The Parallels of Social Constructivism

Bab 5

The Parallels of Social Constructivism

1. Introduction

Today’s burgeoning intellectual climate views all human knowledge as a social construction. Mathematics, the last bastion of certainty in knowledge, has been trying to resist this current of thought. But as we have seen, more and more authors are joining the stream and viewing mathematics as a social construction. Of course social constructivism, the particular philosophy expounded above, is just one account of mathematics from this perspective. Not many alternatives are yet formulated in the philosophy of mathematics, but beyond, parallel views of mathematics and knowledge abound in other disciplines. This chapter explores some of these parallels, showing how overwhelming the intellectual current is becoming.

2. Philosophical Parallels

A. Sceptical Philosophy

The most central claim of social constructivism is that no certain knowledge is possible, and in particular no certain knowledge of mathematics is possible. Concerning empirical knowledge, this thesis is one that is subscribed to by many philosophers. These include continental sceptics beginning with Descartes; British empiricists such as Hume; American pragmatists such as James and Dewey; some modern American philosophers such as Goodman, Putnam, Quine and Rorty; and modern philosophers of science, including Popper, Kuhn, Feyerabend, Lakatos and Laudan.

Throughout a number of these strands of thought there is agreement that empirical knowledge of the world is a human construction. Beyond those cited, this view is shared by Kant and his followers, who see any knowledge of the world as shaped by innate mental categories of thought.

Scepticism concerning absolute empirical knowledge has grown to be the dominant view. However, until Lakatos (1962) the full extension of scepticism to mathematics was not made. Since then, it has gained partial acceptance, whilst remaining controversial. Social constructivism is an attempt to extend Lakatos’ sceptical approach systematically to a philosophy of mathematics.

However, social constructivism does not entail the fullest form of scepticism, such as cartesian doubt. For it accepts the existence of the physical world (whilst denying certain knowledge of it) and accepts the existence of language and the social group. Both the physical and the social worlds play an essential part in the social constructivist epistemology. As a commentator on Wittgenstein says: ‘Doubt presupposes mastery of a language-game.’ Kenny (1973, page 206) Social constructivism is sceptical about the possibility of any certain knowledge, particularly in mathematics, but it is not sceptical about the necessary pre-conditions for knowledge.

B. The Private Language Problem

One challenge for the social constructivist account of subjective knowledge is the ‘private language’ problem. If an individual’s concepts are personal constructions, how are they able to communicate using a shared language? Why should different mathematicians understand the same thing by a concept or proposition, when their meanings are personally unique? May not each have a private language, to refer to his or her own private meanings?

Social constructivism overcomes this problem through the interpersonal negotiation of meanings to achieve a ‘fit’. Support for this position, if not the precise form of argument, is widespread. Wittgenstein (1953) first answered the problem, arguing that private languages cannot exist. A number of philosophers commenting on his work, such as Kenny (1973) and Bloor (1983), support the rejection of private languages, as do others including Ayer (1956) and Quine (1960). With regard to mathematics, the private language problem is also considered soluble, for example by Tymoczko (1985) and Lerman (1989), both arguing from a position close to social constructivism.

The solution of the private language problem by social constructivism reflects a substantial body of philosophical opinion. Generally, it is argued that the shared rules and ‘objective pull’ of inter-personal language use makes it public, consistent with social constructivism.

C. Knowing and the Development of Knowledge

The social constructivist philosophy of mathematics treats knowledge as the result of a process of coming to know, including the social processes leading to the justification of mathematical knowledge. Thus it attaches great weight to knowing and the development of knowledge, in addition to its product, knowledge. This emphasis, although far from universal, is to be found in the works of a number of philosophers, including Dewey (1950), Polanyi (1958), Rorty (1979), Toulmin (1972), Wittgenstein (1953) and Haack (1979).

Other authors have looked to an evolutionary model to account for the growth and development of knowledge. This includes the genetic epistemology of Piaget (1972, 1977), and the evolutionary epistemologies of Popper (1979), Toulmin (1972) and Lorenz (1977).

The majority of modern philosophers of science view it as a growing and developing body of knowledge either detached from history (Popper, 1979) or embedded in human history (Kuhn, Feyerabend, Lakatos, Toulmin and Laudan).

Educational thinkers have also stressed the processes and means of knowledge acquisition, as a basis for the curriculum, including, most notably, Schwab (1975) and Bruner (1960).

The process of coming to know relates to practical knowledge and the applications of knowledge. Ryle (1949) established that practical knowledge (‘knowing how’) belongs to epistemology as well as declarative knowledge (‘knowing that’). Sneed (1971) proposes a model of scientific knowledge which incorporates the range of intended applications (models) as well as the core theory. This model has been extended to mathematics by Jahnke (Steiner, 1987). Such approaches admitting practical knowledge or its applications into the traditional domain of knowledge thus parallel aspects of the social constructivist proposals.

The social constructivist account of the nature and genesis of subjective knowledge of mathematics is to a large extent based on the radical constructivism of Glasersfeld (1984, 1989). This has parallels in the thought of Kant, and even more so, Vico, as well as with the American pragmatists and modern philosophers of science cited above.

Thus there is a growing current of thought in modern philosophy which gives a central place in epistemology to considerations of the human activity of knowing and the evolution of knowledge, as in social constructivism.

D. The Divisions of Knowledge

A key tenet of social constructivism, following Lakatos, is that mathematical knowledge is quasi-empirical. This leads to the rejection of the categorical distinction between a priori knowledge of mathematics, and empirical knowledge. Other philosophers have also rejected this distinction, most notably Duhem and Quine (1951), who hold that because the assertions of mathematics and science are all part of a continuous body of knowledge, the distinction between them is one of degree, and not of kind or category. White (1950) and Wittgenstein (1953) also reject the absoluteness of this distinction, and a growing number of other philosophers also reject the water-tight division between knowledge and its empirical applications (Ryle, 1949; Sneed, 1971; Jahnke).

