Selasa, 27 Desember 2016
Social
Constructivism and Subjective Knowledge
- Prologue
This
chapter faces a difficult task: that of showing the relationship between
subjective and objective knowledge of mathematics in social constructivism. The
task is difficult for a number of reasons. It skirts the edge of psychologism,
and it needs to conjoin two different languages, theories and modes of thought
that apply to two different realms, the subjective and the objective. Beyond
this, the epistemology underpinning social constructivism is quite slippery to
grasp, since it is claimed that there is no realm where a determinate entity
‘knowledge’ basks in tranquillity. Knowledge, perhaps analogous to
consciousness, is seen as an immensely complex and ultimately irreducible
process of humankind dependent on the contributions of a myriad of centres of
activity, but also transcending them. Science fiction authors (Stapledon, 1937)
and mystical philosophers (Chardin, 1966) have groped for a vision of how the
consciousnesses of individual human beings can meld into a greater whole. But
these provide too simplistic a vision to account for knowledge and culture as
dynamic, cooperative dances uniting millions of thinking and acting but
separate human beings. The seduction of idealism is great: to say that
knowledge exists somewhere in an ultimate form, possibly growing and changing,
but that all our representations of knowledge are but imperfect reflections.
The pull to view human knowledge attempts as parts of a convergent sequence
that tends to a limit in another realm, is almost irresistible.
Once these simplifying myths are rejected, as they
are by social constructivism, there is the complex task of accounting for
knowledge. It is social, but where is the social? Is it a moving dance, a cloud
of pirouetting butterflies, which when caught is no more? Books do not contain
knowledge, according to this account. They may contain sequences of symbols,
carefully and intentionally arranged, but they do not contain meaning. This has
to be created by the reader, although books may guide the reader to create new
meanings. This is subjective knowledge, the unique creation of each individual.
Yet by some miracle of interaction, the way human beings use this knowlege in
their transactions fits together.
The concept of the individual, the knowing subject,
in Western thought is another problem. Since Locke, or earlier, the subject is
a tabula rasa, and gradually knowledge is inscribed on its blank page by
experience and education. But the form and content of mind cannot be separated
like this, and there is no universal form of knowledge that can be written in
our minds. The view that follows is that knowledge has to be created anew in
the mind of every human being, and solely in response to their active efforts
to know. Consequently, objective knowledge is all the while being born anew.
Thus knowledge is more like a human body, with its every cell being replaced
cyclically, or like the river: never the same twice! This is why I called the
epistemology involved slippery!
These are some of the problems that the present
chapter raises and tries to tackle.
- The Genesis of Subjective Knowledge
The
fundamental problem to be accounted for in the growth of subjective knowledge
concerns the acquisition of knowledge of the external world. How can an
individual acquire knowledge of the external world by means of sensory organs
alone? The external world includes other human beings, so acquisition of
knowledge of the external world includes knowledge of human beings, their
actions, and ultimately their speech. Only when we have accounted for the
acquisition of speech can we begin to consider how the substantive structures
of mathematics are acquired. We begin, therefore, by inquiring how any
subjective knowledge is acquired.
- The Construction of Subjective Knowledge
How
does the individual acquire knowledge of the external world? Human beings have
incoming sense impressions of the world, as well as being able to act
physically on the world, and thus in some measure are able to control aspects
of the environment. Clearly subjective knowledge is acquired on the basis of
interaction with the external world, both through incoming sense data and
through direct actions. What is also clear is that these interactions are
necessary but not sufficient for the acquisition of knowlege of the external
world. For the sense data are particulars. Whereas our knowledge is evidently
general, since it includes general concepts (universals), and it allows for
anticipation and the prediction of regularities in our experience. Therefore
some further mechanism is required to account for the generation of general knowledge
of the world of our experience, on the basis of particular items of information
or experiences.
This is precisely the problem that the philosophy of
science faces, but expressed at the subjective level. Namely, how can we
account for (and justify) theoretical scientific knowledge on the basis of
observations and experiment alone? The solution proposed is the same. The
development of subjective knowledge, like that of science, is
hypothetico-deductive. The answer proposed is that the minds of individuals are
active, conjecturing and predicting patterns in the flow of experience, and
thus building theories of the nature of the world, although these may be
unconsciously made theories. These conjectures or theories serve as guides for
action, and when they prove inadequate, as inevitably they do, they are
elaborated or replaced by new theories that overcome the inadequacy or failing
of the previous theory. Thus our subjective knowledge of the external world
consists of conjectures, which are continually used, tested and replaced when
falsified.
Thus the account of the formation of subjective
knowledge is a recursive one. Our knowledge of the world of our experience
consists of private conjectures or theories, which order the world of our
experience. These theories are based on two factors. First, our immediate
experiences of the world, including interactions with it, as perceived and
filtered through our theories. Second, our previously existing theories. Thus
the formation of our subjective theories is recursive in that it depends
essentially on these theories, albeit in an earlier state.
This account mirrors that of Popper (1959), but at
the level of subjective as opposed to objective knowledge. However, it is clear
that Popper intends his account of science to apply only to objective
knowledge, and furthermore, he has nothing to say on the genesis of scientific
theories. As a purely subjective view of knowledge, this view is elaborated by
Glasersfeld (1983, 1984, 1989) as ‘radical constructivism’.
‘The world we live in’ can be understood also as the
world of our experience, the world as we see, hear and feel it. This world does
not consist of ‘objective facts’ or ‘things-in-themselves’ but of such
invariants and constancies as we are able to compute on the basis of our
individual experience. To adopt this reading, however, is tantamount to
adopting a radically different scenario for the activity of knowing. From an
explorer who is condemned to seek ‘structural properties’ of an inaccessible
reality, the experiencing organism now turns into a builder of cognitive
structures intended to solve such problems as the organism perceives or
conceives. Fifty years ago, Piaget characterised this scenario as one could
wish: ‘Intelligence organises the world by organising itself’ (Piaget, 1937).
