Notes on the civil applications of mathematics

Int. J. Man-Machine Studies (1976) 8, 501-515

Notes on the civil applications of mathematics BRIAN MELVILLE

Birmingham, U.K. (Received 10 June 1976) Mathematics derives its meaning from its civil applications. It enters civil-life as a response to requests for conceptual control in the domination of nature. It is incorporated in a technical cognitive interest. An interesting question arises as to whether mathematics can be related to knowledge-constitutive interests concerned with emancipation. Some of the dangers of a technical-context mathematics entering emancipatory language-games are pointed out. Atkin's contributions to mathematical sociology are examined critically. He offers a piece of mathematics suitable for modelling deprivations as objects of reflection in the social life-world. The way this can enter emancipatory language-games presupposes a consensus theory of truth and a theory of communicative competence. "Nerzhin, his lips tightly drawn, was inattentive to the point of rudeness; he did not even bother to ask what exactly Verenyov had written about this arid branch of mathematics in which he himself had done a little work for one of his courses. Suddenly he felt sorry for Verenyov. Topology belonged to the stratosphere of human thought. It might conceivably turn out to be of some use in the twenty-fourth century, but for the time being . . . . " Alexander Solzhenitsyn

The First Circle (Chapter Nine)

I Mathematical Sociology is a disreputable subject. Some sociologists tend to associate statistics and similar sorts o f inductive mathematics with the book-keeping period of their discipline's history, and quite out of keeping with the renewal of interest in theory and epistemology (Rex~ 1974a, b). However the application of deductive mathematics to the study o f social processes usually degenerates into a high-Carnapian language-game, in which the "theoretical language" comprises roughly that part o f mathematics relevant to empirical science, as well as the terms that cannot be introduced with the aid o f observational predicates (Abell, 1974). According to this viewpoint, theories are finite series of postulates formulated in the theoretical language, and can be understood as a logical conjunction of these postulates. Some of these possess an empirical interpretation as they are united with observational predicates through "rules of correspondence". These rules permit the deduction o f observational propositions from the theoretical propositions, or vice versa (Carnap, 1956). However attractive this meta-theory o f science may be to mathematicians and those concerned with the manipulation o f objectified processes, it will not find favour as a meta-theory for the study o f social processes by those who are trying to reconstruct social theory as a critical theory of society (Wellmer, 1974) or as a hermeneutic-dialectical sociology (Radnitsky, 1970, Part II). 501

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In this note, I shall resurrect Wittgenstein's notion that the meaning of mathematics is contained in its "civil applications" (Wittgenstein, 1967), as a basic argument against simplistic notions that mathematics can explain civil life. However the notion of "civil applications" is too broad to classify our perceptions of human knowledge, and so I shall introduce the concept of "knowledge-constitutive human interests" (Habermas, 1972, Appendix). It turns out that mathematics enters civil applications predominantly through a technical cognitive interest; an interest in the prediction and control of objectified processes; an interest which some social theorists would regard as inappropriate for reflecting on society (Habermas, 1976a, b). An interesting question arises as to whether mathematics can ever enter civil life through an emancipatory cognitive interest; that is an interest in emancipating the human subjects whom the "laws" of social theory are about. An affirmative answer can be given to this question, but typically the mathematics does arise primarily from the contexts appropriate to an interest in technical control. However in Atkin's (1974) contribution to mathematical sociology, there arises out of tile selfreflection of mathematics on its methods in the physical sciences, a mathematical methodology which is "naturally" suited for modelling the objects of self-reflection in the social life-world (Lebenswelt, Schutz, 1972). The structure of this piece of mathematics is such that we can say it is the first piece of mathematics to enter its civil applications with emancipatory intentions as its appropriate object. In a possibly perverse reading of Atkin, I shall argue that mathematics does not provide a meta-language for the discussion of social processes. On the contrary, the application of Atkin's contributions in emancipatory contexts presupposes (in Atkin's terminology) an_ extra vertex-namely the Theory of Communicative Competence (Habermas, 1970a, b; McCarthy, 1973). I! To shift from "everyday" discourse in the social life-world through reflection and to establish generalizations about that social life-world in the form of theoretical concepts presupposes a social use of theory. Sociologists need to examine the contingent connections between the latent domain and the domain of everyday concepts. It is not clear that this task requires a meta-language to mediate what Wittgenstein (1958) calls language-games. It supposes that the intention to enquire behind the meaning apparent in the immanent logic of each language-game is the tendency to become theory. What sort of theory is involved in describing the contingent connections between essence and appearance as a mathematical theory? What sort of human interests restrict our discussion of latent ontology through such a medium 9. Certainly Wittgenstein, in the course of his philosophical development, abandoned the idea of the unique philosophical language together with that of the ideal common language (transcendentally) prior to all natural language games. However, it would be a mistake to over-contrast the picture theory of meaning of the Tractatus with the use theory of meaning in the Philosophical Investigations (Wittgenstein, 1961, 1958). Wittgenstein was no logical atomist, in which "atomic facts in the world" were related to "names in logical sentences". Wittgenstein does not offer an extensional foundation for semantic analysis, for the objects of the Tractatus are not like things in the empirical world which can be individuated extensionally. He claimed that words are capable of