A further parallel is found in ‘post-structuralist’ and ‘post-modernist’ philosophers, such as Foucault (1972) and Lyotard (1984), who take the existence of human culture as their starting point. Foucault claims that the divisions of knowledge are modern constructs, defined from certain social perspectives. Throughout history, he argues, the different disciplines have changed. Their objects, concepts, accepted rules of thought and aims have evolved and changed, even amounting, in extreme cases, to discontinuities. Knowledge, in his view, is but one component of ‘discursive practice’, which includes language, social context and social relations. In evidence, he documents how certain socially privileged groups, such as doctors and lawyers, have established discourses creating new objects of thought, grouping together hitherto unconnected phenomena defined as delinquent behaviour or crime. Elsewhere, Foucault (1981) shows how a new area of knowledge, the discourse of human sexuality, was defined by church and state, to serve their own interests.

Lyotard (1984) considers all human knowledge to consist of narratives, whether literary or scientific. Each disciplined narrative has its own legitimation criteria, which are internal, and which develop to overcome or engulf contradictions. He describes how mathematics overcame crises in its axiomatic foundations due to Godel’s Theorem by incorporating meta-mathematics into an enlarged research paradigm. He also claims that continuous differentiable functions are losing their pre-eminence as paradigms of knowledge and prediction, as mathematics incorporates undecidability, incompleteness, Catastrophe theory and chaos. Thus a static system of logic and rationality does not underpin mathematics, or any discipline. Rather they rest on narratives and language games, which shift with organic changes of culture.

These thinkers exemplify a move to view the traditional objective criteria of knowledge and truth within the disciplines as internal myths, which attempt to deny the social basis of all knowing. This new intellectual tradition affirms that all human knowledge is interconnected through a shared cultural substratum, as social constructivism asserts.

Another post-structuralist is Derrida, who as well as supporting this view, argues for the ‘deconstructive’ reading of texts:

In writing, the text is set free from the writer. It is released to the public who find meaning in it as they read it. These readings are the product of circumstance. The same holds true even for philosophy. There can be no way of fixing readings…

Anderson et al. (1986, page 124)

This offers a parallel to the social constructivist thesis that mathematical texts are empty of meaning. Meanings must be constructed for them by individuals or groups on the basis of their knowledge (and context).

E. The Philosophy of Mathematics

Various modern philosophers of mathematics have views consistent with some if not all of the theses of social constructivism. Here we draw together some of the points of contact between them and social constructivism.

Some philosophers emphasize the significance of the history and empirical aspects of mathematics for philosophy. Kitcher (1984) erects a system basing the

justification of mathematical knowledge on its empirical basis, with the justification transmitted from generation to generation by the mathematical community.¹ An empirical or quasi-empirical justification of mathematical knowledge, drawing on mathematical practice, is also adopted by N.D.Goodman, Wang, P.Davis, Hersh, Wilder, Grabiner, Tymoczko (all in Tymoczko, 1986), Tymoczko (1986a), Stolzenberg (1984), MacLane (1981), McCleary and McKinney (1986), and Davis (1974). Thus a move away from the traditional aprioristic view or justification of mathematics, as advocated by social constructivism, is widespread.

A number of other contributory theses of social constructivism are espoused by philosophers of mathematics. The conventionalist viewpoint is implicit in several of these authors’ work. Those who make it explicit include Stolzenberg (1984), as well as Bloor, Quine and Wittgenstein, cited above, and others mentioned in Chapter 2. In addition, the thesis that the objects of mathematics are reified constructions is proposed by both Davis (1974) and Machover (1983).

Beyond such piecemeal comparisons, two philosophers who have anticipated much of the social constructivist philosophy of mathematics are Bloor (1973, 1976, 1978, 1983, 1984) and Tymoczko (1985, 1986, 1986a). Both argue that objectivity in mathematics can best be understood in terms of social acceptance, and draw upon the seminal work of Wittgenstein and Lakatos.

Although no new paradigm is yet fully accepted, social constructivism sits comfortably in a growing quasi-empiricist tradition. Beyond this, a few contemporary philosophers are beginning to propose approaches to the philosophy of mathematics similar to and coherent with social constructivism.

3. Sociological Perspectives of Mathematics

A. Cultural and Historical Approaches

Several authors offer historico-cultural accounts of the nature of mathematics, treating the relationships between the social and cultural groups involved in mathematics, and the genesis and nature of mathematical knowledge. These include Crowe (1975), Mehrtens (1976), Restivo (1985), Richards (1980, 1989), Szabo (1967), Wilder (1974, 1981) and Lakatos (1976). These authors have offered theories of the development of mathematical knowledge, which relate it to its social, historical and cultural context. In particular, they theorize how the micro social context (i.e. interactions in small groups), in the case of Szabo and Lakatos, or the macro social context, in the case of Crowe, Mehrtens, Restivo, Richards and Wilder, influence the development of mathematical knowledge.

Studies of the micro social context concern negotiation within groups of individuals, leading to the acceptance of logical argument or mathematical knowledge, as well as concepts. Such theories reflect the quasi-empiricist account of the acceptance of knowledge, on the empirical level. Lakatos (1976) offers an account of this type with his conjectured 7 stage pattern of mathematical discovery. As an empirical conjecture this fits here, because it represents an historical parallel with aspects of quasi-empiricism and social constructivism, at the micro social level. Szabo (1967) argues that the deductive logic of Euclid derives from pre-socratic dialectics, with conversation serving as the model. Again, this fits with the social constructivist account.

Studies of the macro social context offer theories of the structural patterns, social relationships or ‘laws’ in the development of mathematical knowledge in history and culture. Many of these are social constructionist accounts, consistent with conventionalism, and hence social constructivism, albeit in a different realm. In this bracket can be included a new breed of histories of mathematics acknowledging its fallibility (Kline, 1980) and its multi-cultural social construction (Joseph, 1990).

Historical and cultural studies of mathematics with a bearing on the philosophy of mathematics draw strength and inspiration from the comparable ‘externalist’ approaches to the philosophy of science, such as those of Kuhn (1970) and Toulmin (1972). Such historical approaches, as well as the philosophy of science, provide parallels and support for social constructivism. Likewise, when the social constructi¬ vist account is supplemented with empirical hypotheses, a theory of the history of mathematics results, as in the quasi-empiricism of Lakatos (1976).