What determines the value of the conceptual structures is their experimental
adequacy, their goodness of fit with experience, their viability as means for
the solving of problems, among which is, of course, the never-ending problem of
consistent organisation that we call understanding.
The world we live in, from the
vantage point of this new perspective, is always and necessarily the world as
we conceptualize it. ‘Facts’, as Vico saw long ago, are made by us and our way
of experiencing, rather than given by an independently existing objective
world. But that does not mean that we can make them as we like. They are viable
facts as long as they do not clash with experience, as long as they remain
tenable in the sense that they continue to do what we expect them to do.
(Glasersfeld, 1983, p. 50–51)
Constructivism is a theory of knowledge with roots
in philosophy, psychology and cybernetics. It asserts two main principles…(a)
knowledge is not passively received but actively built up by the cognizing
subject; (b) the function of cognition is adaptive and serves the organization
of the experiential world, not the discovery of ontological reality.
(Glasersfeld,
1989, page 162)
This
view accounts for the development of subjective knowledge of the external
world. It explains how an individual constructs subjective knowledge, notably a
theoretical model of a portion of the external world which fits that portion,
and how this knowledge or model develops, improving the fit. It does this
without presupposing that we construct true knowledge matching the given
portion of the world, which would contradict much modern thought, especially in
the philosophy of science. Thus the theory provides an account of how external
reality serves as a constraint in the construction of subjective knowledge, a
constraint that ensures the continued viability of the knowledge. What the
theory does not yet do, is to account for the possibility of communication and
agreement between individuals. For the sole constraint of fitting the external
world does not of itself prevent individuals from having wholly different,
incompatible even, subjective models of the world.
Such differences would seem inescapable. However,
this is not the case. Suitably elaborated, the social constructivist view also
provides an account of the development of knowledge of the world of people and
social interaction, and the acquisition of language. The very mechanism which
improves the fit of subjective knowledge with the world also accounts for the
fit with the social world, including patterns of linguistic use and behaviour.
Indeed, the experiential world of the cognizing subject which Glasersfeld
refers to, does not differentiate between physical or social reality. Thus the
generation and adaptation of personal theories on the basis of sense data and
interactions equally applies to the social world, as the following account
shows.
Individuals, from the moment of birth, receive sense
impressions from, and interact with, the external and social worlds. They also
formulate subjective theories to account for, and hence guide, their
interactions with these realms. These theories are continually tested through
interaction with the environment, animate and inanimate. Part of this mental
activity relates to other persons and speech. Heard speech leads to theories
concerning word (and sentence) meaning and use. As these theories are
conjectured, they are tested out through actions and utterances. The patterns
of responses of other individuals (chiefly the mother or guardian, initially)
lead to the correction of usage. This leads to the generation of an ever
growing set of personal rules of language use. These rules are part of a
subjective theory, (or family of theories) of language use. But the growth of
this theory is not monotonic. The correction of use leads to the abandonment of
aspects of it, the adaptation of the theory and hence the refinement of use.
This subjective knowledge of language is likely to be more procedural than
prepositional knowledge. That is, it will be more a matter of ‘knowing how’
than ‘knowing that’ (Ryle, 1949).
The acquisition of language involves the exchange of
utterances with other individuals in shared social and physical contexts. Such
interaction provides encounters with rule governed linguistic behaviour. In
other words, it represents the confrontation with, and accommodation to,
socially accepted or objective features of language. The acquisition of
linguistic competence results from a prolonged period of social interaction.
During this period, by dint of repeated utterances and correction, individuals
construct subjective theories or personal representations of the rules and
conventions underpinning shared language use. The viability of these theories
is a function of their mode of development. Quine refers to the ‘objective
pull’, which brings about adequate levels of agreement between individuals
utterances and behaviour:
Society, acting solely on overt manifestations, has
been able to train the individual to say the socially proper thing in response
even to socially undetectable stimulations.
(Quine, 1960, pages 5–6)
Halliday
(1978) describes linguistic competency in terms of mastery of three
interlocking systems, namely the forms, the meanings and the (social) functions
of language. Of these, language forms and functions are publicly manifested
systems, which thus lend themselves to correction and agreement. Whilst the
system of meanings is private, the other systems ensure that where they impact
on public behaviour, there is a pull towards agreement.
The uniformity that unites us in communication and
belief is a uniformity of resultant patterns overlying a chaotic subjective
diversity of connections between words and experience. Uniformity comes where
it matters socially...Different persons growing up in the same language are
like different bushes trimmed and trained to take the shape of identical
elephants. The anatomical details of twigs and branches will fulfil the
elephantine form differently from bush to bush, but the overall outward results
are alike.
(Quine,
1960, page 8)
What
has been provided is an account of how individuals acquire (or rather
construct) subjective knowledge, including knowledge of language. The two key
features of the account are as follows. First of all, there is the active
construction of knowledge, typically concepts and hypotheses, on the basis of
experiences and previous knowledge. These provide a basis for understanding and
serve the purpose of guiding future actions. Secondly, there is the essential
role played by experience and interaction with the physical and social worlds,
in both the physical actions and speech modes. This experience constitutes the
intended use of the knowledge, but it provides the conflicts between intended
and perceived outcomes which lead to the restructuring of knowledge, to improve
its fit with experience. The shaping effect of experience, to use Quine’s
metaphor, must not be underestimated. For this is where the full impact of
human culture occurs, and where the rules and conventions of language use are
constructed by individuals, with the extensive functional outcomes manifested
around us in human society.
