CEPA eprint 2989

The nature of mathematics: Towards a social constructivist account

Ernest P. (1992) The nature of mathematics: Towards a social constructivist account. Science & Education 1: 89–100. Available at http://cepa.info/2989
Table of Contents
Introduction
The philosophy of mathematics
Mathematics as a social construction
Objections to this view
Strengths of the social constructivist account
Some implications for education
Conclusion
References
Two dichotomies in the philosophy of mathematics are discussed: the prescriptive – descriptive distinction, and the process – product distinction. By focusing on prescriptive matters, and on mathematics as a product, standard philosophy of mathematics has overlooked legitimate and pedagogically rewarding questions that highlight mathematics as a process of knowing which has social dimensions. In contrast the social-constructivist view introduced here can affect the aims, content, teaching approaches, implicit values, and assessment of the mathematics curriculum, and above all else, the beliefs and practices of the mathematics teacher.
Introduction
I wish to apply two dichotomies to a discussion of the philosophy of mathematics: the prescriptive-descriptive distinction, and the process – product distinction. For much of this century, the philosophy of mathematics has focussed on mathematical knowledge as a product, and has eschewed the process aspect of its epistemology. Knowledge as a product has been the focus of much of the philosophy of mathematics, not the process of coming to know. Secondly, the most visible accounts of the nature of mathematics and mathematical knowledge have been prescriptive – legislating how mathematics should be understood – rather than providing accurately descriptive accounts of the nature of mathematics. Although there are good historical reasons for these misconceptions, until they are overcome accounts of the nature of mathematics do violence to its reality. Once knowing, as well as knowledge is admitted as a legitimate concern of epistemology, it is possible to be more accurately descriptive. This leads to the acknowledgement of the human role in the creation of mathematics, that it is in fact a social construct. Once this view is adopted, it can be seen as relating to a number of other currents of contemporary thought. Such a case stands on its own merits, without a consideration of the educational implications. But once this case has been made, it can be seen to have immense educational implications. For a view of mathematics as a way of knowing and as a social construct can powerfully affect the aims, content, teaching approaches, implicit values, and assessment of the mathematics curriculum, and above all else, the beliefs and practices of the mathematics teacher.
The philosophy of mathematics
The problem and task of the philosophy of mathematics is to account for the nature of mathematics. This is a special case of the task of epistemology, namely to account for human knowledge. The standard approach to this task is to assume:
Assumption 1: There is a set of propositions that represents human knowledge in any field, and a set of procedures for verifying these propositions (or at least providing a warrant for their assertion).
Epistemology traditionally distinguishes between a posteriori and a priori knowledge, according to whether these procedures involve empirical verification or not (respectively). Mathematical knowledge is a priori knowledge because its propositions are established by means of logic, without recourse to empirical data. Since the truths of mathematics are established irrespective of the facts of the world, no data can overturn them. Thus mathematical knowledge is the most certain of all forms of knowledge.
Traditionally the philosophy of mathematics has seen its task as providing a foundation for the certainty of mathematical knowledge. That is, providing a system into which mathematical knowledge can be cast, which provides a systematic way of establishing its truth.
Assumption 2: The role of the philosophy of mathematics is to provide a systematic foundation for mathematical knowledge, that is for mathematical truth.
This assumption can be illustrated with the well known philosophical schools: logicism, formalism and intuitionism. Each of these schools utilises deductive logic as a means of warranting mathematical knowledge. Mathematical propositions which count as knowledge are either axioms, which are assumed or stand on their own merits, or the deductive consequences of axioms. Logicism rests its axioms on logic, intuitionism claims its axioms are self-evident, and formalism denies meaning or truth to its axioms at the object language level, but claims they are self-evident at the meta-language level. Each school justifies its rules of logic similarly. The schools are thus able to offer a programme for the warranting of mathematical knowledge, by re-casting it in their own way.
Ab initio the quest for certainty in mathematics is problematic, because of the precision that is needed in defining mathematical truth. We can distinguish between three truth-related concepts used in mathematics:
There is the truth of a mathematical statement relative to a background theory: the statement is satisfied by some interpretation or model of the theory.There is the logical truth or validity of a mathematical state ment relative to a background theory: the statement is satisfied by all interpretations or models of the theory.There is the provability of a mathematical statement with the aid of assumptions drawn from a background theory: there is a finite logical proof of the statement from the axioms of the theory.
The second and third of these senses are what is usually meant by mathematical truth (the ambiguity matters little, since in most cases these two senses are demonstrably equivalent, by the appropriate completeness and soundness theorems). In a naive sense truths are statements which accurately describe a state of affairs – a relationship – in some realm of discourse. In this view, the terms which express the truth name objects in the realm of discourse, and the statement as a whole describes a true state of affairs, the relationship that holds between the denotations of the terms. A mathematical truth in this sense is rare. Mathematical truths are made up of terms which usually do not name unique individuals, and the statement as a whole describes a structural relation which holds between whichever objects are named by its terms, in any appropriate interpretation (that is one satisfying the axioms of the background theory). This is uncontroversial. What it shows is that the concept of truth employed in mathematics no longer has the same meaning as either the everyday, naive notion of truth, or its equivalent, as was used in mathematics in the past.