B. The Sociology of Knowledge

A number of sociologial theses offer parallels to social constructivism.

Knowledge as a social construction

First of all, there is the ‘social constructionist’ thesis that all knowledge is a social construction. There is a tradition in the sociology of knowledge supporting and elaborating this thesis, including such theorists as Marx, Mannheim, Durkheim, Mead, Schutz, Berger and Luckman, and Barnes (although the first few named in this list assert that some knowledge, notably mathematics, can be free from social bias). This is the dominant view in the sociology of knowledge, contrasting with the main traditions in philosophy which claim that there is certain knowledge of the world from observations (empiricism) or through abstract thought (idealism).

In the sociology of knowledge there are variations in the degree of relativism ascribed to knowledge. In the extreme case, all human knowledge is seen as relative to social groups and their interests, and physical reality itself is regarded as a social construction. More moderate positions regard knowledge (and not reality) as a social construction, and accept an enduring world as a constraint on the possible forms of knowledge. For example, Restivo (1988a) argues that although the new sociology of science regards knowledge as a social construction, it is better aligned with realism than with simple relativism, with which it has no necessary connection. Such positions are parallel to social constructivism in the assumptions they adopt, although they remain sociological as opposed to philosophical theories. Their existence

suggests the potential fruitfulness of a sociological version of social constructivism, to account for the social structures and development of mathematics.

The ‘strong programme’ in the sociology of knowledge

Bloor (1976) has laid down criteria (the tenets of the ‘strong programme’) that a sociology of knowledge should satisfy if it is to provide a sociologically acceptable account of knowledge. Briefly, these require that for adequacy a theory of knowledge should account for: (i) the social genesis of knowledge; (ii and iii) both true and false knowledge and beliefs symmetrically; and (iv) itself (reflexivity).

Although designed for the sociology of knowledge, it is interesting to apply these criteria to social constructivism. Re (i): the account given evidently accounts for the social genesis of mathematical knowledge. Re (ii): it can be said that social constructivism accounts for beliefs and knowledge without regard to their truth or falsity. For the generation of knowledge by the hypothetico-deductive method has no implications concerning its truth. Social constructivism accounts both for the adoption of new, and for the rejection of old beliefs and knowledge when falsified, or for other reasons, denied acceptance. Like sociological accounts of knowledge, social constructivism is symmetrical in these explanations, in terms of social acceptance, and not in terms of a ‘match’ with a transcendent reality.

Re (iv): Although social constructivism is primarily a philosophy of mathematics, it can be extended to account for itself, at least in part. For it is based on a number of epistemological and ontological assumptions, from which conclusions are inferred. As such it has similar status to that which it ascribes to mathematics, namely an hypothetico-deductive theory, except for differences in subject matter and rigour. Both start with a set of plausible but conjectural assumptions (albeit concerning different realms of knowledge), from which consequences are inferred. In addition, any justification for social constructivism must reside in its social acceptance, directly paralleling its account of mathematics. Finally, social constructivism rejects the analytic-empirical distinction, and views all knowledge as inter-related. Consequently, it is legitimately applicable throughout realms of human knowledge, including to itself. Thus social constructivism may be said to be reflexive, since a parallel account can be applied to itself.

Overall, social constructivism largely satisfies the criteria of the ‘strong programme’. This contrasts with absolutist philosophies, which treat truth quite differently from falsehood, failing to satisfy (ii) and (iii), as well as being unable to satisfy (iv). Whilst in terms of traditional philosophy, this is of limited significance, it suggests that a sociological parallel will satisfy the criteria, auguring well for an over-arching social constructivist theory.

Knowledge is value laden

Third, there is the value laden-ness of knowledge. Values are the basis for choice, and  so to be value-laden is to represent the preference or interest of a social group. Values can be manifested explicitly, as in a conscious act of choice, or tacitly, as in unconscious compliance or acceptance. For example, Polanyi (1958) argues that many of the shared values of the scientific community, such as the endorsement of the scientific consensus, are tacit. However, the traditional view of science and scientific knowledge is that it is logical, rational, objective, and hence value-free.² Both social constructivism and the sociology of knowledge reject this view, for different reasons. The sociology of knowledge asserts that all knowledge is value-laden, because it is the product of social groups, and embodies their purposes and interests.

Social constructivism denies that mathematical knowledge is value-free. First, because it rejects the categorical distinction between mathematics and science, and as is increasingly accepted by philosophers of science, science is value-laden. Second, because it posits a shared linguistic basis for all knowledge, which since it serves all human purposes, it is imbued with human values. The mathematical use of language, formal and informal, attempts to eradicate values, by adhering to objective logical rules for the definition and justification of mathematical knowledge. However, the use of the hypothetico-deductive (i.e. axiomatic) method means that values are involved in the choice of hypotheses (and definitions). Beyond this, there are values implicit in logic and the scientific method.

Although mathematics was thought to epitomize value-free objectivity, alongside the sociology of knowledge social constructivism rejects this belief, arguing that objectivity itself is social, and that consequently mathematical knowledge is laden with human and cultural values.

The reification of concepts

Fourth, there is the reification of concepts, in which they become autonomous, objective things-in-themselves. In sociology, this mechanism was first proposed by Marx, by analogy with the genesis of religion.

…the productions of the human brain appear as independent beings endowed with life, and entering into a relation both with one another and with the human race.

(Marx, 1967, page 72)

He argues that the form of products becomes reified and fetishized into an abstract thing: money, value or commodity (Lefebvre, 1972). Subsequent theorists in this tradition of thought, such as Lukacs, have extended the range of operation of reification to a much broader range of concepts.

Evidently the social constructivist thesis concerning the reification of newly defined concepts has a strong parallel in Marxist sociology. This parallel has been extended to mathematics by Davis (1974) and others such as Sohn-Rethel, as Restivo (1985) reports.