Bauersfeld describes this theory as the triadic
nature of human knowledge: the subjective structures of knowledge, therefore,
are subjective constructions functioning as viable models, which have been
formed through adaptations to the resistance of ‘the world’ and through
negotiations in social interactions’
(Grouws
et al., 1988, page 39)
The
theory has a number of implications for communication, for the representation
of information, and for the basis and location of objective knowledge. With
regard to communication, the theory imposes severe limits on the possibility of
communicating meanings by linguistic or other means. Since the subjective
meanings of individuals are uniquely constructed (with certain constraints
accommodated in view of their genesis), it is clear that communication cannot
be correctly described as the transfer of meanings. Signals can be transmitted
and received, but it is impossible to match the meanings that the sender and
recipient of the signals attach to them, or even talk of such a match. However,
the ways in which linguistic competence is acquired, mean that a fit between
sender and receiver meanings can be achieved and sustained, as evidenced by
satisfactory participation in shared language games. This view of communication
is fully consistent with the Communication Theory of Shannon (cited in
Glasersfeld, 1989).
However, there is something I want to call
objective, which enters into communication. This is not the informational
content of messages, but the preexisting norms, rules and conventions of
linguistic behaviour that every speaker meets (in some form) when entering into
a linguistic community. These, in Wittgenstein’s (1953) term, are a ‘form of
life’, the enacted rules of linguistic behaviour shared (at least in
approximation) by speakers. These rules, represent the constraints of the world
of interpersonal communication, which permit the possibility of a fit between
senders’ and receivers’ meanings. Such a fit will depend on the extent to which
the actors are drawn from communities which share the same norms of linguistic
competence, as well as on the success of the individuals in reconstructing
these norms for themselves. These norms or rules are objective, in the sense
that they are social, and transcend individuals. However, at any one time, they
are located in the regularities of the linguistic behaviour of the group,
sustained by the subjective representations of them, in the minds of the
individual group members.
A further consequence of the view of subjective
knowledge growth concerns the extent to which meanings are inherent in symbolic
representations of information, such as a book or a mathematical proof.
According to the view proposed, such meanings are the constructions of the
reader. (This view is in essence, Derrida’s deconstructive approach to textual
meanings; Anderson et al., 1986). The linguistic rules, conventions and norms
reconstructed by a reader during their acquisition of language constrain the
reader to a possible interpretation whose consequences fit with those of other
readers. In other words, there is no meaning per se in books and proofs. The
meanings have to be created by readers, or rather, constructed on the basis of
their existing subjective meanings. Within a given linguistic community, the
readers’ private meaning structures are constructed to fit the constraints of
publicly manifested linguistic rules. Thus it is the fit between the readers’
subjective theories of language, brought about by a common context of
acquisition, including shared constraints, rather than an inherent property of
text that brings about a fit between interpretations. However, the social
agreement within a community as to how a symbolism is to be decoded constrains
individuals’ meaning constuctions, giving the sense that there is informational
content in the text itself.
This is consistent with the account given of
objective knowledge in the previous chapter. For it was stated that public
representations of subjective knowledge are just that. Knowledge, truth and
meaning cannot be attributed to sets of marks or symbols. Only the assignment
of meanings to a set of marks, or a symbol system, which ultimately has to be
done by an individual, results in the knowledge or meaning of a published
document. As in communications theory, decoding is essential if meaning is to
be attributed to a set of broadcast codes.
The social constructivist account of subjective
knowledge is also consistent with the conventionalist account of the basis of
mathematical, logical and linguistic knowledge given in previous chapters. For
according to the constructivist view, the growth of the subjective knowledge of
an individual is shaped by interactions with others (and the world). This
shaping takes place throughout a linguistic community, so that the constraints
accommodated by all of its individuals allow shared participation in language
games and activities. These constraints are the objective, publicly manifested
rules and conventions of language. On the basis of these constraints
individuals construct their own subjective rules and conventions of language.
It can be said that these ‘fit’ (but do not necessarily match) since they allow
for shared purposes and interchange which satisfy the participants to any
degree of refinement desired.
One problem that arises from the account of the
constructivist epistemology that has been given is that it seems to necessitate
cumbersome circumlocutions. Knowledge is no longer ‘acquired’, ‘learned’, or
‘transmitted’, but ‘constructed’ or ‘reconstructed’ as the creative subjective
response of an individual to certain stimuli, based on the individual’s pre-existing
knowledge which has been shaped to accommodate rules and constraints inferred
(or rather induced) from interactions with others. Whilst the latter account is
the accurate one from the constructivist viewpoint, it is convenient to retain
the former usage on the understanding that it is merely a façon de parler,
and an abbreviation for the latter.
An analogy for such usage is provided by the
language of analysis in mathematics. To say that a function f(x) (defined on
the reals) approaches infinity is acceptable, provided this is understood to
have the following more refined meaning (provided in the nineteenth century).
Namely, that for every real number r there is another s such if x>s, then
f(x)>r. This reformulation no longer says that the function literally
approaches infinity, but that for every finite value, there is some point such
that thereafter all values of the function exceed it. The two meanings
expressed are quite different, but the convention is adopted that the first
denotes the second. The rationale for this is that an abbreviated and
historically prior mode of speech is retained, which in all contexts can be
replaced by a more precise definition. Likewise, we may retain the use of
terminology of knowledge transmission in situations where there is no danger of
confusion, on the understanding that it has constructivist meaning, which can
be unpacked when needed.
In summary, it has been argued that: (a) subjective
knowledge is not passively received but actively built up by the cognizing
subject, and that the function of cognition is adaptive and serves the
organization of the experiential world of the individual (Glasersfeld, 1989).
(b) This process accounts for subjective knowledge of the world and language
(including mathematics), (c) Objective constraints, both physical and social,
have a shaping effect on subjective knowledge, which allows for a ‘fit’ between
aspects of subjective knowledge and the external world, including social and
physical features, and other individuals’ knowledge, (d) Meanings can only be
attributed by individuals, and are not intrinsic to any symbolic system.