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referring to "objects in the world" only in so far as they are possible components of propositions. Much of his analysis revolves round the ambiguous notion of "structure" implied in the truth-functional structure of systems of propositions "experienced" as logical compulsions, and the practical compulsions of the validity-claiming non-truthfunctional structure of elementary propositions comprising the system. The importance of detaching Wittgenstein from the logical atomists at this point is that one cannot further analyse an elementary proposition by splitting it up into its component names, for the concept of a simple object (in the Tractatus) is more like an instantiation of an irreducible property. The difficulties of making sense of the world in an ideal language conceived as philosophical logic are making themselves felt (Winch, 1969; Ishiguro, 1969). The sense of these difficulties resulted in a theory of meaning which was contextdependent (Wittgenstein, 1958). It is from this vantage point that we can see the Tractatus theory "as a curious combination of a basically correct theory about names, of a mistaken assimilation of complex things and facts, and of a wrong and unnecessary claim about the independence of elementary propositions" (Ishiguro, 1969). In his later work Wittgenstein conceives language-games as monadic unities, but " . . . . transfers the inclusiveness of the unique language divided into a multiplicity of language-games to the individual language-games themselves... Wittgenstein rediscovered the "perfect order" of the ideal language in the perfect order of ordinary languages (of admittedly complex structure). The unbroken intersubjectivity of these languages is that of the logical ideal language itself" (Wellmer, 1974). This sort of perspective would not admit the presence of any (transcendentally) prior language-game, and the prospect for mathematics acting as such a language-game would appear to be bleak. Indeed it seems that as Wittgenstein moved to describing human thought as a motley or collection of intedocking languagegames, he tended to see mathematics not as any inclusive language-game in its own right, but also as a motley--the motley of mathematics. Far from acting as a theoretical language for the description of other language-games, Wittgenstein conceives the motley of mathematics as having meaning only in the civil non-mathematical roles of the concepts investigated (Shwayder, 1969; Richardson, 1976). Two dominant themes in his thinking are t h a t " . , mathematics is an asorted kit of instruments for the conceptual control of civil language and life at large, and the conception of mathematical proof as the perspicuous demonstration of essentials" (Shwayder, 1969). Successful mathematizing would expose and confirmthe conceptual connections in our social life-world. Mathematics is built on the presuppositions of "common-sense" and has contingent implications. If there is a role for mathematics, it rests largely in the conceptual connections in other language-games. This is not the same thing as saying that mathematics can act as a (transcendentally) prior language-game. Wittgenstein polemicizes against the Frege, Hilbert, and Russell-Whitehead type programmes to anchor mathematics to the bedrock of civil language-games through the medium of philosophical logic. To be sure, this philosophical logic is more broadly conceived than would be admitted nowadays under the title of mathematical logic. But according to these views, logic and mathematics are connected to the familiar activities of measuring, counting, and inferring, and are ultimately theories about propositions which formulate facts in the social life-world. However Wittgenstein " . . . thought that the whole logistical programme foundered on a misconception about the relation between mathematics and language. The misconception involved the assumption that

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mathematics with logic could be rearranged as an autonomous discipline. The systems of the Grundgesetze and Principia Mathematica seemed like Muscovite monuments to this mistake... The main difficulty was that neither Frege nor Russell were able to say what they thought the ultimate relation between mathematics and language really was" (Shwayder, 1969). Russell and Frege were driven to conceiving the laws of logic as very general propositions about the world, perhaps about what we ought to take as true. In contrast, Wittgenstein thought that the truths of logic were not general propositions about language, but rather reflections on how we use language. Mathematical or logical proof is now conceived as "demonstration". We exhibit how things are in our language. We do not exhibit the logical derivation of propositions. On this account the proposals to unify mathematics within philosophical logic were absurd, the concern with "consistency" irrelevant, and Goedelian-type proofs on undecidability a sort of technical trick which reveals the syntactic structure of language-games as interlocking through the requirement of an extra axiom from outside the system. Proof as perspicuous demonstration, and not as logical derivation, can be viewed as an attempt to make logical compulsion intelligible. Mathematics unfolds the properties of familiar notions. It is a difficult kind of reflective, intuitive, non-observational knowledge. Unfortunately Wittgenstein restiicts his mathematical examples to mechanics, kinematics, and geometry. The fascination with gears and mechanisms, where the diagram defining the connections is a picture which conveys the sense of rigidity perceived in the machinery. The grammar of compulsion is conveyed through the rigidity of the geometrical picture. Similarly in topology, untying knots becomes a perspicuous demonstration of what it makes sense to show. So the task of mathematics is to demonstrate what is sensible by showing how to apply a rule, by putting ordinary civil concepts (characteristically having a non-mathematic use) into memorable relations. Mathematics is postulated with language, not the other way round. The correlation of the motley of human thought with a motley of mathematics leaves little room for the notion of a pure mathematics. For Wittgenstein, pictures and models are not themselves propositions, but instruments of conceptual control used to regulate what is to count as a proposition and to perspicuously demonstrate connections amongst systems of propositions. The illusion of a pure mathematics is due to an inadmissable indulgence in implicit realism--the switching back and forth between empirical and conceptual questions--inadmissable if a pure mathematics is to fend for itself. In any theory of language-games, the conceptual control provided by pure mathematics belongs to the human processes of enlightenment. Pure mathematics can register a sense of mastery of conceptual control in that one has apprehended the range of the application of a concept. I think that Wittgenstein might have argued that the differential and integral calculus illustrate the application of the concept "infinite", and Goedel's work on inconsistency reveals mastery of the notions of "recursive calculation". To detach mathematics as a separate language-game is to fail to see that the material and temporal potentiality of capability and opportunity in the social life-world is not the same thing at all as the timeless adverbial potentiality of logical and mathematical possibility. If mathematics is concerned with human enlightenment as an instrument of conceptual control, then the "logic" of mathematical discovery is a structure of proofs and refutations in human discourse (Lakatos, 1963-1964). Primitive conjectures lead to attempted proofs. Counter-examples to the conjecture lead, through the examination of the perspicuous demonstration we call proof, to the "deduction" of guilty "hidden lemmas".