C. The Sociology of Mathematics

The sociology of mathematics is a substantial field of study concerning the social development and organization of mathematics, as the survey in Restivo (1985) indicates. In contrast with the philosophy of mathematics, it is concerned to offer empirical theories of the growth, development and organisation of mathematical knowledge. To achieve this, it tries to account for mathematics and mathematical knowledge as a social construction³ (unlike the traditional philosophies of mathematics). Consequently, the social constructivist philosophy of mathematics offers a parallel with sociologial accounts, but whereas the former is concerned with a logical and conceptual analysis of the conditions of knowledge, the latter is concerned with the social determinants of the actual body of knowledge.

One of the aims of social constructivism is to offer a descriptive philosophy of mathematics, as opposed to the prescription of the traditional philosophies. Thus parallel accounts of mathematics from sociological, as well as historical and psychological perspectives should be possible. Therefore this section offers a preliminary sociology of mathematical knowledge.

A social constructivist sociology of mathematics

From a sociological perspective, mathematics is the name given to the activities of, and knowledge produced by a social group of persons called mathematicians. When linked to social history by a definition like this, the term ‘mathematics’ has an organic, changing denotation, as does the set of mathematicians.

‘[M]athematics’ by 1960, consisted of various subgroups working, to some extent, within different cognitive and technical norms, on different orders of phenomena and different types of problems. What had changed, with some exceptions—such as computing—was the relative numerical strength and status within the overall discipline of groups carrying particular norms.

(Cooper, 1985, page 7)

Subjects (e.g. mathematics) will be regarded not as monoliths, that is as groups of individuals sharing a consensus both on cognitive norms and on perceived interests, but rather as constantly shifting coalitions of individuals and variously sized groups whose members may have, at any specific moment, different and possibly conflicting missions and interests. These groups may, nevertheless, in some arenas, all successfully claim allegiance to a common name, such as ‘mathematics’.

(Cooper, 1985, page 10)

These complexities form a backdrop of the brief, conjectural sociological account of mathematics that follows, in line with social constructivis

(i)                 Mathematicians. At any one time, the nature of mathematics is determined primarily by a fuzzy set of persons: mathematicians. The set is partially ordered by the relations of power and status. The set and the relations on it are continually changing, and thus mathematics is continuously evolving. The set of mathematicians has different strengths of membership (which could in theory be quantified from 0 to 1). This includes ‘strong’ members (institutionally powerful or active research mathematicians) and ‘weak’ members (teachers of mathematics). The ‘weakest’ members could simply be numerate citizens. The notion of a fuzzy set usefully models the varying strengths of individuals’ contribution to the institution of mathematics. Mathematical knowledge is legitimated through acceptance by the ‘strongest’ members of the set. In practice the set of mathematicians is made up of many sub-sets pursuing research in sub-fields, each with a similar sub-structure, but loosely interconnected through various social institutions (journals, conferences, universities, funding agencies).

(ii) Joining the set. Membership of the set of mathematicians results from an extended period of training (to acquire the necessary knowledge and skills) followed by participation in the institutions of mathematics, and presumably the adoption of (at least some) of the values of the mathematics community (Davis and Hersh, 1980; Tymoczko, 1985). The training requires interaction with other mathematicians, and with information technology artefacts (books, papers, software, etc.). Over a period of time this results in personal knowledge of mathematics. To the extent that it exists, the shared knowledge of mathematics results from this period of training in which students are indoctrinated with a ‘standard’ body of mathematical knowledge. This is achieved through common learning experiences and the use of key texts, which have included Euclid, Van der Waerden, Bourbaki, Birkhoff and MacLane, and Rudin, in the past. Many, probably most students fall away during this process. Those that remain have successfully learned part of the official body of mathematical knowledge and have been ‘socialized’ into mathematics. This is a necessary, but not sufficient condition for entry into the set of mathematicians (with a membership value significantly greater than 0). The ‘standard’ body of knowledge will have a shared basis, but will vary according to which subfields the mathematician contributes.

(iii) Mathematicians’ culture. Mathematicians form a community with a mathematical culture, with sets of concepts and prior knowledge, methods, problems, criteria of truth and validity, methodology and rules, and values, which are shared to a varying degree. A number of authors have explored the culture and values of mathematics, including Bishop (1988), Davis and Hersh (1980) and Wilder (1974, 1981). Here we will undertake a more limited inquiry, restricted to the different realms of discourse and knowledge of mathematicians, and their associated values. The analysis given here is three-fold, proposing that mathematicians operate with knowledge on the three levels of the syntax, semantics and pragmatics of mathematics. This is based on the classificatory system of Charles Morris (1945) who distinguishes these three levels in language use. In his sytem the syntax, semantics and pragmatics of a language refer to the formal rule system (grammar and proof), the system of meanings and interpretations, and the nexus of human rules, purposes and decisions concerning language use, respectively. In constructing this system, Morris added to the formal logical levels of syntax and semantics a further level of pragmatics, inspired by pragmatism.

There is also a parallel with the three interlocking systems of language distinguished by Halliday (1978), consisting the forms, meanings and functions of language. In the sociology of mathematics, Restivo (1985) distinguishes the syntactical and semantic properties of an object (following Hofstadter), paralleling the syntax-semantics distinction. Hersh (1988) makes an analogous distinction between the ‘front’ and ‘back’ of mathematics. Restivo (1988) also distinguishes between ‘social’ and ‘technical’ talk of mathematics, paralleling the distinction between the third level of pragmatic considerations and the first two levels taken as one, respectively. Thus precursors of these three levels, in various forms, are to be found in the literature.

The three levels of mathematical discourse proposed are as follows. First of all, there is the level of syntax or formal mathematics. This consists of rigorous formulations of mathematics, consisting of the formal statement and proof of results, comprising such things as axioms, definitions, lemmas, theorems and proofs, in pure mathematics, and problems, boundary conditions and values, theorems, methods, derivations, models, predictions and results in applied mathematics. This level includes the mathematics in articles and papers accepted for conferences and journal publication, and constitutes what is accepted as official mathematics. It is considered to be objective and impersonal, the so-called ‘real’ mathematics. This is the level of high status knowledge in mathematics, what Hersh (1988) terms ‘the front’ of mathematics. This level is not that of total rigour, which would require exclusive use of one of the logical calculi, but of what passes in the profession for acceptable rigour.