- The Construction of Mathematical Knowledge
It
has been argued that linguistic knowledge provides the foundation (genetic and
justificatory) for objective mathematical knowledge, both in defending the
conventionalist thesis, and subsequently as part of the social constructivist
philosophy of mathematics. What is proposed here is the parallel but distinct
claim that linguistic knowledge also provides the foundation, both genetic and
justificatory, for the subjective knowledge of mathematics. In a previous
section we saw how social (i.e. objective) rules of language, logic, etc.,
circumscribe the acceptance of published mathematical creations, allowing them
to become part of the body of objective mathematical knowledge. Thus we were
concerned with the subjective origins of objective knowledge. In this section
the focus is on the genesis of subjective mathematical knowledge, and it will
be argued that the origins of this knowledge lie firmly rooted in linguistic
knowledge and competence.
Mathematical knowledge begins, it can be said, with
the acquisition of linguistic knowledge. Natural language includes the basis of
mathematics through its register of elementary mathematical terms, through
everyday knowledge of the uses and interconnections of these terms, and through
the rules and conventions which provide the foundation for logic and logical
truth. Thus the foundation of mathematical knowledge, both genetic and
justificatory, is acquired with language. For both the genetic basis of
mathematical concepts and propositions, and the justificatory foundation of
prepositional mathematical knowledge, are found in this linguistic knowledge.
In addition, the structure of subjective mathematical knowledge, particularly
its conceptual structure, results from its acquisition through language.
One of the characteristics of mathematical knowledge
is its stratified and hierarchical nature, particularly among terms and concepts.
This is a logical property of mathematical knowledge, which is manifested both
in public expositions of objective mathematical knowledge and, as will be
claimed here, in subjective mathematical knowledge. We consider first the
hierarchical nature of objective mathematical knowledge.
It is acknowledged that concepts and terms, both in
science and mathematics, can be divided into those that are defined and those
taken as primitive and undefined, in any theory (see, for example, Popper,
1979; Hempel, 1966; Barker, 1964). The defined terms are defined using other
terms. Ultimately, after a finite number of defining links, chains of
definition can be chased back to primitive terms, or else the definitions would
be based on, and lead to, an infinite regress4. On the basis of the division of
terms into primitive and defined, a simple inductive definition of the level of
every term within an hierarchical structure can be given. Assuming that each
concept is named by a term, this provides an hierarchy of both terms and
concepts. Let the terms of level 1 be the primitive terms of the theory.
Assuming that the terms of level n are defined, we define the terms of level
n+1 to be those whose definitions include terms of level n, but none of any
higher level (although terms of lower level may be included). This definition
unambiguously assigns each term of an objective mathematical theory to a level,
and hence determines an hierarchy of terms and concepts (relative to a given
theory).
In the domain of subjective knowledge, we can, at
least theoretically, divide concepts similarly, into primative observational
concepts, and abstract concepts defined in terms of other concepts. Given such
a division, an hierarchical structure may be imposed on the terms and concepts
of a subjective mathematical theory precisely as above. Indeed Skemp (1971)
offers an analysis of this sort. He terms observational and defined concepts
primary and secondary concepts, respectively. He bases the notion of conceptual
hierarchy upon this distinction in much the same way as above, without
assigning numbers to levels. His proposals are based on a logical analysis of
the nature of concepts, and their relationships. Thus the notion of a
conceptual hierarchy can be utilized in a philosophical theory of subjective
knowledge without introducing any empirical conjecture concerning the nature of
concepts.
To illustrate the hierarchical nature of subjective
mathematical knowledge, consider the following sample contents, which exemplify
its linguistic origins. At the lowest level of the hierarchy are basic terms
with direct empirical applications, such as ‘line’, ‘triangle’, ‘cube’, ‘one’,
and ‘nine’. At higher levels there are terms defined by means of those at lower
levels, such as ‘shape’, ‘number’, ‘addition’ and ‘collection’. At higher
levels still, there are yet more abstract concepts such as ‘function’, ‘set’,
‘number system’, based on those at lower levels, and so on. In this way, the
concepts of mathematics are stratified into a hierarchy of many levels.
Concepts on succeeding levels are defined implicitly or explicitly in terms of
those on lower levels. An implicit definition may take the following form:
numbers consist of ‘one’, ‘two’, ‘three’, and other objects with the same
properties as these. ‘Shape’ applies to circles, squares, triangles, and other
objects of similar type. Thus new concepts are defined in terms of the implicit
properties of a finite set of exemplars, whose membership implicitly includes
(explicitly includes, under the new concept) further exemplars of the
properties.
It is not the intention to claim here that there is
a uniquely defined hierarchy of concepts in either objective or subjective
mathematical knowledge. Nor is it claimed than an individual will have but one
conceptual hierarchy. Different individuals may construct distinct hierarchies
for themselves depending on their unique situations, learning histories, and
for particular learning contexts. We saw in the previous section that different
individuals’ use of the same terms in ways that conform to the social rules of
use does not mean that the terms denote identical concepts or meanings (such an
assertion would be unverifiable, except negatively). Similarly, such conformity
does not mean that individuals’ conceptual structures are isomorphic, with
corresponding connections. All that can be claimed is that the subjective
conceptual knowledge of mathematics of an individual is ordered hierarchically.
It is conjectured that the generation of a hierarchy
of increasingly abstract concepts reflects a particular tendency in the genesis
of human mathematical knowledge. Namely, to generalize and abstract the shared
structural features of previously existing knowledge in the formation of new
concepts and knowledge. We conjecture the existence of some such mechanism to
account for the genesis of abstract concepts and knowledge (as was noted
above). At each succeeding level of the conceptual hierarchy described, we see
the results of this process. That is the appearance of new concepts implicitly
defined in terms of a finite set of lower level terms or concepts.
This abstractive, vertical process contrasts with a
second mode of mathematical knowledge generation: the refinement, elaboration
or combination of existing knowledge, without necessarily moving to a higher
level of abstraction. Thus the genesis of mathematical knowledge and ideas
within individual minds is conjectured to involve both vertical and horizontal
processes, relative to an individual’s conceptual hierarchy. These directions
are analogous with those involved in inductive and deductive processes,
respectively. We discuss both these modes of knowledge generation in turn,
beginning with that described as vertical.
