgenstein’s sense is also replaced by performativeness in language games.

The essential point is as follows. Science or scientific explanation consists in defining the relationship between symbols of a formal system. There is no performativeness in scientific texts themselves. Scientific explanation is separate from the natural system itself. Therefore the objects of explanations are not natural phenomena but are documents in the formal system: described (reported) finite collections of symbols. Then explanation consists in constructing the proof sequence from the axiom to the finite collection of symbols as documents, which means defining a specific formal language by specific axioms (Figure 2).

For example, let us consider the case of the Fibonacci series. Generally we believe that the Fibonacci series represents the process of cell prolifer-

Formal Syfitem

Model + l -- + l ** --- + Documents

(Axioms) (Object of Explanation) \ &

Y Proof Sequence (Explanation)

FK:. 2. Schematic diagram of scientific explanation. The explanation consists in constructing a relationship between documents or finite symbols of a formal system. The explanation itself has nothing to do with what we call meaning.


ation. However, though the series refers to the specific natural phenomenon, they are not the same. After all, the transition rule

a t+l=ut+at-l,

where a’ represents the number of cells at the tth time step, does not explain the behavior of cell proliferation but only explains finite collections of symbols such as

u 1,2,3,5 ,... ),(12,15,27,42 ,...) ,... }.

Here I define uf as the number of cells, but the relationship between the expressions (1) and (2) d oes not include the concept of cell. Performative- ness cannot participate in the scientific explanation, and so there is no relation connecting the expressions (1) and (2) with the concept of cells in a formal system. The scientific explanation is completed only in the formal system, where the axiom illustrated by (1) and the documents illustrated by (2) constitute the proof sequence. Here we can conceal performativeness in following rule of mathematics.

Most scientists purport to examine whether documents such as (2) “correctly” abstract the real phenomena in a natural system. However, that is nonsense. To do so, one would have to know the meaning of the real phenomenon. That is impossible in principle. Whether the symbols in (2) refer to cell proliferation or not is not dependent on correct abstraction but is decided a posteriori by performativeness in the language game of biologists.

Hence what we call the relationship between a natural phenomenon (a word in a natural system) and the documents in a formal system cannot be talked about in the explanation itself. If what we call the degree of abstraction from the natural phenomenon is decided in the explanation, we have to estimate that relationship in assuming the meaning of a word in the natural system. Then we have to symbolize or describe the meaning of the word, as if a word could be regarded as an abbreviation of the meaning (most people regard such meaning as explanation). Hence we fall into infinite regression of explanation. However, we must not claim that we cannot describe or abstract the natural phenomena by finite expression, merely because the formal description of the meaning of a natural phenomenon must be infinite. The unprogrammability [7-121 is not related to infinite description at all.

Figure l(c) illustrates the difference between explanation as defined here and explanation as generally understood. In Figure l(c), explanation


as defined here is in the formal sysem. On the other hand, explanation in the general sense is shown by the broken lines connecting the formal system with the words. The broken lines shown in Figure l(c) must exist according to naive realists. Therefore the former version of explanation is possible and has nothing to do with infinite regression. We can explain in the formal system without doubting the idea of explanation itself.

However, even if we accept explanation as defined here, it is not satisfactory to talk about the explanation of a biological system only in terms of the concept of possibility. The explanation of biological system is possible but not necessary. This will be discussed in the next section.

3. EXPLANATION OF LIFE IS POSSIBLE BUT NOT NECESSARY

We can use mathematics and formal languages, not because there are meanings and/or foundations for them, but because all defined symbols and rules are performative in language games [14]. In this context, explanation consists only in defining rules and/or axioms of formal language, and it has nothing to do with reference in natural systems. I here repeat that explanation in this sense is possible.

However, thz explanation of biological systems, or life, though possible, is not necessary. In speaking about biological systems at all, we have to describe the wholeness which consists of the system itself and its environment. Note that environments can by no means be described positively or concretely, and that they just refer to the outside of the system. However, in the context of the preceding section, such an idea may seem false because it looks similar to the idea that environments refer to the outside of a formal system. One who clings to such an idea must believe in meaning or reality outside of the formal system in the same sense as that of naive realism (for my purposes, naive realism is defined by the belief that if X can be described, then X must exist), and he must believe that meaning in natural systems is not subject to finite description, and that the explanation connecting with the natural system falls into infinite regression.