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The hidden lemma is then made explicit in the statement of the conjecture, thus arriving at a proof-generated condition. But this condition is dependent on the structure of mathematics as a process of conjecture and refutation in human discourse. To separate the sub-strata of meanings in civil-life of concepts refined in proof-generated conditions as a basis of a free-floating mathematics detached from civil-life, is to ignore the constitution and conditions of human knowledge. A sociology which relies on mathematics to oscillate between the latent structure of society and ordinary language-games (Abell, 1974) is in danger of forgetting that mathematics also oscillates between the empirical and logical worlds. To forget this is to forget that mathematics mediated as it is in modern universities and research establishments can be the subject of sociological investigation and critique (Thomas, 1972).

III Most of Wittgenstein's examples indicate that mathematics enter civil-life through what later critical theorists would call a technical cognitive interest (Habermas, 1972, Appendix) If we reformulate the idea of human history, as a history of a symbol-using species, in which the concept of self-reflection by the species on its history becomes the main objection to the idea of one-dimensional necessary and irreversible social evolution of a reified species-subject, then we arrive at the notion of human thought as the abandoned stages of reflection. Reflecting on the nature of human knowledge is a task strewn with pitfalls, but it seems reasonable to ground human knowledge from the perspectives of the human interests involved. The way we apprehend reality seems to be grounded in three categories of knowledge-constitutive interests. Firstly, there is a technical cognitive interest, in which the interest in knowledge enters as the manipulation and control of the phenomena in question. Prototypically physics is the embodiment of this interest. Modern control theory regarding the manipulation of objectified processes, in which the logic of explanation is also the logic of prognosis is a paradigm for this knowledgeinterest. Secondly hermeneutic cognitive interests refer to the practical knowledge gained from the elaboration of the meanings of the phenomena in question. Textual criticism and its generalizations into historical and cultural explanation exemplify bermeneutics (Gadamer, 1975; Wolff, 1975). Finally emancipatory cognitive interests are concerned with emancipating the human subject, whom the laws of knowledge are about. Characteristically Freud's model of the psycho-analytic encounter between analyst and analysand, and Marx's analysis of the class framework of capitalist society for the emancipation of the proletariat represent this category of knowledge-interest (Habermas, 1972). However we must not confuse the psychological and epistemological meaning of "interest". It is not important that any part of science is developed because of the subjective interest of the scientist in its final use. What is important is that the very logical and meta-theoretical structure of theories predetermines their possible application. For example, nomological knowledge can de used in a technical application, but a hermeneutic type of interest such as historical or sociological knowledge can only be used if it affects the self-understanding of the speaking subjects involved. The attempt to relate cognition and interest in this way is to show that human knowledge is not something which is developed independently of life-practices. This serves to warn us of a pure mathematics and rekindle interest in Wittgenstein's notion that mathematics derives its meaning from life-practice or its civil applications. Critical theorists would

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doubt the presence of a realm of pure theory. What can be asserted is that there are structural differences between action and experience on the one hand, and an area of theoretical discourse where the main task is to check out truth claims. It is against this background, that we can make sense of Wittgenstein's profound attempt to reduce mathematics to a motley in the kalaidescope of human thought. An older generation of critical theorists would have maintained that mathematics as a tool of the Enlightenment enters civil life in the form o f a technical cognitive interest. "The programme of the Enlightenment was the disenchantment of the world; the dissolution of myths and the substitution of knowledge for fancy". Mathematics is a major tool of abstraction and "abstraction... treats its objects as did fate, the notion of which it rejects: it liquidates them." But the objects are known in so far as they can be manipulated. The potentiality of objects is turned to human ends. Mathematical representation becomes universal interchangeability. "What is different is equalized. That is the verdict which critically determines the limits of possible experience. The identity o$ everything with everything else is paid for in that nothing may at the same time be identical with itself... In its neo-positivist version, science becomes aestheticism, a system of detached signs devoid of any intention that would transcend the system: it becomes the game which mathematicians have for long proudly asserted is their concern". Mathematics is the antithesis of art which alone retains a connection with the banished world of the mythic and enchantment. The unknown is exiled: " . . . when in mathematical procedure the unknown becomes the unknown quantity of an equation, this marks it as well-known even before any value is inserted. Nature before and after the quantum theory is that which is to be comprehended mathematically; even what cannot be made to a g r e e . . , is converted by mathematical theorems in the anticipatory identification of the wholly conceived and mathematical world with truth; enlightenment intends to secure itself against the return of the mythic. It confounds thought and mathematics" (Horkheimer & Adorno, 1973). These grave and heady assertions seek to identify the connections between the growth of the enlightenment conceived as the growth of instrumental reason, the domination of man and nature, the repression of the mythic, and mathematics as the desirable model of human reason. But even if we suspect such a high-flown dialectical analysis, we would still have to incorporate Wittgenstein's insights on the pragmatics of scientific and mathematical languages deriving their meanings from the sub-stratum of civil life, dominated by the intention to manipulate objects. Scientism, or the ideal of a unified science, an ideal partly unified by the use of mathematical language as a means of ordering the social life-world through symbolic abridgements, remains as a driving force of intellectual life in our modern scientific civilization. In turn this force depends on a traditional concept of theory, wanting to derive everything from itself, which succumbs to unacknowledged external conditions. This leads to practical consequences of a restricted scientistic consciousness, concealing the connection between knowledge and interest. Apel subsumes the notion of ideal mathematical languages, entering discourse as "scientism", under the notion of the "abstractive fallacy": " I f we abstract from the pragmatic dimension of symbols, there can be no human subject of the reasoning process. Accordingly, there can be no reflection upon the predetermined conditions of why reason is possible. What we do get is an infinite