Before continuing with the exposition of the
mechanisms underpinning the genesis of mathematical knowledge, a methodological
remark is called for. It should be noted that the conjectures concerning the
vertical and horizontal modes of thought in the genesis of subjective
mathematical knowledge are inessential for social constructivism. It has been
argued that some (mental) mechanism is needed to account for the generation of
abstract knowledge from particular and concrete experience. This is central to
social constructivism. But as a philosophy of mathematics it is not necessary
to analyse this mechanism further, or to conjecture its properties. Thus the
rejection of the following exploration of this mechanism need not entail the
rejection of the social constructivist philosophy of mathematics.
The vertical processes of subjective knowledge
generation involve generalization, abstraction and reification, and include
concept formation. Typically, this process involves the transformation of
properties, constructions, or collections of constructions into objects. Thus,
for example, we can rationally reconstruct the creation of the number concept,
beginning with ordination, to illustrate this process. The ordinal number ‘5’
is associated with the 5th member of a counting sequence, ranging over 5
objects. This becomes abstracted from the particular order of counting, and a
generalization ‘5’, is applied as an adjective to the whole collection of 5
objects. The adjective ‘5’ (applicable to a set), is reified into an object,
‘5’, which is a noun, the name of a thing-in-itself. Later, the collection of
such numbers is reified into the set ‘number’. Thus we see how a path can be
constructed from a concrete operation (using the ordinal number ‘5’), through
the processes of abstraction and reification, which ultimately leads (via the
cardinal number ‘5’) to the abstract concept of ‘number’. This account is not
offered as a psychological hypothesis, but as a theoretical reconstruction of
the genesis of subjective mathematical knowledge by abstraction.
What is proposed is that by a vertical process of
abstraction or concept formation, a collection of objects or constructions at
lower, pre-existing levels of a personal concept hierarchy become ‘reified’
into an object-like concept, or noun-like term. Skemp refers to this ‘detachability’,
or ‘the ability to isolate concepts from any of the examples which give rise to
them’ (Skemp, 1971, page 28) as an essential part of the process of abstraction
in concept formation. Such a newly defined concept applies to those lower level
concepts whose properties it abstracts, but it has a generality that goes
beyond them. The term ‘reification’ is applied because such a newly formed’
concept acquires an integrity and the properties of a primitive mathematical
object, which means that it can be treated as a unity, and at a subsequent
stage it too can be abstracted from, in an iteration of the process.
The increasing complexity of subjective mathematical
knowledge can also be attributed to horizontal processes of concept and
property elaboration and clarification. This horizontal process of object
formation in mathematics is that described by Lakatos (1976), in his
reconstruction of the evolution of the Euler formula and its justification.
Namely, the reformulation (and ‘stretching’) of mathematical concepts or
definitions to achieve consistency and coherence in their relationships within
a broader context. This is essentially a process of elaboration and refinement,
unlike the vertical process which lies behind ‘objectification’ or
‘reification’.
Thus far, the account given has dwelt on the genesis
and structure of the conceptual and terminological part of subjective
mathematics. There is also the genesis of the propositions, relationships and
conjectures of subjective mathematical knowledge to be considered. But this can
be accommodated analogously. We have already discussed how the elementary
truths of mathematics and logic are acquired during the learning of
mathematical language. As new concepts are developed by individuals, following
the hierarchical pattern described above, their definitions, properties and
relationships underpin new mathematical propositions, which must be acquired
with them, to permit their uses. New items of prepositional knowledge are
developed by the two modes of genesis described above, namely by informal
inductive and deductive processes. Intuition being the name given to the
facility of perceiving (i.e., conjecturing with belief) such propositions and
relationships between mathematical concepts on the basis of their meaning and
properties, prior to the production of warrants for justifying them. Overall,
we see, therefore, that the general features of the account of the genesis of
mathematical concepts also holds for prepositional mathematical knowledge. That
is we posit analogous inductive and deductive processes, albeit informal, to
account for this genesis.
In summary, this section has dealt with the genesis
of the concepts and propositions of subjective mathematical knowledge. The
account given of this genesis involves four claims. First of all, the concepts
and propositions of mathematics originate and are rooted in those of natural
language, and are acquired (constructed) alongside linguistic competence.
Secondly, that they can be divided into primitive and derived concepts and
propositions. The concepts can be divided into those based on observation and
direct sensory experience, and those defined linguistically by means of other
terms and concepts, or abstracted from them. Likewise, the propositions consist
of those acquired linguistically, and those derived from preexisting
mathematical propositions, although this distinction is not claimed to be clear
cut. Thirdly, the division of concepts, coupled with the order of their
definition, results in a subjective (and personal) hierarchical structure of
concepts (with which the propositions are associated, according to their
constituent concepts). Fourthly, the genesis of the concepts and propositions
of subjective mathematics utilizes both vertical and horizontal processes of
concept and proposition derivation, which take the form of inductive and
deductive reasoning.
These claims comprise the social constructivist
account of the genesis of subjective mathematical knowledge. However, in
providing the accounts, examples have been given, especially concerning the
third and fourth of these claims, which may have the status of empirical
conjectures. The hierarchical nature of subjective mathematical knowledge can
be accepted, without relying on such empirical conjectures. Likewise the
existence of the horizontal process of subjective concept refinement or
prepositional deduction, by analogy with Lakatos’ logic of mathematical
discovery, can be accepted in principle. This leaves only the vertical
processes of abstraction, reification or induction to account for, without
assuming empirical grounds. But some such procedure is necessary, if subjective
knowledge is to be constructed by individuals on the basis of primitive
concepts derived from sense impressions and interactions, or elementary
mathematical propositions embedded in language use, as we have assumed. For it
is clear that relatively abstract knowledge must be constructed from relatively
concrete knowledge, to account for the increasing abstraction of the subjective
knowledge of mathematics. Hence, as with the horizontal process, the existence
of this vertical process is needed in principle, irrespective of the fact that
some of the details included in the account might be construed as empirical
conjectures. For this reason, these details were characterized as inessential
to the central thesis of social constructivism.