Secondly, there is the level of informal or semantic mathematics. This includes heuristic formulations of problems, informal or unverified conjectures, proof attempts, historical and informal discussion. This is the level of unofficial mathematics, concerned with meanings, relationships and heuristics. Mathematicians refer to remarks on this level as ‘motivation’ or ‘background’. It consists of subjective and personal mathematics. It is considered to be low status knowledge in mathematics, what Hersh (1988) terms ‘the back’ of mathematics.

Third, there is the level of pragmatic or professional knowledge of mathematics and the professional mathematical community. It concerns the institutions of mathematics, including the conferences, places of work, journals, libraries, prizes, grants, and so on. It also concerns the professional lives of mathematicians, their specialisms, publications, position, status and power in the community, their work places and so on. This is not considered to be mathematical knowledge at all. The knowledge has no official status in mathematics, since it does not concern the cognitive content of mathematics, although aspects of it are reflected in journal announcements. This is the level of ‘social talk’ of mathematics (Restivo, 1988).

These three levels are the different domains of practice within which mathematicians operate. As languages and domains of discourse they form a hierarchy, from the more narrow, specialized and precise (the level of syntax), to the more inclusive, expressive and vague (the level of pragmatics). The more expressive systems can refer to the contents of the less expressive systems, but the relation is asymmetric.

The hierarchy also embodies some of the values of mathematicians. Namely, the more formal, abstract and impersonal that the mathematical knowledge is, the more highly it is valued. The more heuristic, concrete and personal mathematical knowledge is, the less it is prized. Restivo (1985) argues that the development of abstract mathematics follows from the economic and social separation of the ‘hand’ and ‘brain’. For abstract mathematics is far removed from practical concerns. Since the ‘brain’ is associated with wealth and power in society, this division may be said to lead to the above values.

The values described above lead to the identification of mathematics with its formal representations (on the syntactical level). This is an identification which is made both by mathematicians, and philosophers of mathematics (at least those endorsing the absolutist philosophies). The valuing of abstraction in mathematics may also partly explain why mathematics is objectified. For the values emphasize the pure forms and rules of mathematics, facilitating their objectification and reification, as Davis (1974) suggests.⁴ This valuation allows the objectified concepts and rules of mathematics to be depersonalized and reformulated with little concerns of ownership, unlike literary creations. Such changes are subject to strict and general mathematical rules and values, which are a part of the mathematical culture. This has the result of offsetting some of the effects of sectional interests exercised by those with power in the community of mathematicians. However, this in no way threatens the status of the most powerful mathematicians. For the objective rules of acceptable knowledge serve to legitimate the position of the elite in the mathematical community.

Restivo (1988) distinguishes between ‘technical’ and ‘social’ talk of mathematics, as we saw, and argues that unless the latter is included, mathematics as a social construction cannot be understood. Technical talk is identified here with the first and second levels (the levels of syntax and semantics), and social talk is identified with the third level (that of pragmatics and professional concerns).

Denied access to this last level, no sociology of mathematics is possible, including a social constructionist sociology of mathematics. However, social constructivism as a philosophy of mathematics does not need access to this level, although it requires the existence of the social and language, in general. An innovation of social constructivism is the acceptance of the second level (semantics) as central to the philosophy of mathematics, following Lakatos. For traditional philosophies of mathematics focus on the first level alone.

Sociologically, the three levels may be regarded as distinct but inter-related discursive practices, after Foucault. For each has its own symbol systems, knowledge base, social context and associated power relationships, although they may be hidden. For example, at the level of syntax, there are rigorous rules concerning acceptable forms, which are strictly maintained by the mathematics establishment (although they change over time). This can be seen as the exercise of power by a social group. In contrast, the absolutist mathematician’s view is that nothing but logical reasoning and rational decision-making is relevant to this level. Thus a full sociological understanding of mathematics requires an understanding of each of these discursive practices, as well as their complex inter-relationships. Making these three levels explicit, as above, is a first step towards this understanding.

D. Sociological Parallels of Social Constructivism

The above suggests that social constructivism may offer a potentially fruitful parallel sociological account of mathematics. Such a parallel, highly compatible with social constructivism is already partly developed by Restivo (1984, 1985, 1988) and others. Although sociological parallels do not add weight to social constructivism in purely philosophical terms, they offer the prospect of an interdisciplinary social constructivist theory, offering a broader account of mathematics than a philosophy alone. Mathematics is a single phenomenon, and a single account applicable to each of the perspectives of philosophy, history, sociology and psychology is desirable, since it reflects the unity of mathematics. If successful, such an account would have the characteristics of unity, simplicity and generality, which are good grounds for theory choice.⁵

4. Psychological Parallels

A. Constructivism in Psychology

Constructivism in psychology can be understood in narrow and broad senses.⁶ The narrow sense is the psychological theory of Piaget and his school. Piaget’s epistemological starting point resembles that of social constructivism in its treatment of subjectve knowledge. His epistemological assumptions are developed into the philosophy of radical constructivism by von Glasersfeld, as we have seen. However, Piaget’s psychological theory goes far beyond its epistemological starting point. Fully articulated, it is a specific empirical theory of conceptual development, with particular concepts and stages. It also assumes the narrow Bourbakiste structural view of mathematics, which is not compatible with social constructivism.

The Bourbaki group have been developing and publishing a unified axiomatic formulation of pure mathematics for about fifty years in Elements de Mathematique (see for example, Kneebone, 1963). Their formulation is structuralist, based on axiomatic set theory in which three ‘mother-structures’ are defined: algebraic, topological and ordinal, providing the foundation for pure mathematics. As a view of mathematics, the Bourbaki programme may be criticised as narrow. First, because it excludes constructive mathematical processes, and second, because it represents mathematics as fixed and static. Thus it reflects the state of mathematics during a single era (mid twentieth-century), although this is denied in Bourbaki (1948). It is incompatible with social constructivism because of this narrowness, and because it is a foundationist programme, and hence is implicitly absolutist.