I have shown here that explanation is restricted to formal systems. Hence explanation cannot be impossible due to infinite regression.

Let us now evaluate the significance of the statement that explanation is possible but not necessary in formal systems. Let us consider the concept of time, for example. Most scientists believe that time exists in formal systems; however, documents or finite symbols of formal system themselves have no concept of time. Documents just include the specific order (total or partial order) called time or space. I here call them clocklike time and clocklike space. When one introduces only clocklike time and space


to explain some documents, one assumes that space is independent of time.

For example, symbols in documents refer to the number of cells in an embryo. Here we have to identify the embryo with the number of cells. Insofar as the number of cells is not constant in documents, we must introduce clocklike time or space in the explanation of the formal system and explain the growth, in which the number of cells increases with the progression of the clocklike time. The clocklike time in the formal system is not of course real, but is virtual, and is introduced as a tool of self-consistent explanation in order to identify the documents of the formal system.

As is very well known, biological systems are generally regarded as hierarchical structures; however the notion of hierarchy itself is not very well understood. Now let us consider the following abstract dynamical hierarchical system:

Upperlevel: .** +D+ D + 0 +... ul UJ

. . . +dk- d,,, - d;+, --*en.

. . . + fk - fk,, +.*a

Lower level: * * .

(3)

Suppose that D represents the set of the cell’s states or the intracellular distribution of biochemical substrates, and that R( D, D) represents the horn set of functions from D to D. The functions in P( D, D) represent enzyme switches transforming the intracellular state di into d,j. These assumptions are natural or concordant with the biological documents (see Figure 3) [7, 81. Is it possible to describe this structure self-consistently?

For the sake of convenience I denote an enzyme switch as an element of Y( D, D); however, the statement that an enzyme switch has a function is just an interpretation in the formal system. In documents there are just enzyme sequences, denoted as fk, fk+-,l, fk+s,. . . . Enzymes as symbols have shape or morphology but no function. How can we say that an enzyme switch has a specific function? Imagine that #‘(D, D) represents just a set of enzymes, not a set of functions from D to D. Here we can say that a cell state d, is transformed by an enzyme switch fk into j,(d,) = d k+l, in so far as we can assume a map

D+X(D,D). (4)

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Upper

Lower

level

FK:. 3. Schematic diagram of dynamical hierarchical structure in organisms. Here the

upper-level sequence represents the transformation of cell’s states, and the lower-level one

represents the transformation of enzyme switches, which functions according to a key-lock

relation.

The statement fx-( dk) = d,, , is interpreted to mean that the enzyme fk assigns d, to dksl. Can we conclude that the enzyme fk has a specific function? We cannot, because function in a biological system is different from what we call function in natural language. In general, function suggests performativeness in a natural system. We say that an object has a function if it refers to environments and/or is performative in a universe. On the other hand, here we talk about function in a formal system. Hence it has to be rigorously defined. However, in a later section we xvi11 see that this is impossible. The task of defining a function rigorously in a formal system can be replaced by that of defining the foundation of the function. Note that this approach is very similar to Lufevre’s [15] and Flege’s [14]. The latter assumed that a word has meaning and tried to describe the foundation of a word by the recursive form of thou and I. Therefore the function fk had to be based on the elements of the set D. This is the reason why we need a map (4).

Now define a map D+ fl( D, D), using a categorical commutable diagram. First, the evaluation function ev : D x Y?( D, D) -+ D is defined as

ev(GTfk) =&(4) ( = 4+1)$ (5)

and the transition rule g : D x D 4 D is defined as


For these two maps, there exists a unique map E: D x D+ D x H( D, D), such that the following diagram commutes:

DxY(D,D) *

id,, X 2 : I c\ I

DxD -\,

D 52

where id,] represents an identity map D + D. Hence we can here define a map D-t X( D, D) in the form of 2. However, even when we obtain the map 2, it is not satisfactory. We also need a map X( D, D) * D. This map is required by the biological documents of formal system, because it is reported that the three-dimensional structures of an enzyme switch are decided by the energy of the cell’s state [16].