- Subjective Belief in the Existence of Mathematical Objects
The
account given above of the development of individuals’ knowledge of the
external world is that it is a free construction of the individual subject to
the constraints of the physical and social worlds. The individual directly
experiences these worlds and has his or her conjectural maps of these worlds
confirmed as viable or demonstrated to be inadequate on the basis of the
responses to their actions. The consequence of this is that the individual
constructs personal representations of these worlds, which are unique and
idiosyncratic to that individual, but whose consequences fit with what is
socially accepted. Such a fit is due to the shared external constraints which
all individuals accommodate (more or less), and in particular, the constraint
of viable negotiation of meanings and purposes in social intercourse. Thus,
according to this account, individuals’ construct their own subjective
knowledge and concepts of the external and social worlds, as well as that of
mathematics, so that they fit with what is socially accepted.
These self-constructed worlds represent reality to
the individuals who have made them, be it physical or social reality. Since the
same mechanism lies behind the construction of mathematics as the other
representations, it is not surprising that it too seems to have a measure of
independent existence. For the objects of mathematics have objectivity, in that
they are socially accepted constructs. Other socially constructed concepts are
known to have a powerful impact upon our lives, such as ‘money’, ‘time’
(o’clock), ‘the North Pole’, ‘the equator’, ‘England’, ‘gender’, ‘justice’ and
‘truth’. Each of these is, undeniably a social construct. Yet each of these
concepts has as tangible an impact as many concretely existing objects.
Consider ‘money’. This represents an organizing
concept in modern social life of great power, and more to the point, of
undeniable existence. Yet it is clearly an abstract human-made symbol of
conventional, quantified value, as opposed to some aspect of the physical
world. Let us explore ‘money’ further. What is it that gives money its
existence? There are two features on which its ontological status is based.
First, it is socially accepted, which gives it objectivity. Second, it is
represented by tokens, which means that it has tangible concrete reference.
Now consider the analogy with the objects of
mathematics. These have objectivity, being socially accepted. In addition, the
primitive concepts of mathematics, such as ‘square’ and ‘7’, have concrete
examples in our perceptions of the physical world. So far, the analogy is good.
The defined concepts of mathematics do not fit so well with the analogy, for
they may only have concrete applications indirectly, via chains of definition.
Although there is an analogy between these abstract objects of mathematics and
the more abstracted applications of money (budgeting, financial forecasting,
etc.), this is stretching it a bit far. What can be said is that the analogy
between ‘money’ and mathematical objects lends some plausibility to subjective
belief in the latter objects. They are both objective social constructions and have
concrete manifestations.
Of course mathematics has a further feature
supporting this belief. This is the necessary relationships between its
objects, due to their strict logical relationships in deductive systems.
Logical necessity attaches to the objects of mathematics through their defining
relationships, their inter-relationships and their relationships with
mathematical knowledge. This lends necessity to the objects of mathematics (a
feature that money lacks).
In a nutshell, the argument is this. If an
individual’s knowledge of the real world, including its conventional
components, is a mental construct constrained by social acceptance, then belief
in such constructs evidently can be as strong as beliefs in anything.
Subjective knowledge of mathematics, and acquaintance with its concepts and
objects is also a mental construct. But like other socially determined
constructs, it has an external objectivity arising from its social acceptance.
The objects of mathematics also have (i) concrete exemplifications, either
directly (for the primitive mathematical concepts), or indirectly (for the
defined mathematical concepts); and (ii) logical necessity, through their
logical foundations and deductive structure. These properties are what give
rise to a belief in the objective existence of mathematics and its objects.
Traditionally, knowledge has been divided into the
real and the ideal. It is common to accept the reality of the external world
and our scientific knowledge of it (scientific realism). It is also common to
accept the ideal existence of (objective) mathematics and mathematical objects
(idealism or platonism). This dichotomy places physical and scientific objects
in one realm (Popper’s world 1) and mathematical objects in another (subjective
knowledge of them in world 2, objective knowledge in world 3). Thus it places
mathematical and physical objects in different categories. The social
constructivist thesis is that we have no direct access to world 1, and that
physical and scientific objects are only accessible when represented by
constructs in world 3 (objective concepts) or in world 2 (subjective concepts).
Thus our knowledge of physical and mathematical objects has the same status,
contrary to traditional views. The difference resides only in the nature of the
constraints physical reality imposes on scientific concepts, through the means
of verification adopted for the two types of knowledge (scientific or
mathematical). The similarity, including the social basis of the objectivity of
both types of knowledge, accounts for the subjective belief in the existence of
mathematical objects (almost) just as for theoretical physical objects.7
- Relating Objective and Subjective Knowledge of Mathematics
The
relationship between subjective and objective knowledge of mathematics is
central to the social constructivist philosophy of mathematics. According to
this philosophy, these realms are mutually dependent, and serve to recreate
each other. First of all, objective mathematical knowledge is reconstructed as
subjective knowledge by the individual, through interactions with teachers and
other persons, and by interpreting texts and other inanimate sources. As has
been stressed, interactions with other persons (and the environment),
especially through negative feedback, provides the means for developing a fit
between an individuals subjective knowledge of mathematics, and socially
accepted, objective mathematics. The term ‘reconstruction’, as applied to the
subjective representation of mathematical knowledge, must not be taken to imply
that this representation matches objective mathematical knowledge. As has been
said, it is rather that the subjective knowledge ‘fits’, to a greater or lesser
extent, socially accepted knowledge of mathematics (in one or more of its manifestations).