However, the Bourbaki programme is not a philosophy of mathematics, and does not need to defend itself against this criticism. For it can be seen merely as a programme, albeit ambitious, to reformulate the structural part of mathematics. But Piaget views Bourbaki as revealing the nature of mathematics. Thus this criticism can be validly directed at Piaget’s implicit philosophy of mathematics, rendering the details of his psychological theory incompatible with social constructivism. For he takes the three ‘mother-structures’ of Bourbaki as a priori, and assumes that they are an integral part of the psychological development of individuals. This is evidently an error, due to a misinterpretation of the significance of Bourbaki.

Other aspects of Piaget’s theory do offer a parallel to social constructivism. For example, the notion of ‘reflective abstraction’, which allows mental operations to become objects of thought in their own right, accommodates the social constructivist thesis of mathematical objects as reifications. However, much of Piaget’s developmental psychology, such as his stage theory, goes beyond any parallel with social constructivism, and is extensively criticized on both psychological (Bryant, 1974; Brown and Desforges, 1979; Donaldson, 1978), and mathematical grounds (Freudenthal, 1973).

The broad sense of constructivism in psychology is what Glasersfeld (1989) refers to as ‘trivial constructivism’, based on the principle that knowledge is not passively received but actively built up by the cognizing subject. This broad sense encompasses many different psychological theories including the personal construct theory of Kelly (1955), the information processing theory of Rumelhart and Norman (1978), the schema theory of Skemp (1979) and others, the social theory of mind of Vygotsky (1962), as well as the basis of the constructivism of Piaget and his followers. This list indicates some of the diversity of thought that falls under the broad sense of constructivism. What these authors share is a belief that the acquisition and develop¬ ment of knowledge by individuals involves the construction of mental structures (concepts and schemas), on the basis of experience and reflection, both on experience and on mental structures and operations. Many, but not all psychologists in this group accept that knowledge grows through the twin processes of assimilation and accommodation, first formulated by Piaget.

On the basis of their epistemological assumptions alone, both the broad and narrow senses of constructivism offer a psychological parallel to social constructivism. The auxiliary hypotheses of individual constructivist psychologies, such as Piaget’s, may be incompatible with the social constructivist philosophy of mathematics. But the potential for a psychological theory of mathematics learning paralleling social constructivism clearly exists.

A number of researchers are developing a constructivist theory of mathematics learning, including Paul Cobb, Ernst von Glasersfeld and Les Steffe (see, for example, Cobb and Steffe, 1983; Glasersfeld, 1987; Steffe, Glasersfeld, Richards and Cobb, 1983). As they appear to have rejected many of the problematic aspects of Piaget’s work, such as his stages, much of their theory can be seen as parallel to social constructivism, on the psychological plane. However it is not clear that all of their auxiliary assumptions, such as those involved in accounting for young children’s number acquisition, are fully compatible with social constructivism.

No attempt will be made to develop a psychological parallel to social constructivism here, although in the next sections we consider briefly some of the key components of such a theory.

B. Knowledge Growth in Psychology

Following Piaget, schema theorists such as Rumelhart and Norman (1978), Skemp (1979) and others, have accepted the model of knowledge growth utilizing the twin processes of assimilation and accommodation. These offer parallels to the social constructivist accounts of subjective and objective knowledge growth. For knowledge, according to this account, is hypothetico-deductive. Theoretical models or systems are conjectured, and then have their consequences inferred. This can include the applications of known procedures or methods, as well as the elaboration, application, working out of consequences, or interpretation of new facts within a mathematical theory or framework. In subjective terms, this amounts to elaborating and enriching existing theories and structures. In terms of objective knowledge, it consists of reformulating existing knowledge or developing the consequences of accepted axiom systems or other mathematical theories. Overall, this corresponds to the psychological process of assimilation, in which experiences are interpreted in terms of, and incorporated into an existing schema. It also corresponds to Kuhn’s (1970) concept of normal science, in which new knowledge is elaborated within an existing paradigm, which, in the case of mathematics, includes applying known (paradigmatic) procedures or proof methods to new problems, or working out new consequences of an established theory.

The comparison between assimilation, on the psychological plane, and Kuhn’s notion of normal science, in philosophy, depends on the analogy between mental schemas and scientfic theories. Both schemas (Skemp, 1971; Resnick and Ford, 1981) and theories (Hempel, 1952; Quine 1960) can be described as interconnected structures of concepts and propositions, linked by their relationships. This analogy has been pointed out explicitly by Gregory (in Miller, 1983), Salner (1986), Skemp (1979) and Ernest (1990), who analyzes the parallel further.

The comparison may be extended to schema accommodation and revolutionary change in theories. In mathematics, novel developments may exceed the limits of ‘normal’ mathematical theory development, described above. Dramatic new methods can be constructed and applied, new axiom systems or mathematical theories developed, and old theories can be restructured or unified by novel concepts or approaches. Such periods of change can occur at both the subjective and objective knowledge levels. It corresponds directly to the psychological process of accommodation, in which schemas are restructured. It also corresponds to Kuhn’s concept of revolutionary science, when existing theories and paradigms are challenged and replaced.

Piaget introduced the concept of cognitive conflict or cognitive dissonance (which will not be distinguished here). In the social constructivist account of mathematics, this has a parallel with the emergence of a formal inconsistency, or a conflict between a formal axiom system and the informal mathematical system that is its source (Lakatos, 1978a). This is analogous to cognitive conflict, which occurs when there is conflict between two schemas, due to inconsistency or conflicting outcomes. In psychology, this is resolved through the accommodation of one or both of the schemas. Likewise in mathematics, or in science, this stimulates revolutionary developments of new theories.

Overall, there is a striking analogy between theory growth and conflict in the social constructivist philosophy of mathematics and schema theory in psychology, and underlying it, between theories and schemas. Unlike the situation in the philosophy of mathematics, schema theory, as sketched above, represents the received view in psychology, lending support to a psychological parallel for social constructivism.