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hierarchy of meta-languages, meta-theories containing (and concealing) the reflexive competence of man as a reasoning subject" (quoted in Habermas, 1973). To be sure, the denial of reflection as such forms no part of mathematics, but the history of mathematics is littered with a continuous rewriting of mathematics in such a form as to conceal the history of its derivation as a form of knowledge divorced from the external conditions which gave it life (Thomas, 1972). Mathematics is intimately connected with a traditional concept of theory which derives from ancient classical philosphy (Kolakowski, 1972). It was essential to ancient philosophers with ontologies based on the distinction between a realm of Being (logos) purged of inconstancy and uncertainty, and the realm of the mutable and perishable (doxa), Through contemplation (theoria), the philosopher abandons himself to the immortal order, and "he cannot help bringing himself into accord with the proportions of the cosmos and reproducing them internally" (Habermas, 1972, Appendix). The manifestation of these proportions is viewed mathematically, and the soul enters the order of the cosmos through relating the power of reflection to the beauty of its order. Our present day conceptualizations of modern science borrow the old ontology of a world of appearance independent of the knower, and the basic theoretical attitude which apprehends those structures of reality which permit the explanation of singular phenomena. The triumph of nominalist attitudes requires the use of conceptual instruments which describe ideal states through the symbolic abridgements of mathematics. However, there is a crucial difference between ancient philosophy and modern technical science. Whereas the ancients were concerned with the dialectic between the internal reproduction of the cosmos and ethical life, modern science reduces essences to the world of appearance viewed as an objectified world suitable for technical control (Marcuse, 1968, Chapter 2). Mathematics is now less concerned with its status as metaphysics in the apprehension of reality, and becomes crucially identified with entering practice (praMs) as the instrumental subjection of nature. For a science with a technical cognitive interest, "theories comprise hypotheticp-deductive connections of propositions, which permit the deduction of law-like hypothesis with empirical content. The latter can be interpreted as statements about the covariance of observable events; given a set of initial conditions, they make predictions possible. Empirical-analytical knowledge is thus possible predictive knowledge" (Habermas, 1972, Appendix). For such knowledge, the appearance of objectivity is grounded in basic mathematical statements seemingly purged of reflective content, but concealing the varying ways in which civil-life enters mathematics predominantly as requests for the clarification of instrumental control. Consider the classic linear plant, which can be modelled by a linear state-space equation system with time-invariant matrix operators. Each component of the plant is connected with a mathematical symbol as isomorphism. A mathematician can bluntly declare t h a t " . . , control theory does not deal with the real world, but only with mathematical models of certain aspects of the real world; therefore the tools as well as the results of control theory are mathematical" (Kalman, 1969). However this way of talking exhibits all the abuses of the "abstractive fallacy". Although linear mathematics can resolve flagrant ambiguities in control-engineering language-games, it disguises that the meaning of the choice of linear vector spaces only acquires that prejudged meaning from the traditions of control engineering. Linear mathematics may well deduce important theories about controllability, observability, and reachability, which explain the success