Secondly, subjective mathematical knowledge has an
impact on objective knowledge in two ways. The route through which individuals’
mathematical creations become a part of objective mathematical knowledge,
provided they survive criticism, has been described. This represents the avenue
by means of which new creations (including the restructuring of pre-existing
mathematics) are added to the body of objective mathematical knowledge. It also
represents the way in which existing mathematical theories are reformulated,
inter-related or unified. Thus it includes creation not only at the edges of
mathematical knowledge, but also throughout the body of mathematical knowledge.
This is the way that subjective knowledge of mathematics explicitly contributes
to the creation of objective mathematical knowledge.8 However, there is also a
more far-reaching but implicit way in which subjective mathematical knowledge
contributes to objective mathematical knowledge.
The social constructivist view is that objective
knowledge of mathematics is social, and is not contained in texts or other
recorded materials, nor in some ideal realm. Objective knowledge of mathematics
resides in the shared rules, conventions, understandings and meanings of the
individual members of society, and in their interactions (and consequently,
their social institutions). Thus objective knowledge of mathematics is
continually recreated and renewed by the growth of subjective knowledge of
mathematics, in the minds of countless individuals. This provides the
substratum which supports objective knowledge, for it is through subjective
representations that the social, the rules and conventions of language and
human interaction, is sustained. These mutually observed rules, in their turn,
legitimate certain formulations of mathematics as accepted objective
mathematical knowledge. Thus objective knowledge of mathematics survives
through a social group enduring and reproducing itself. Through passing on
their subjective knowledge of mathematics, including their knowledge of the
meaning to be attributed to the symbolism in published mathematical texts,
objective knowledge of mathematics passes from one generation to the next.
This process of transmission does not merely account
for the genesis of mathematical knowledge. It is also the means by which both
the justificatory canons for mathematical knowledge, and the warrants
justifying mathematical knowledge itself are sustained. Kitcher (1984) likewise
claims that the basis for the justification of objective mathematical knowledge
is passed on in this way, from one generation of mathematicians to the next,
starting with empirically warranted knowledge.
As a rational reconstruction of mathematical history
to warrant mathematical knowledge, Kitcher’s account has some plausibility.
Like Kitcher, social constructivism sees as primary the social community whose
acceptance confers objectivity on mathematical knowledge. However, unlike
Kitcher, social constructivism sees the social as sustaining the full rational justification
for objective mathematical knowledge, without the need for historical support
for this justification. According to social con¬ structivism, the social
community which sustains mathematics endures smoothly over history, with all
its functions intact, just as a biological organism smoothly survives the death
and replacement of its cells. These functions include all that is needed for
warranting mathematical knowledge.
It should be made clear that the claim that
objective mathematical knowledge is sustained by the subjective knowledge of
members and society does not imply the reducibility of the objective to the
subjective. Objective knowledge of mathematics depends upon social
institutions, including established ‘forms of life’ and patterns of social
interaction. These are sustained, admittedly, by subjective knowledge and
individual patterns of behaviour as is the social phenomenon of language. But
this no more implies the reducibility of the objective to the subjective, than
materialism implies that thought can be reduced to, and explained in terms of
physics. The sum of all subjective knowledge is not objective knowledge.
Subjective knowledge is essentially private, whereas objective knowledge is
public and social. Thus although objective knowledge of mathematics rests on
the substratum of subjective knowledge, which continually recreates it, it is
not reducible to subjective knowledge.
As a thought experiment, imagine that all social
institutions and personal interactions ceased to exist. Although this would
leave subjective knowledge of mathematics intact, it would destroy objective
mathematics. Not necessarily immediately, but certainly within one lifetime.
For without social interaction there could be no acquisition of natural
language, on which mathematics rests. Without interaction and the negotiation
of meanings to ensure a continued fit, individual’s subjective knowledge would
begin to develop idiosyncratically, to grow apart, unchecked. The objective
knowledge of mathematics, and all the implicit knowledge sustaining it, such as
the justificatory canons, would cease to be passed on. Naturally no new
mathematics could be socially accepted either. Thus the death of the social
would spell the death of objective mathematics, irrespective of the survival of
subjective knowledge.
The converse also holds true. If, as another thought
experiment we imagine that all subjective knowledge of mathematics ceased to
exist, then so too would objective knowledge of mathematics cease to exist. For
no individual could legitimately assent to any symbolic representations as
embodying acceptable mathematics, being deprived of the basis for such assent.
Therefore there could be no acceptance of mathematics by any social group. This
establishes the converse relationship, namely that the existence of subjective
knowledge is necessary for there to be objective knowledge of mathematics.
Of course it is hard to follow through all the
consequences of the second thought experiment, because of the impossibility of
separating out an individual’s subjective knowledge of language and
mathematics. Knowledge of language depends heavily on the conceptual tools for
classifying, categorising and quantifying our experience and for framing
logical utterances. But according to social constructivism these form the basis
for mathematical knowledge. If we delete these from subjective knowledge in the
thought experiment, then virtually all knowledge of language and its conceptual
hierarchy, would collapse. If we leave this informal knowledge and only debate
explicit knowledge of mathematics (that learned as mathematics and not as
language), then subjective knowledge of mathematics could be rebuilt, for we
would have left its foundations intact.
In summary, the social constructivist thesis is that
objective knowledge of mathematics exists in and through the social world of
human actions, interactions and rules, supported by individuals’ subjective
knowledge of mathematics (and language and social life), which need constant
re-creation. Thus subjective knowledge re-creates objective knowledge, without
the latter being reducible to the former. Such a view of knowledge is endorsed
by a number of authors. Paul Cobb, argues from a radical constructivist
perspective that:
the view that cultural knowledge in general and
mathematics in particular can be taken as solid bedrock upon which to anchor
analyses of learning and teaching is also questioned. Instead it is argued that
cultural knowledge (including mathematics) is continually recreated through the
coordinated actions of the members of a community.
(Cobb,
1988, page 13)
Paulo
Freire has elaborated an epistemology and philosophy of education that places
individual consciousness, in the context of the social, at the heart of
objective knowledge. He ‘recognize(s) the indisputable unity between
subjectivity and objectivity in the act of knowing.’ (Freire, 1972b, page 31)
Freire argues, as we have done, that objective knowledge is continually created
and re-created as people reflect and act on the world.