Therefore we have to assume that an enzyme switch is also transformed by a cell state, which means the description of the function of a cell state in the formal system. Hence we have to define the foundation for the function of d, as well as the foundation for the function of fk (Figure 4). Using a commuting diagram such as

X(D,D)iY(D,D) )X(D,D), IL

(8)

we have to obtain a map .X( D, D) + D. Insofar as we use the diagram (8), we require

D=~(fl(D,D),X(D,D)), (9)

because an element of X( D, D) must be transformed by the cell’s state. The isomorphism (9) represents that for all pe.@(fl( D, D),Y( D, D)) there exists a map 1c/ : T( D, D) --t D with $ ’ : D + H'( D, D) such that the

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. . . -D

FIG. 4. In order to define the function &EY(D, D) which assigns rl,~D to dk+l~D (surrounded by solid line), we have to describe the foundation of the function in the form of a

map (denoted by solid arrow). On the other hand, in order to define the function dkE D which transforms fke H( D, D) (surrounded bp broken line), we need the foundation of the

function represented by the open arrow.

P

I

h /A D

I -=%‘( D, D) H( D, DtD ‘CL

where all squares commute. We obtain #-r”$ = id,(, n). Similarly $“$-I = id, is also obtained. Finally,

DeY”(D, D) (11) is obtained in describing two sequences which [Equation (3)] and which involve the concept documents.

interact with each other of function in biological

Therefore, when we explain dynamical hierarchical structure in biological documents, we have to assume D = 2’( D, D). This statement admits of the following two important interpretations. First, D = X( D, D) represents that the velocity of particles at the upper level is equivalent to that of particles at the lower level. Here we can compare such a dynamical

following diagram commutes:

id

X(D, D) n(r,, X(D, D)

278 YUKIO-PEG10 GUN]1

hierarchical structure to the neural networks, and then the upper-level dynamics corresponds to the intercellular dynamics in terms of neurons, and the lower-level dynamics to the intracellular dynamics in terms of biochemical substrates like CAMP [7-111. As is well known, the velocities of particles at these levels cannot be assumed equal, because the velocity of CAMP movement is smaller than that of electric impulses propagated through neurons. Hence we cannot assume D = Y( D, D). This means that we cannot always ignore intracellular dynamics when we describe intercellular dynamics, and that we cannot describe the dynamical hierarchical system represented by the sequence (3) unless D = Z’( D, D).

In general, when one describes intercellular dynamics, one defines the range of interaction and defines a local transition rule in terms of the cellular interaction, which can operate synchronously on the whole space. In this case synchronous operation means that the velocity of observation propagation is assumed to be infinite, which means that we ignore the finite velocity of intracellular particle motion (i.e. the observation propagation of local observers) [19, 11, 121. If we recognize intracellular dynamics, the definition of a local transition rule with cellular interaction which is universal to the whole space is impossible. Strictly speaking, we can describe the specific local rule for a specific temporal and spatial local site; however, though possible, that is not necessary. The isomorphism D = A“( D, D) is also possible but not necessary, because we arbitrarily chose a specific relation between the velocity of upper-level dynamics and that of lower-level dynamics in the structure (3) in defining that isomorphism. This feature will be discussed in detail in the next section.

Second, note that D = I?( D, D) is regarded explicitly as the essential property of autopoiesis by Varela [5, 61. Motivated by Spencer-Brown’s discussion [17], he thought that operands cannot be separated from operators, which gives rise to the problem of self-reference. [If we regard D as data or semantics, Z’( D, D) can be regarded as program or syntax. The mixture between syntax and semantics leads to self-contradiction.] Hence he emphasized that self-contradiction in logic must be obviated in a newly formulated formal language. Constructing a new formal language without self-contradiction means finding a topological space D which satisfies D = X( D, D), where D is unknown.