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of the engineer in controlling the linear plant. It brings analytical awareness to what control engineers already knew, and clarifies the comparative lack of success with nonlinear systems. It also suppresses the history of the mutual interacting development of linear mathematics and practical engineering. It emphasizes the attitude prevalent among mathematicians that their concerns are self-subsisting and independent from civil-life. Wittgenstein may well have declared that the activity of being a mathetical control theorist is attached for its meaning to the activity of being a control engineer. The two subjects have developed with interactive connections. It doesn't make much sense to separate them. Proof as perspicuous demonstration of controllability shows what is already known. This is only, possible because of the intimate connections between the two subjects, and because the two groups can reflect on matters of mutual concern. Now it may well be that in the nexus of tradition that the suppression of the reality of the reflexive competences engaged in technified control leads to no adverse effects on the development of the respective subject matters. But what if the paradigm of technical control is carried over into the fields of socio-economic processes, and social control is viewed as technical control ? What will be concealed in mathematical control models of economic planning? What is the meaning of mathematical symbols in these contexts? Who can reflect on them, if these sciences have nothing to say about the ways in which reflective competence could enter their subject matter? IV A thesis that mathematics is a "motley" suggests that mathematics can also enter language-games devoted to emancipating mankind. However, it may be instructive to reflect on the fatal consequences which may ensue, when such language-games borrow from a mathematics devoted to technical control. Consider Marx's critique of classical political economy in which he argues that the composition of labour-time destroys the semblance of freedom, in which the free labour contract conceals the specific class content of social coercion. This critique is in no way methodologically below the level of what may be expected from a reflexive critical theory of society. "It rests upon the same intuition of the limited nature of the formal presuppositions of objective knowledge" (O'Neill, 1972, Chapter 16). But the critique does require the use of simple mathematics to illustrate the points. The last part of Marx's life was a hard struggle with algebra, differential calculus, and numerical examples. Indeed Engels was unable to refrain from adding a note on Marx's behalf in the preface to Capital, volume II" "Firmly grounded as Marx was in algebra, he did not get the knack of handling figures, particularly commercial arithmetic... But knowledge of the various methods of calculation and exercise in daily practical commercial arithmetic are by no means the same, and consequently Marx got tangled up in his computations". However limited Marx's mathematics, it was sufficient for him to elaborate capitalist reproduction schemes. T h e s e . . . "schemes play a closely defined and specific role in his analysis.., and they are designed to solve a single probleni and no other. Their function is to explain how and why an economic system based on "pure" market a n a r c h y . . , does not lead to continuous chaos and constant interruptions.., but instead functions "normally"--that is with a big crash in the form of an economic crisis breaking out (in Marx's time) once every seven or ten y e a r s . . . The function of the reproduction schemes is thus to prove that it is possible for the capitalist mode of production to exist at all" (Mandel, 1975). To be sure, Marx moves

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the discussion of classical political economy into what we would nowadays call the theory of economic growth. Here knowledge of the "various methods of calculation" (implying ultimately a knowledge of control theory) may have led Marx to the Von-Neumann model of economic growth and "a huge cut might have been made in the history of economic theory" (Morishima, 1973). It is pointless to bemoan that Marx never discovered that technical control theory could enable capitalist societies to "steer the economy". At best it only provides insights into the rise of Keynesian theories of state intervention in a mixed economy. Much more serious is the endorsement by Marx's students of a scientific praxis t h a t " . . , overlooks the alienation of objectivity separated from its subjective sources in the historical decision to treat nature . . . . 'mathematically'" (O'Neill, 1972). Control-theoretic experimentation with systems of expanded reproduction lead to an economistic view, in which the political control of society is obtained through a "socialist" technical control of production. The decision to treat nature "mathematically", to discover objective laws of economic development as objective prognosis, when taken in conjunction with the reduction of systems of human interaction to systems of human labour, lead to those outcomes we identify as Stalinist praxis (Habermas, 1974, Chapter 4). Perhaps Marx's critique requires a different form of mathematics than that devoted to technical control--a form in which crisis and conflict constitute the essence, living in a tense and unstable relation with the domain of appearance.

V "Space" is a deictic concept--that is, it is a concept crucial for our understanding of "pointing", "demonstrating", and for referring to objects as "objects". The selfreflection of mathematics requires a mastery of the concept, which results in homology theory. Such a theory is a procedure that associates with every topological space an algebraic object expressing some of the topologic properties of that space; that is, it is a classification procedure. If the amount of topologic information is restricted, then the transmission into algebraic terms will be less involved and possibly more apparent (Atkin, 1974a, Appendix C). Nowadays the history of this struggle to gain mastery over the concept of"space" is suppressed, and the advanced mathematics student approaches homology theory directly with a grounding in the Eilenberg-Steenrod axioms (Eilenberg & Steenrod, 1952). However, the simplicial homology theory was the first one to be completed and deals with a restricted, but interesting case, the homology theory of polyhedra. The main tool of this early algebraic topology was the concept of "incidence". For each algebraic dimension of an oriented simplicial complex, an incidence matrix could be constructed, enabling the Betti numbers and torsion coefficients to be calculated. The conjecture that these easily determined constants associated with the incidence matrices were topological invariants was "proven" later along so-called classical lines by Alexander (1926). If incidence matrices were tools for perspicuous demonstration, then group theory became the language for handling the gradedstructure of the topological space--a crucial move, if one wishes to discuss the restrictions which may attach to other algebraic objects associated with a complex in a cohomology theory. Atkin's q-analysis is a procedure for generalizing the zero-order Betti number to higher dimensions of the complex, but it is a procedure which retains the general features