C. Reification and Concept Formation

The social constructivist philosophy of mathematics distinguishes two modes of concept development, vertical processes of concept formation, involving the reification of concepts into objects, and horizontal processes. These can be elaborated as part of a psychological parallel to social constructivism.

We may conjecture that psychological concept formation involves both vertical and horizontal processes. The vertical processes include the standard processes of concept formation, namely the generalization and abstraction of shared features of earlier formed concepts to form new concepts. Beyond this, we conjecture the existence of a psychological mechanism or, tendency which transforms mental procedures or processes into objects. This mechanism changes a property, a construction, a process, or an incomplete collection into a mental object, a complete thing-in-itself. What is represented as a process, a verb or an adjective becomes represented as a noun. This is ‘reification’ or ‘objectification’. Psychologically, much concept formation has this character. Even in the act of coordinating different perceptions of an external object, in sensory concept formation, we reify the set of perceptions into the concept, an enduring object-representation in a schema.

There is some parallel between this conjectured ‘Vertical’ mechanism and Piaget’s notion of reflective abstraction, the process whereby an individual’s operations, both physical and mental, become represented cognitively as concepts. Thus reflective abstraction includes concept reification, although the former is a broader notion.

A number of other researchers have proposed psychological theories dealing specifically with concept reification (Skemp, 1971). Dubinsky (1988, 1989) includes ‘encapsulation’ as part of his explication of the notion of reflective abstraction. Encapsulation converts a subjective mathematical process into an object, by seeing it as a total entity. Sfard (1987, 1989) has been testing a theory of mathematical concept development, in which operational concepts are transformed into structural concepts, by a process of reification. Both these researchers have empirical data consistent with the hypothesis that a process of encapsulation or reification occurs in vertical concept formation. Thus there is evidence for a psychological process of vertical concept formation, parallel to the social constructivist account, and accounting for subjective belief in platonism.

D. Individualism in Subjective Knowledge

A central feature of the social constructivist theory is that subjective knowledge comprises idiosyncratic personal meanings, concepts and knowledge structures. These are subject to the constraints imposed by the external and social worlds, but this leaves room for considerable variation. A psychological version of this thesis would predict that significant variations in concepts and knowledge should occur between individuals, both within a single culture, and even more so in inter-cultural comparisons. This hypothesis seems to be confirmed empirically, although there is, of course, the methodological problem of comparing private meanings. Any evidence about individuals’ personal meanings and knowledge must be based on inference and conjecture, for subjective knowledge is, by definition, unavailable for public scrutiny.

A number of different psychological approaches provide evidence of the uniqueness of individuals’ subjective knowledge. First of all, there is research on errors in mathematics learning (Ashlock, 1976; Erlwanger, 1973; Ginsburg, 1977). From the patterns observed, it is clear that many errors are systematic and not random. The range of errors observed in learners suggests that they are not taught, and that learners construct their own idiosyncratic concepts and procedures. Secondly, researchers are finding that ‘alternative conceptions’ (i.e., idiosyncratic personal concepts) are also very widespread in science (Abimbola, 1988; Driver, 1983; Pfundt and Duit, 1988). Thirdly, researchers have tried to represent learners’ cognitive structures in mathematics, using a variety of data-gathering methods. Their findings have included spontaneous (i.e., untaught) sequences of procedures in learning arithmetic (Steffe et al., 1983; Bergeron et al., 1986), and unpredictable growth in the links in personal concept hierarchies (Denvir and Brown, 1986).

These approaches illustrate the broad base of empirical and theoretical support for a psychological version of social constructivism. Individuals do seem to construct unique personal meanings and conceptual structures. There are, however, patterns to be found in these constructions across individuals (Bergeron et al., 1986), presumably reflecting the similar mental mechanism generating subjective knowledge, and the comparable experiences and social contexts of individuals.

E. Social Negotiation as a Shaper of Thought

A central thesis of social constructivism is that the unique subjective meanings and theories constructed by individuals are developed to ‘fit’ the social and physical worlds. The main agency for this is interaction, and in the acquisition of language, social inter-action. This results in the negotiation of meanings, that is the correction of verbal behaviour and the changing of underlying meanings to improve ‘fit’. Briefly put, this is the conjectured process by means of which the partial inner representation of public knowledge is achieved.

This thesis is close to the social theory of mind of Vygotsky (1962) and his followers. Vygotsky’s theory entails that for the individual, thought and language develop together, that conceptual evolution depends on language experience, and of particular relevance to social constructivism, that higher mental processes have their origin in interactive social processes (Wertsch, 1985).

Vygotsky’s point is not that there are hidden cognitive structures awaiting release through social interaction. His point is the radical one that they are formed through social interaction. Development is not the process of the hidden becoming public, but on the contrary, of the public and inter-subjective becoming private.

(Williams, 1989, page 113)

Thus Vygotsky’s social theory of mind offers a strong parallel with social constructivism, one that can also be found elsewhere in psychology, such as Mead’s (1934) symbolic interactionism. A further development in this direction is the Activity Theory of Leont’ev (1978), with perceives psychological motives and functioning as inseparable from the socio-political context. Possibly less radical is the move to see knowing as bound up with its context in ‘Situated cognition’ (Lave, 1988; Brown et al., 1989), although Walkerdine (1988, 1989) proposes a fully social constructionist psychology of mathematics. Social constructionism as a movement in psychology is gaining in force, as Harre (1989) reports, and is replacing the traditional developmental or behaviourist paradigms of psychology with that of social negotiation. Harre goes so far as to propose that inner concepts such as self-identity are linguistic-related social constructions.

F. Psychological Parallels

A number of psychological parallels of social constructivism have been explored, including ‘constructivism’ and ‘social constructionism’. Many reflect a dominant view in psychology, contrasting with the controversial position of social constructivism in the philosophy of mathematics. Thus it seems likely that a psychological version of social constructivism, enriched with appropriate empirical hypotheses, could offer a fruitful account of the psychology of mathematics.