Is it possible to avoid self-contradiction in the formal language? As long as X( D, D) represents the function in the formal system, we have to define the foundation of the function, and as mentioned before we cannot give a specific foundation for the function even in a formal system. It is possible to assume that D coincides with X( D, D) up to isomorphism; however, that is just one of the possibilities. It is possible but not necessary. In this sense, the problem of self-contradiction need not be

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solved. Not proving is proving. Self-contradiction itself must be conserved in formal description. However, self-contradiction itself entails description by bad language. Hence, we have to talk about this contradiction, where we have to describe bad language by good language [ll, 121.

Rosen pointed out that in the Newtonian paradigm perturbation reflects only the input state; however in biological systems it is reported that perturbation reflects both the input state and the operator, the metabolic map itself [l, 21. Then he introduced the repair map to construct a new metabolic map, as

(12) where f, 4 represent the metabolic and repair maps, and X and Y represent the sets of input and output states respectively.

In order to introduce a map 6: Y +Z( X, Y ), Rosen uses an evaluation function ev: Z( X, Y) + Y, fixing an element x E X, and then assumes that there exists an inverse function ev-’ = 4. On substituting X = Y = D, this means D = fl( D, D). Here we repeat that we cannot give the foundation on which a specific repair map can be described. After perturbation, we cannot identify the newly repaired metabolic map a priori, but can identify it a posteriori. Any definitions of repair maps are just definitions of possibilities. A repair map is not necessary. The fact that it is possible but not necessary explicitly suggests the indeterminacy of a priori description in principle.

In explaining dynamical hierarchical structure documented in a formal system, which essentially involves the concept of function, we cannot avoid the problem of self-reference. If one neglects this problem, one stays in the Newtonian paradigm. If one intends to solve this problem, one has to assume D = G’?( D, D) explicitly or implicitly. I choose the third way, in which it is found that even D = ,rP( D, D) is just one of the possible explanations. However we explain the documents, we cannot neglect that the defined axiom or model can refer to other possible models. In the third approach we have to estimate how to describe “referring to other possibilities.” Of course, if we concretely describe other possible models, “referring to other possibilities” itself can be beyond many models. It should be remembered that we cannot talk about the outside positively.

4. PART AND WHOLE; TIME-SPACE COMPLEMENTARITY

Here we discuss the significance of a form that is possible but not necessary in detail. In the context of Kripke [13], the problem of naming


should be discussed with respect not only to possibility but also to necessity. Naming is a performance in a language game, and its performativeness is judged among words in a natural system. The word “pen” is performative when I say “Give me a pen” and then you give me a pen. If you need an object referred by the word “pen” and you call that object a glass, the word “glass” is not performative and naming a glass in this case is impossible. Naming is decided with respect to whether it is possible or not.

However, using “pen” for the object generally referred to by the word “pen” does not require the unique environment where the word “pen” is performative. For example, imagine the case that you and I are lost in a desert, and also imagine that we have to draw a map on the ground without a pen. I say “Give me a pen,” and then you give me a stone with which I can draw a map on the ground. Hence the word “pen” is still performative. As shown in this special case, we cannot identify the reference of the word “pen,” which means that there are many possibilities or various environments where the word “pen” is performative. Therefore naming a pen is possible but not necessary.

On the other hand, if I name my cat “Tama,” then the word “Tama” refers to nothing but my cat. It is possible and necessary.

In fact, Kripke proposed both possible truth (or falsity) and necessary truth (or falsity). A proposition which is possibly true can be replaced by the proposition that a logician does not believe that the proposition is false. On the other hand, the proposition which is necessarily true can be replaced by the proposition that a logician believes that the proposition is true. In other words, in the case of possible truth a logician does not find the foundation of truth at all, and in the case of necessary truth a logician thinks that he can believe the foundation of truth. In this sense the concept of necessity refers to the foundation of description.

Kripke showed that with regard to the problem of naming we should discuss not only possibility but also necessity, in environments where we realize possible names. The property of necessity can be compared to Giidel’s theorem of incompleteness. In Giidel’s theorem, introducing metalanguage beyond syntax and semantics leads to the fact that we cannot ignore the possibility that there are other semantics contradicting the axioms of the very syntax we are using. We cannot neglect the circumstance in which the specific semantics is defined or identified by ourselves as observers. Hence the property of necessity has something to do with Giidelness [18].

