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.
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.