5. Conclusion: a Global Theory of Mathematics

Social constructivism is a philosophy of mathematics, concerned with the possibility, conditions and logic of mathematical knowledge. As such, its acceptability depends on philosophical criteria. It has been shown to have more in common with some other branches of philosophy, than with traditional philosophies of mathematics, for it inescapably raises issues pertaining to empirical knowledge, and to the social and psychological domains. Despite raising such issues, no empirical assumptions concerning the actual history, sociology or psychology of mathematics have been made.

Due to the multidisciplinary nature of the issues raised, there is also the prospect of a unified social constructivist account of mathematics. The aim of this section is to propose an overall social constructivist theory of mathematics, incorporating its philosophy, history, sociology and psychology. These are distinct disciplines, with different questions, methodologies and data. What is proposed is an overarching social constructivist meta-theory of mathematics, to provide schematic explanations treating the issues and processes in each of these fields, to be developed to suit the characteristics and constraints of that field. This would result in parallel social constructivist accounts of:

1      the history of mathematics: its development at different times and in different cultures;

2      the sociology of mathematics; mathematics as a living social construction,

with its own values, institutions, and relationship with society in the large;

3      the psychology of mathematics: how individuals learn, use and create mathematics.

The goal of providing such a meta-theory of mathematics is ambitious, but legitimate. Theoretical physics is currently seeking to unify its various theories into a grand theory. In the past century other great strides have been made to unify and link sciences. There have been ambitious schemes to document a shared methodology and foundations, such as the International Encyclopedia of Unified Science. The history of mathematics likewise provides many examples of theoretical unification.What is claimed here is that this is also a desirable goal for the philosophy of mathematics.

There are a number of reasons why such a project is worthwhile. First of all, as mathematics is a single discipline and social institution, it is appropriate to coordinate different perspectives of it, for the unity of mathematics should transcend the divisions between disciplines. A meta-theory which reflects this unity gains in plausibility, and reflects the characteristics of a good theory, namely agreement with the data, conceptual integration and unity, simplicity, generality and, it is to be hoped, fertility.

Secondly, beyond this general argument is the fact of the strong parallels between the social constructivist philosophy and the history, sociology and psychology of mathematics demonstrated above. These are not coincidental, but arise from genuinely interdisciplinary issues inherent in the nature of mathematics as a social institution.

Thirdly, in exploring these parallels one factor has recurred, the greater acceptability of the parallel theses in general philosophy, sociology, psychology and the history of mathematics, than in the philosophy of mathematics. In these fields, many of these theses are close to the received view or a major school of thought. In particular, social constructionist views in sociology and psychology have a great deal of support. This contrasts strongly with the position in the philosophy of mathematics, where absolutist philosophies have dominated until very recently. Thus the call for a social constructivist meta-theory of mathematics is stronger from the surrounding fields than from the traditional philosophy of mathematics.

Fourthly, one of the theses of social constructivism is that there is no absolute dichotomy between mathematical and empirical knowledge. This suggests the possibility of a greater rapprochement than hitherto, between the logical concerns of philosophy, and the empirical theories of history, sociology and psychology. An overarching social constructivist meta-theory of mathematics would offer such a rapprochement. Such a theory is therefore proposed, in the spirit of developing the self-consistent (i.e., reflexive) application of social constructivism.⁷

Notes

1    There is an interesting analogy between deductive proof and Kitcher’s (1984) justification of mathematical knowledge, which might be its source. Just substitute axioms for his basis and inference as the means of deriving each stage from the next.

2    An example of this ‘standard’ view is that of Scheffler: ‘science is a systematic public enterprise, controlled by logic and empirical fact whose purpose is to formulate the truth about the empirical world’. (Brown et al., 1981, page 253)

3    A leading exponent of social constructionism as a sociology of mathematics is Sal Restivo (1984, 1985, 1988). (In addition Restivo, 1984, offers valuable insights into social constructivism as a philosophy of mathematics). David Bloor (1976, 1983) has made major contributions to both the sociology and philosophy of mathematics as a social construction.

4    Restivo (1985, page 192) also suggests, following Struik, that the separation of form from content in the objectification of mathematical knowledge is a product of the prevailing social conditions. The argument is that idealism results from, and provides a solution to problems in social outlook, during periods of social decline, such as the disintegration of the western Roman empire, and the enfeeblement of empire. Similarly Koestler (1964, page 57) suggests that Plato’s idealism was a

response to the decline of Greece. An interesting analogy might be drawn with the development of the rigid philosophy of logical positivism in post-Great war Austria and Germany.

5    The strength of the sociological parallels might be used to direct a charge of sociologism against social constructivism, claiming that it is a sociological theory of mathematical knowledge, which although avoiding overtly empirical matters, remains essentially sociological. My response is that the primary focus is on the general conditions and justification of mathematical knowledge, which is the proper concern of the philosophy of mathematics.

6    ‘Constuctivism’ has many meanings. Below two senses of ‘constructivism’ are distinguished in psychology. In the philosophy of mathematics, ‘constructivism’ encompasses intuitionism and similar schools of thought. The psychological and philosophical senses are quite distinct (Lerman, 1989). ‘Social constructivism’ introduces another sense into the philosophy of mathematics. Social constructivism is also applied in the sociology of mathematics, by Restivo. ‘Constructivism’ also denotes a movement in the history of modern art, with proponents such as Gabo, Pevsner and Tatlin. The account of judicial reasoning of Ronald Dworkin (in his 1977 book ‘Taking Rights Seriously’) is termed ‘constructivist’, according to the Fontana Dictionary of Modern Thought.

Doubtless further ‘constructivist’ schools of thought exist in other disciplines. What they seem to share is the metaphor of construction: the product involved is built up by a synthetic process from previously constructed components.

7    An outcome of the social constructivist meta-theory of mathematics might be to demystify the philosophy of mathematics. For if the meta-theory is possible, then the strict demarcation of the disciplines may be seen as the reification, mystification and even the fetishization of philosophy and mathematics. The force with which the inviolability of the boundaries has been asserted (e.g. by

logical positivists and empiricists) resembles a social taboo. It is surely in the interests of knowledge to offer a rational challenge to such a taboo, even if it is against the interests of the professionals who have created the mystique.