As mentioned in the last section, even when it is restricted to a formal system, describing biological hierarchical structure in the form of local transition rules entails a form which is possible but not necessary. Because

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of the finite velocity of observation progation, we cannot assume synchronous updating (operation) of the whole space. Even when we introduce asynchronous updating [lq], the order of the asynchronous updating rule cannot be defined a priori. Asynchronous updating is just an approximation.

Now I interpret the property “not necessary” in terms of the relationship between time and space, because the local transition rule itself originally designates clocklike time structure at a local site, universal- ly with respect to time. In view of this, when we adapt local clocklike time to the whole space, the property of necessity arises from the relationship between the definitions of clocklike time and space. Therefore, “not necessary” results from the spatial structure as long as one first defines the local clocklike time. I here concentrate on such a time-space complementarity. This problem is inevitable in describing organizing unity or wholeness. If we observe a hand separated from whole human body in vitro, we cannot find that it is a living human hand. A part cannot in principle be separated from the whole. Indeed, if we constitute the whole by gathering various properties of various parts, the concept of wholeness is by no means found and is not needed. Wholeness or unity must not be deduced, but just be referred to by local description or the description of a part. Though the concept of wholeness should result from a specific local description, we cannot express the wholeness just by concrete local description. Thus reference to wholeness can be replaced by the statement that local description is not necessary. Therefore we can talk about the relationship between part and whole, by using the term “not necessary.”

In most scientific documents, we are faced with ordered structures called time series, and then we give the definition of a part in the form of local dynamics, including what we call clocklike time. Scientific explanation means constructing the relationship among symbols of documents self-consistently, and then introducing dimensional concepts to obviate the contradictions (also see the discussion about embryos in the last section). Hence space is introduced only because there exists contradiction that is not obviated by the introduction of the single order time. If the rule for clocklike time to explain various statements of documents in a formal system cannot be uniquely decided, we cannot help introducing another order, called space. However, we generally believe that space is independent of time, not because that is a logical requirement in all cases, but because the time-space relationship originated from fundamental physics, which is based on reversible time.

In classical mechanics, space can be defined separately from the definition of time, because there is a hidden assumption of infinite velocity of


observation propagation in Newton’s third law (the balance of acting and reacting forces) [9, 11, 121. Hence local description, in the form of forces not including clocklike time, can be regarded as the gradient of a potential, which is nothing but the property of wholeness. In classical mechanics only external force is related to the clocklike time, and so it can be separated from space.

However, the general physics of condensed matter can also be regarded as an extension of classical mechanics, in so far as it is a reversible (or conservative) system. In these models, local clocklike time is defined in terms of the gradient of a potential, and detailed balance is also assumed. Hence the definition of local clocklike time includes not only the shift in the direction of clocklike time but also in that of clocklike space. Note that the spatial components of a property compensate each other because of detailed balance. Therefore, although the definition of time appears to be dependent on space at first glance, space is explicitly separated from time. Hence, in general reversible or conservative systems, we can suppose that time is independent of space.

This aspect is also similar to that in discrete systems. In cellular automata (discrete time, space, and state; synchronous updating) reversible systems are proposed by Fredkin and Toffoli [20] and by Margolus [21]. Margolus’s neighborhood is defined by the function { 0, l}” + { 0, I}” such that (uf”,ofz: - ) - f(uf, uf+i), where af ~(0,l) and f is a bijection. Here the components of the spatial property denoted by i in the definition of clocklike time (the function f itself) are virtually omitted in the discrete version of detailed balance [Figure 5(a)]. On the other hand, Fredkin proposed a reversible rule induced by a rule of elementary cellular automata[22],intheform u~“=f(u~_,,u~,~~+~)-uf-‘,whereuf~{O,l} and f is a rule defined in elementary cellular automata. Such a rule has symmetric structure with respect to clocklike time. Omitting complemen- tary terms in uf_ 1, u:- ‘, and ui, i, it is found that the definition of clocklike time effectively includes no spatial components but just temporal

(A) (B)

tc”pace q i z&l @ FIG. 5. Schematic diagram illustrating a reversible transition rule in cellular automata.