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of the language of homology theory. The topologic properties extracted by q-analysis transmit information about the local structure of a simplicial complex, whereas the invariants of homology theory are concerned with global structure. The main advantage of this procedure would appear to rest in the principle that restrictions in the "free" movement of co-simplices can be related more directly to the local structure of a simplicial complex (Atkin, 1973, 1974a). Atkin has applied these tools to the study of objectified processes, in which the specific dynamic patterns studied by the physicist in the realm of appearance "are tied to" an underlying spatial structure which can be represented geometrically and algebraically as a simplicial complex. If the underlying spatial structure is restricted to a Euclidean threedimensional space, then interest will, by necessity, be only concerned with the dynamics of the observables. Much of the mathematics entering civil-life as a technical interest in control automatically assumes a relatively simple complex as a "backeloth" to the dynamics of the observables. Not only is awareness of the backcloth suppressed, but it can be conveniently ignored by diverting the whole scientific concern to the dynamics of the co-simplices (observables), until such time when this procedure is felt a barrier to intellectual progress. Atkin shows how the history of physics can be reinterpreted as in terms of a cohomology of observations, in which an implicit (and suppressed) backeloth is restored. The nature of the backcloth "permits" the possible behaviour of the observables. Physics is now "formulated in such a manner as to show the heart of its methods, as opposed to the fruits of its labours" (Atkin, 1965, 1971, 1972, 1974a, Chapter 5). It is tempting to translate this model into the study of social processes. Surely, it will be argued, the data collected by our social accounts minded statisticians are observables or "patterns" expressing the properties of the latent backcloth. A knowledge of the obstruction in the backcloth can offer some explanation on the restrictions in the movement of social patterns. The main advantage of Atkin's model lies in the suggestion of a latent social structure, giving rise to restrictions in the "free" movement of the cosimplices--the social patterns. This represents a considerable conceptual advance over the current employment of mathematical models in the study of social processes. These conventional models are concerned primarily with the dynamics of social patterns (in terms of other social patterns) with a view to controlling these social patterns. Mathematically there are no restrictions to the movements of such patterns. However, paradoxically, conventional models are calibrated against a historical sequence of patterns (data), in which a latent social structure obstructed the movement of the patterns in the data (for example, see Fisher (1971) on the neo-classical production function). Also the social scientist can note that conventional models are used by scientific experts in state bureaucracies, government funded research establishments, and monopoly capital corporations to maintain and extend the dominant value systems (Habermas, 1971, Chapters 4--6). There is a deep irony in contemplating on a world, inwhieh mathematical expertise builds on the data output from coercive social systems (i.e. there is an"obstruction" to the "free" movement of social patterns), develops an ideology of social control viewed as technical control of social indicators, legitimizes the rigidity of the social structure by focussing attention on patterns instead of structure, and embeds itself in the social structure as a further obstruction at the cost of maintaining it in essentials. At the theoretical level, Atkin's model restores the notion of a social structure which generates the social patterns. There is no example of a predictive technique (in the control-theoretic sense) in his work. Instead the notion of an implicit structure as an

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object of contemplation (at least) is restored. Here an emancipatory social scientist has a mathematical model for reflecting on the latent social structure behind the social indicators. The model focusses attention on the need to change social structure (and hence human relations) before a "free" change in those patterns can be obtained. To change society one must deal with essentials (latent structure), instead of adjusting appearances (social patterns). However this does raise the question of what can constitute the latent domain for a social scientist. The identification of an appropriate backcloth for the study of social patterns presupposes an understanding of what social structure is, and that its meaning has been recovered in terms of social theory. Atkin has noted two difficulties in translating the model over to the study of social processes (Atkin, 1972). Firstly the physicist is not concerned with a latent domain which is the problem facing a social scientist. The physicist's "static backcloth" is a deietic concept which defines the perceptual field in what it makes sense to say about the "objects" of his world. Laws of indeterminacy and conservation in the "objective" world of physics can be expressed as a co-cyle law in an appropriate cohomology theory. There are no known laws of conservation in social processes, unless they can be experienced by individuals as nemesis. The question of an appropriate perceptual field leads to the second difficulty of defining the social structure which generates the perceptions of individuals in society. Atkin side-steps this difficultyby suggesting that the "'base complex is essentially fluid". However this raises the question of the status of this assertion. To be sure, there is a trivial reason for the fluidity of a base complex in the assumption that we can change social structure. On the other hand, what is problematical is the consciousness of individuals of the coercive effects of that structure in ways which would predispose them to change or maintain that structure. The fluidity of the complex has to be put to an empirical test in terms of the experience of socialized individuals; for to change a social structure implies the awareness of its current deficiencies in individuated members of that structure. Superficially, Atkin offers a so-called meta-language of "structure" to those wishing to approach the study of social processes through macro-descriptions of social structure. However, this strueturalist language may serve to obscure the problem of social lifeworld constitution if it suppresses reflexive competence through geometrical objectification. In the name of a rigourous knowledge, the meaning of knowledge can become irrational in the sense that a structuralist knowledge can be said to describe reality. Univocal relations as isomorphism between geometry and the world have to be recovered in terms of their meaning in ordinary language games, unless the object of the exercise is to suppress the reflexive competence of the human subject. The objects of the social lifeworld are like the objects of the Tractatus; they cannot be individuated extensionally; they are more like instantiations of human practice. But to what extent is there an underlying physicalist practice in Atkin's stress on the importance of well-definedsets for the construction of complexes, urging the importance of Russell's Theory of Types for greater clarity? To be sure, greater clarity is nearly always required; but this constitutes a category mistake if this is viewed as making the "soft" social sciences "hard" and "objective" like physics. Presumably because he is not a social scientist, Atkin has not posed the problem of the social meaning of clarity in the well-definedrelation, which determines the complex. This is a great pity; for what makes Atkin's model an appropriate means for modelling the "objects of reflection" in the social life-world derives from the mathematics. A topological space consists of a collection of open sets of some