Even the received view of epistemology (see, for
example, Sheffler, 1965) can be interpreted as logically founding objective
knowledge on subjective knowledge. For this view defines knowledge (rather more
narrowly than it has been used above) as justified true belief. Belief includes
what has been termed subjective knowledge, in this chapter. In mathematics,
justified true belief can be interpreted as consisting of assertions that have
a justification necessitating their acceptance (in short, a proof). According
to the social constructivist philosophy, such mathematical statements are
socially accepted, on the basis of their justification, and thus constitute
objective mathematical knowledge. Thus, in the terms of this chapter,
‘knowledge is justified true belief’ translates into ‘objective knowlege of
mathematics is socially acccepted subjective knowledge, expressed in the form
of linguistic assertions’. According to this translation, objective knowledge
of mathematics depends logically on subjective knowledge, because of the order
of definition.
The social constructivist view of mathematics places
subjective and objective knowledge in mutually supportive and dependent
positions. Subjective knowledge leads to the creation of mathematical
knowledge, via the medium of social interaction and acceptance. It also
sustains and re-creates objective knowledge, which rests on the subjective
knowledge of individuals. Representations of objective knowledge are what allow
the genesis and re-creation of subjective knowledge. So we have a creative
cycle, with subjective knowledge creating objective knowledge, which in turn
leads to the creation of subjective knowledge. Figure 4.1 shows the links
between the private realm of subjective knowledge and the social realm of
objective knowledge each sustaining the creation of the other. Each must be
publicly represented for this purpose. Thereupon there is an interactive social
negotiation process leading to the reformulation of the knowledge and its
incorporation into the other realm as new knowledge.
Of course there
are powerful constraints at work throughout this creative cycle. These are the physical
and social worlds, and in particular the linguistic and other rules embodied in
social forms of life.
Figure
4.1: The Relationship between Objective and Subjective Knowledge of Mathematics
- Criticism of Social Constructivism
The
account of social constructivism has brought together three philosophical
perspectives as a basis for a unified philosophy of mathematics. These are
quasiempiricism, conventionalism and radical constructivism. As a consquence,
two types of criticism can be directed at social constructivism. First of all,
there is criticism directed at the assumptions and philosophical stance adopted
by one of these tributary philosophies. For example, there are the problems in
accounting for logical necessity in mathematics from the social perspective of
conventionalism. An attempt has been made to anticipate and answer such
criticisms. To those who reject the conventionalist assumptions such arguments
will not be convincing. However, no further criticisms of this type, that is,
directed at the tributary philosophies, will be addressed. For in addition to
having been treated above, they can also be found in the relevant literature.
Secondly, there are criticisms that can be directed
at the novel synthesis that is provided by social constructivism. Two
criticisms in this category will be considered. These concern the use of
empirical assumptions in social constructivism, and the tension between the
subsumed theories of conventionalism and radical constructivism.
Throughout the exposition, it has been argued that
the genesis as well as the justification of knowledge is the proper concern of
the philosophy of mathematics. Consequently both these contexts have been
discussed, but an attempt has been made to distinguish between them carefully,
and to avoid or to demarcate carefully any empirical assumptions, especially
concerning the genesis of mathematical knowledge. This was a criticism directed
at Lakatos in Chapter 2, that in his account of the conditions of the genesis
of mathematical knowledge, he introduced an historical (i.e. empirical)
conjecture. It may be felt that the social constructivist account errs
similarly. However, I believe that this would indicate that a clearer
exposition of social constructivism is needed, rather than necessitating a
rejection of the entire philosophy.
A more substantial criticism arises from a possible
tension, or even inconsistency, between the subsumed theories of
conventionalism and radical constructivism. For the former gives primacy to the
social, comprising accepted rules and conventions underpinning the use of
language and objective knowledge of mathematics. This reflects a ‘form of
life’, constituting accepted social and verbal behaviour patterns. The latter,
gives primacy to the knowing subject, an unreachable monad constructing
hypo-thetical world-pictures to represent experiences of an unknowable reality.
To an adherent of one but not the other philosophy, their conjunction in social
constructivism may seem to be an unholy alliance, for neither of the two foci
is given precedence. Rather each is the centre of a separate realm. The knowing
subject is at the centre of the private realm of individuals and subjective
knowledge. This realm assumes a real but unknowable world, as well as the
knowing subject. But this realm is not enough to account for objective
knowledge, let alone for humanity. For humankind is a social animal, and
depends essentially on interchange and language. The social realm takes this as
its basis, including social institutions and social agreements (albeit tacit).
This realm assumes the existence of social groups of human beings. However,
this perspective seems weak in terms of the interior life and consciousness it
ascribes to individuals.
Thus although the primacy of focus of each of
conventionalism and radical constructivism is sacrificed in social
constructivism, their conjunction in it serves to compensate for their
individual weaknesses, yet this conjunction raises the question as to their
mutual consistency. In answer it can be said that they treat different domains,
and both involve social negotiation at their boundaries (as Figure 4.1 illustrates).
Thus inconsistency seems unlikely, for it could only come about from their
straying over the interface of social interaction, into each other’s domains.
The separateness of the private and social realms,
together with their separate theoretical accounts, has another consequence. It
means that the parts of the social constructivist account could be modified
(e.g., the account of subjective knowledge) without changing the whole
philosophy. This suggests that the philosophy lacks a single overarching
principle. However, there are unifying concepts (or metaphors) which unite the
private and social realms, namely construction and negotiation. For both
subjective and objective knowledge are deemed to be human constructions, built
up from pre-existing knowledge components. The second unifying concept is that
of social negotiation. This not only plays a central role in the shaping of
subjective and objective knowledge. It also plays a key role in the
justification of mathematical knowledge according to social constructivism,
from the quasiempiricist component.