Each box is occupied by state value 1 or 0. (a) Margolus neighborhood, (b) Fredkin

construction. Left: Output states (shaded boxes) are computed using input states (vacant boxes). Right: On removing boxes used for the definition of clocklike time (stippled boxes),

there are remaining boxes (blank). Each remaining box denies the other with respect to the

spatial component. The thick arrow represents the denial relation.


components (i.e., clocklike time or temporal shift is represented here by uj” + uf [Figure 5(b)]. H ence Fredkin’s construction also gives rise to the separation between time and space.

On the other hand, though in irreversible systems there is no foundation for which space can be separated from time, we always assume such separation. Originally, the arrow of time, or irreversibility, resulted from spatial interaction: the so-called microscopic boundary condition. Hence the definition of clocklike time explicitly involves the spatial properties, and it is dependent on spatial structure. However, whenever we formalize any irreversible system, initial conditions are chosen. This is an essential problem. As long as we recognize that the definition of clocklike time depends on space, certain states (initial and boundary conditions) must be arbitrarily chosen. The origin of the initial condition itself must be investigated.

As long as the definition of clocklike time includes spatial components or microscopic boundary conditions, we have to decide how to choose initial and/or boundary conditions. For example, if one defines the clocklike time in elementary cellular automata as “if u:_~ = 1 and ui = 1 and

4+1 = 1, then uf” = 1,” one cannot choose any initial conditions except for ai=” = 1 for i = 1,. . , n (n represents system size). Then can we define the space structure similarly dependent on the definition of time in any case? Before answering this question, we have to examine whether we can define the irreversible clocklike time including spatial properties.

The definition of clocklike time including spatial properties in fact means the formal description of communication. In cellular automata, .;+I = f( uf_ I,af,uf+,), ith as o t 1 ,e assumed that the state values at local sites i - 1 and i + 1 cannot be changed while the state transition happens for a time unit At = 1 at a local site i. Empirically we often observe that state values are simultaneously changed at two local sites. But we suppose that we observe the change at one local site and not at the other (At -+ 0), and hence we can formalize the clocklike time with spatial interaction.

On the one hand, in order to observe asynchronous updating at two local sites, we have to assume At +O. On the other hand, we have to observe two other local sites while At --* 0, which means infinite velocity of observation propagation. However, as discussed in the previous section, we cannot neglect intracellular dvnamics in defining intercellular dynamics in the form ui+’ = f(ui_ ‘t

I>“i>uf+, ). So the velocity of observation propagation here has to be finite [lo-121. Hence, even if we can assume At -+ 0 in order to describe the local transition rule in the form of uf” = f( uj_ , > a;> u;+ 1 ), the function f itself is not defined as a map, but must be a one-to-many mapping.

The definition of clocklike time, or the local transition rule, must be a one-to-many mapping; however, we cannot describe one-to-many map-


pings in terms of the forward clocklike time. Therefore we now understand that the existence of a form which is possible but not necessary is different from nonuniversality in time and space. One-to-many mapping in principle requires the positive aspect in “not necessary.” In this sense, we can replace the term “possible but not necessary” by unprogrammability.

Now the property of “not necessary” cannot be expressed by local transition rules. The property of “just referring” should be positively expressed by the definition of clocklike space, which includes the concept of backward time.

Finally we come to time-space complementarity [lo-121. First we have to express the fact that the local transition rule, or the definition of clocklike time, is defined and simultaneously just refers to other possibilities (the outside of the transition rule itself). Second, reference to the outside in the definition can be replaced by a one-to-many mapping; hence it is expressed in the backward clocklike time. Now recall the definition of irreversible clocklike time as

time : from past to future (in forward time)

using spatial interaction (13) and synchronous updating.

Therefore we have to define complementally the clocklike space as,

space : from future to past (in backward time)

using temporal interaction (14) and asynchronous updating.

This complementarity between time and space is the counterpart of the relationship between forward and backward dynamics in the model with autonomously emerging boundary [12]. In elementary cellular automata, the definition (13) is expressed by

and the definition (14) is expressed by

af,ai+, , 1

t ai+]= g(af_,,a~,aj+‘),

(15)

(16) where the function g operates asynchronously on the whole space. Note that this clocklike time and space, especially the backward time, are not real but virtual in the formal system. In order to express one-to-many


mapping, the combination of forward and backward dynamics is as shown in Figure 6. The one-to-many mappings that arise are themselves real (of course, this reality should be distinguished from the reality in naive realism) in the formal system. Hence, backward time is virtual in two senses.