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(mathematical) object. In Atkin's complexes, the "holes" in the space are combinatorally related (in some sense) to the "missing sets" as defined by the chosen relation. It is almost "trivial" in the mathematical sense, that restricted topologic information leads to restricted algebraic machinery. But it is not trivial in ordinary language-games if the intangible holes in a complex are related to a clear untrivial problem in the social lifeworld when captured by a well-defined relation. The intangibility of holes can only be prevented if the "holes" relate to the experience of untrivial deprivations, modelled "naturally" via the mathematics. Certainly examples of a superficial structuralist practice can be found in some of Atkin's practice, such as land-use structure determining human activity patterns. But even when Atkin generalizes the concept of social networks (Mitchell, 1969) to an algebraic complex in which we are (or imagine ourselves to be) embedded, the success of this structuralist application in which the possibilities of experience are said to be transmitted through the "social cement" (or chains of social connectivity), depends on the recoverable common-sense meanings of "obstruction" and "free" in ordinary languagegames. Atkin recognizes this in referring to the complex as an analogue for what we intuitively experience. The superiority of Atkin's model over the mathematical sociologists lies in the vistas in which the complex can evoke in emancipatory language-games, whereas the social network remains strictly an object of contemplation. The dangers of an objectivist delusion over the reality of social structure will always remain, unless the validity of claims to appropriate model-use can be assessed in emancipatory languagegames. Fortunately Atkin has successfully demonstrated that his model can render the representational substance of speech-acts with emancipatory intentions. In his study of the political processes and organizational structure of a university, he chooses welldefined relations which depend on the common-sense meanings in ordinary languagegames in the assertion that the traffic in the organizational system is structurally deformed by the organization structure (Atkin, 1974b). Successful mathematizing exposes the "missing committee", but depends on Atkin's membership of a speech community which experienced the deformed life of that community and "knew" what relation to define and model. Here the clarity of mathematics has to be interpreted not as making something formerly ambiguous into something hard and objective in the physicalist sense, but as bringing into light something which was only previously dimly perceived. It is the bringing to analytical awareness of the organizational deficiencies in almost a psycho-analytical sense which determines the success of the application. The tension between the deformed actuality and the enlightened potentiality illustrates the emancipatory interest in this contribution to social knowledge. The application of the model must rest on a consensual foundation in the community that the relations chosen for study are the appropriate ones, that new and perhaps more appropriate relations can be chosen, and that the results of the analysis can be translated meaningfully into ordinary language-games. It is not simply that the meaning of the mathematics is contained in the emancipatory civil application, but the implied opportunity for equal, symmetrical, and uncoercive distribution of opportunities for asking questions, making recommendations, and giving interpretations implied in that application presupposes the emancipatory interest in actualizing the enlightened society; that is, it presupposes the arguments of the theory of communicative competence. Habermas' sweeping attempt to reconstruct the theory of knowledge, as social theory

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as enlightenment in the form of a theory of communicative competence, is based on isolating the speech-act as the basic unit of speech (Searle, 1969) to avoid the difficulties of the early Wittgenstein's struggles with the correspondence theory of truth. The insights of the later Wittgenstein are retained in the sense that ordinary language-games are simultaneously communication and meta-communication. But the special way in which Wittgenstein negates the natural reflexivity of ordinary language-games and the possibilities of self-transcendence guaranteed in them are avoided. For Wittgentsein's " . . . reflexive language analysis could only represent as a demonstrable meaning what was already fully covered by a rule-governed activity" (Wellmer, 1974). Habermas restores the notion that analysis must go beyond the restrictive therapeutic significance assigned by Wittgenstein, to the reordering of language-games which are suspected as a pathologically disturbed form of communication. Habermas' theory implies a consensus theory of truth for assessing and grounding claims to validity. In terms of social theory it implies a reconstruction of society in which "truth" and "justice" can be put to a test of human discourse. It finds little room for a mathematical sociology in the hands of experts based on equating political and technical control. However a theory of communicative competence will always require deictic concepts, and it may well be that mathematics can enter ordinary language-games with emancipatory intentions. If so, it is possible that certain aspects of Atkin's theory and practice can serve as a precursor. Mathematical sociology has always been disputed, but it has not often been disputed by a practice arising out of the self-reflection of mathematics. In all conscience a critical theory cannot become overinterested in matters which stray too far from its own concerns. Certainly mathematics is a human achievement, in which enlightenment is its most significant feature. Yet paradoxically mathematics too frequently enters civil-life as a form of tyranny. To be sure, this can suspend us in the dialectic of controversy. But what is clear is that in our societies, for the first time, it seems possible to detach human behaviour from a normative system linked to the grammar of ordinary language-games, and to integrate it into self-regulated systems of the man-machine type. Mathematics can be suspected of offering symbolic abridgements to an instrumental scientism. Within the new theories of language-games seeking to link truth, reason, and freedom, we need to rewrite the philosophy of mathematics as enlightenment arising out of the self-reflection on its nature and its applications in human practice. I am grateful to Dr R. H. Atkin (Essex University) for numerous conversations on mathematics in human affairs and for teaching me a little about topology. Also J. Lewis (Cambridge University) offered some valuable comments on an earlier draft. Naturally they are not to be held responsible for the assertions put forward in these notes.