Finally, in describing documented biological systems or in explaining the reported form of life in the formal system, we have to express the form, which is possible but not necessary. In this form, the aspect of referring only to the outside of the system should be positively expressed. Hence we need both forward and backward time in explaining the form of life. In this paper, we have introduced the concept of the complementarity between clocklike time and space, and it gives rise to a one-to-

(A) Clock-like Time (Forward-dynamics)

(B) Clock-like Space (Backward-dynamics)

(Cl

One-to-many type mapping a prior1

Degenerated result a posteriori .----_ z o---o *‘---

_ ---- -4

t t+1

Frc. 6. Schematic diagram of the relationship between one-to-many mapping and the combinative expression which consists of the clocklike time (in forward dynamics) and the clocklike space (in backward dynamics). First (a) we operate the forward dynamics (a map defined in forward time), and then (b) operate the backward dynamics (a map defined in backward-time). (c) These two operations virtually give rise to a one-to-many mapping.


many mapping in forward time, or a priori, and the degeneration of possibilities a posteriori. This aspect may be related to what we call “time,” or time empirically recognized. We may reemphasize that both clocklike time and space, and both forward and backward dynamics, bring forth “time.”

It is not yet clear how we can express the concept of complementarity. According to the definition (13) and (14), the complementarity between time and space can be described in the categorical structure called the “adjoint” [23]. In the virtual form, to describe the unprogrammability in a formal system including forward time only, it may be possible to obtain a newly formalized mathematical structure using category theory. However, how to embed the form of life in category theory itself is essentially not a mathematical but a biological problem.

5. CONCLUSION

In the naive sense, biological descriptions cannot be separated from the problem of how to describe functions. That in turn leads to the problem of how we can describe the foundation of the function even in a formal system. This problem has been discussed in the domain of philosophy, and in the view of Wittgenstein any such attempt must fail. The reason why we are faced with unprogrammability in describing the form of life is not that we cannot distinguish operand from operator in biological systems, but that we have to take the foundation of operators into consideration in a formal system. Hence, we cannot avoid the unprogrammability in the description of functions. Instead of solving it, we have to see how to dodge this problem. Note that we can describe a bad formal language, or why a specific description entails a bad language, using a good formal language.

Rosen, Casti, and Varela treated the problem of the stability of description itself, by proposing M-R systems and autopoietic systems [l-6]. However, they proposed to describe the stability of a system in metasys- tern terms, which leads to the problem of the stability of metasystems. The fact that the stability of a system can be unprogrammable should itself be projected into a formal description [7-121. Therefore we have to introduce the two aspects-real and virtual forms-into the formal system even if we distinguish formal systems from natural ones in other contexts.

In order to avoid trivial misunderstanding, I here introduce the term “possible but not necessary,” f o 11 owing Kripke [13]. The aspect of Gijdel’s theorem of incompleteness is expressed by the term “not necessary.” When we talk about the unity of organisms, we have to approach wholeness by a description of a part or a local site; however, wholeness cannot be reduced to the accumulation of parts. The description of a part itself


refers to wholeness, which is immediately related to the term “not necessary.” Strictly speaking, the term “not necessary” cannot be replaced by the term “not universal.” Because a description of a local site in forward time only (in the Newtonian paradigm) includes the feature of prediction, it is not possible to describe a dynamics even at a specific temporal and spatial local site a priori; that is possible only a posteriori.

For this reason, a description of a local site has to be a one-to-many mapping in forward time. Hence unprogrammability is an essential problem in the Newtonian paradigm (science as prediction).

The “possible but not necessary” form suggests other possibilities a priori, and also suggests one-to-many mapping. The result of a choice of possibilities is degenerate only a posterior-i. Therefore, the property “not necessary” has to be talked about in virtual backward time. Simultane- ously we should remark that “not necessary” results from the formal description of the relationship between whole and part, or the dynamical hierarchical structure which consists of both inter- and intracellular dynamics. Hence the property “not necessary” is expressed by the concept of relation between parts, which is nothing but the concept of space. The form which is possible but not necessary is embedded in the virtual structure of the time-space complementarity, which can be replaced by the categorical structure of adjointness [23].

This formalization is proposed in order to conceptualize the origin of irreversible systems, and so it is beyond the paradigm of prediction. It may have ethical power against the temptation of control. As a concrete example, this formalization is effective in constructing the theory or software used in autonomously distributed processing and/or biocomputers. In autonomous distributed processing constructed from biochips we have to compute multiple local subsystems or processors at the same time; however, our capability for computing or logical inference is essentially sequential. Thus we are faced with inevitable unprogrammability in using biocomputers. Therefore we have to give up the thought of perfect usage in the sense of control, and will realize the third way, which is neither just description nor perfect control.

I greatly thank Professor K. Ito, N. Konno, and T. Nakamura for detailed

discussions on various topics.

REFERENCES

1 R. Rosen, Some relational cell models: The metabolic-repair system, in

Foundations of &fathematical Biology (R. Rosen, Ed.), Vol. 2(4), Academic,

New York, 1972.

2 R. Rosen, Organisms as causal systems which are not mechanisms: An essay

into the nature of complexity, in Theoretical Biology and Complexity

(R. Rosen, Ed.), Academic, New York, 1985. 3 J. Casti, The theory of metabolic-repair systems, A&. Math. Comput.

28:113-154 (1988).

4 J. Casti, Newton, Aristotle and the modeling of living systems, in Newton to

Aristotle: Toward a Theory of Models for Liring Systems (J. Casti and A. Karlqrist, Eds.), Birkhauser, Boston, 1989.

5 F. Varela, A calculus for self-reference, lnternat. J. Gen. Systems 2:5-22

(1975). 6 F. Varela, Principles of Biological Autonomy, North Holland, New York, 1979. 7 M. Conrad, Adaptability: The Significance of Variability from Molecule to

Ecosystem, Plenum, New York, 1983. 8 M. Conrad, Microscopic-macroscopic interface in biological information

processing, Biosystems 16:345-363 (1984). 9 K. Matsuno, Protobiology: Physical Basis uf Biology, CRC Press, New York,

1989. 10 Y. Gunji and T. Nakamura, Nondefinite form for autopoiesis, illustrating

automata, Proc. Ann. Inter-nut. Conf. IEEE/EMBS 12(4):1778-1780 (1990).

11 Y. Gunji and T. Nakamura, Time reverse automata patterns generated by Spencer-Brown’s modulator: Invertibility based on autopoiesis, Biosystems, to appear.

12 Y. Gunji and N. Konno, Artificial life with autonomously emerging hound- aries, Appl. Math. Comput., to appear.

13 S. Kripke, Naming and Necessity, Basil Blackwell and Harvard U.P., 1980. 14 L. Wittgenstein, Philosophical Inoestigations, Basil Blackwell & Mott, Oxford,

1953.

15 V. Lufevre, The fundamental structures of human reflections, J. Sot. Biol.

Struct. 10:129-162 (1987).

16 M. Conrad, Molecular computing, in Adoances in Computers (M. Yovits, Ed.),

Vol. 31, Academic, Boston, 1990. 17 G. Spencer-Brown, Laws of Form, George Allen Rc Unwin, London, 1969.

18 R. Smullyan, Foreoer Undecided: A Puzzle Guide to Giidel, Knopf, New York,

1987. 19 Y. Gunji, Pigment color patterns of molluscs as an autonomous process

generated by asynchronous automata, Biosystems 23:317-334 (1990). 20 E. Fredkin and T. Toffoli, Conservative logic, Internat. /. Theoret. Phys.

21:219-253. 21 N. Margolus, Physics-like models of computation, Physica lOD:81-95 (1984).

22 S. Wolfram, Statistical mechanics of cellular automata, Rev. Modern Phys.

55(3):601-644 (1983). 23 Y. Gunji, The form of life. II. Time-space adjointness, Appl. Math. Comput.,

submitted for publication.