References A~ELL, P. (1974). Mathematical sociology and sociological theory. In J. R~x, Eo., Approaches to Sociology. London: Routledge & Kegan Paul. ALEXANDER,J. W. (1926). Combinatorial analysis situs. Transactions of the American Mathematical Society 28, 301-329. ATKIN, R. H. (1965). Abstract physics. Nuovo Cimento, 38, 496. ATKIN, R. H. (1971). Cohomology of observations. In Quantum Theory and Beyond. London: Cambridge University Press. ATION, R. H. (1972). From cohomology in physics to q-connectivity in social science. International Journal of Man-Machine Studies, 4, 139-167.

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B. MELVILLE

ATKIN,R. H. (1973). A Survey of Mathematical Theory. Dept. of Matheniatics, Essex University, mimeo. ATKIN, R. H. (1974a). Mathematical Structure in Human Affairs. London: Heinemann. ATKIN, R. H. (1974b). A Community Study, The University of Essex. Dept. of Mathematics, Essex University, mimeo. CARNAP, R. (1956). The methodological character of theoretical concepts. In H. EE1GL& M. SCRIVEN, EDS., Minnesota Studies in the Philosophy of Science I. EILENBERG,S. • STEENROD,N. (1952). Foundations of Algebraic Topology. Princeton University Press. FISHER, F. M. (1971). Aggregate production functions and the explanation of wages. Rev. Economics and Statistics, 53, 305-325. GADAMER,H. G. (1975). Truth and Method. London: Sheed & Ward. HABERMAS,J. (1970a). On systematically distorted communication. Inquiry, 13, 205-218. HAaERMAS,J. (1970b). Towards a theory of communicative competence. Inquiry, 13, 360-375". HAaERMAS,J. (1971). Toward a Rational Society. London: Helnemann. HABERMAS,J. (1972). Knowledge and Human Interests. London: Heinemann. HABERMAS,J. (1973). A postscript to Knowledge and Human Interest. Philosophy of the Social Sciences, 3, 157-189. HABERMAS,J. (1974). Theory and Practice. London: Heinemann. HABERMAS,J. (1976a). The analytical theory of science and dialectics. In T. W. ADORNO, ED., The Positivist Dispute in German Sociology. London: Heinemann. HABERMAS, J. (1976b). A positivistically bisected rationalism. In T. W. ADORrqO, ED., The Positivist Dispute in German Sociology. London: Heinemann. I-IoRrd-IEIMER,M. & ADORNO,T. W. (1973). Dialectic of Eldightenment. London: Allen Lane. ISnl~URO, H. (1969). Use and reference of names. In P. WINCH, ED., Studies in the Philosophy of Wittgenstein. London: Routledge & Kegan Paul. KALMAN,R. E. (1969). Elementary control theory from the modern point of view. In KALMAN, R. E., FALB,P. L. & ARBm,M. A., EDS., Topics in Mizthematical Systems Theory. New York: McGraw-Hill. KOLAKOWSKI, L. (1972). Positivist Philosophy: from Hume to the Vienna Circle. Middlesex: Penguin. LAKATOS,I. (1963-1964). Proofs and refutations. British Journal for the Philosophy of Science, 14, nos. 53, 54, 55, 56. McCARTHY, T. A. (1973). A theory of communicative competence. Philosophy of the Social Sciences, 3, 135-156. MANDEL,E. (1975). Late Capitalism. London: New Left Books. MARCUSE, H. (1968). Negations: Essays in Critical Theory. London: Allen Lane. MITCHELL,J. C. (1969). Social Networks in Urban Situations. Manchester University Press. MORISmMA, M. (1973). Marx!s E'conomics: a Dual Theory of Value and Growth. Cambridge University Press. O'NEILL, J. (1972). Sociology as a Skin Trade. London: Heinemann. RADIqITSKY,G. (1970). Contemporary Schools of Metascience. Lund: Scandinavian University Books. l~x, J. (1974a). Sociology and the Demystification of the Modern World. London: Routledge & Kegan Paul. REX, J. (1974b). Introduction. In J. REx, ED., Approaches to Sociology. London: Rout!edge & Kegan Paul. RaCHARDSON,J. T. E. (1976). The Grammar of Justification. London: Sussex University Press. Scntrrz, A. (1972). The Phenomenology of the Social World. London: Heinemann. SEAttL1/,J. R. (1969). Speech Acts: an Essay in the Philosophy of Language. Cambridge University Press. SHWAYDER, D. S. (1969). Wittgenstein on mathematics. In P. WrNcn, ED., Studies in the Philosophy of Wittgenstein. London: R.outledge & Kegan Paul. THOMAS, M. (1972). The faith of the mathematician. In T. PAT~iA~, Ed., Counter Course: a Handbook for Course Criticism. Middlesex: Penguin. WELLMER,A. (1974). Critical Theory of Society. New York: Seabury Press.

CIVIL APPLICATIONSOF MATI-IEMATICS

515

WINCH, P. (1969). The unity of Wittgenstein's philosophy. In P. WINCH,ED., Studies in the Philosophy of Wittgenstein. London: Routledge & Kegan Paul. WIT~GENSTEIN,L. (1958). Philosophical Investigations. Oxford: Blackwell. WITTGENSTEIN,L. (1961). Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul. WITTGENSTEIN,L. (1967). Remarks on the Foundations of Mathematics. Oxford: Blackwell. WOLFF, J. (1975). Hermeneutic Philosophy and the Sociology of Art. London:/7,outledge & Kegan Paul.