Whether we assume this new meaning of truth as it is employed in mathematics or not, it is still not possible to establish the certainty of mathematical knowledge. As Lakatos (1978) shows, despite all the foundational work and development of mathematical logic, the quest for certainty in mathematics leads inevitably to an infinite regress. Any mathematical system depends on a set of assumptions, and there is no way of escaping them. All we can do is to minimise them, to get a reduced set of axioms (and rules of proof). This reduced set can only be dispensed with by replacing it with assumptions of at least the same strength. Thus we cannot establish the certainty of mathematics without making assumptions, which therefore is not absolute certainty. Furthermore, if we want to establish that our mathematical systems are safe (i.e. consistent), for any but the simplest systems we are forced to expand the set of assumptions we make, further undermining the certainty of the foundations of mathematics.
What has been argued is that the traditional philosophies of mathematics which have assumed the task of trying to establish the certainty of mathematical knowledge have failed. However the point I wish to make is not only that such attempts are doomed, but that they are wrong headed. Insofar as the philosophical schools of logicism, formalism and intuitionism offer an account of the nature of mathematics, it is prescriptive. That is, mathematics is a body of knowledge which should be seen as logic, intuitionistic mathematics, or consistent formal systems, according to which viewpoint is adopted. More generally, Assumption 2 is prescriptive, in that it requires us to focus on the systematic reconstruction of mathematical knowledge to warrant its assertion. It concerns how mathematics ‘should be seen’, to justify certain philosophical requirements.
The programmes and philosophies based on this assumption are akin to medieval chivalry or the Arthurian Legend. Namely, the pursuit of a lofty ideal – the Holy Grail – by an elite band with no concern for the mundanities of everyday life. Part of Cervantes’ greatness is that his parody revealed this quest for what it was – a social construction of reality!
Should not an account of the nature of mathematics involve some consideration of how it is, as well as how it should be? A tenable philosophy of mathematics surely needs to be descriptive as well as prescriptive. However this question raises again the problem of what constitutes mathematics, and thus questions Assumption 1.
Is mathematics a body of knowledge expressible as a set of propositions, together with a set of logical procedures for their verification? It can be conceded that this is a part of mathematics. But evidently more can be considered. There are the processes of creating, transmitting and modifying such knowledge. Histories of mathematics document the creation and evolution of mathematical concepts and knowledge. Is this not also germane to the philosophy of mathematics?
There are a number of reasons for questioning Assumption 1, some general epistemological considerations, and some specific to mathematics.
First of all, since, as history illustrates, knowledge is perpetually in a state of change, epistemology must concern itself with the basis of knowing, as well as with the specific body of knowledge accepted at any one time. This is the view of pragmatists such as Dewey (1950), as well as modern philosophers of science such as Kuhn (1970) and Lakatos (1978).
Secondly, mathematical knowledge has an empirical basis, contrary to traditional views. This is the view put forward by an increasing number of philosophers, such as Lakatos (1976, 1978), Kitcher (1983) Tymoczko (1986). If this claim is accepted, then it is necessary to consider the process of coming to know in mathematics in considering the basis of mathematical knowledge.
Thirdly, if the task of the philosophy of mathematics is to account for the nature of mathematics, then other things being equal, a fuller descriptive account is to be preferred to a narrower account, provided it is coherent. For the parameters of the more extensive account allow it to better fulfil the task. Indeed, new mathematical structures invented to ‘fit’ other areas have been a source of renewal for mathematics throughout its history. Very powerful philosophical arguments would be needed to exclude whole areas of the domain in question from consideration at the outset of this inquiry, and I cannot find such arguments stated.
Given these three arguments, it seems appropriate to attempt to provide an account of mathematics that includes the process of coming to know as well as its product, that is knowledge. It is also appropriate to attempt to give a descriptive account, which locates mathematics where it is to be found in the world: in a social context. The next section offers a tentative social constructivist account of mathematics which attempts to overcome some of the difficulties described above.
Mathematics as a social construction
Philosophically, the social constructivist view of mathematics proposed here is both conventionalist and empiricist, in that human language, agreement and experience play a role in establishing its truths. Central to this view is the fact that over the course of time, mathematical knowledge changes, just as knowledge in the empirical sciences evolves. At any one given time, mathematics is an intersubjectively agreed, rather than an objective body of knowledge. It includes pragmatic rules governing procedures, as well as a body of propositions and methods.
It is easier to sketch such a view of mathematics in quasi-sociological terms. Thus a more suggestive, but tentative account along these lines is as follows. (For an account of social constructivism as a philosophy of mathematics see Ernest 1990, 1991).
The starting point for any social constructivist account of mathematics is the assumption that the concepts, structures, methods, results and rules that make up mathematics are the inventions of humankind.
At any one time, the nature of mathematics is determined by three fuzzy sets, as well as the relationship between them. The three sets are: a set of information-technology artefacts (books, papers, software, etc); a set of persons (mathematicians); and more intangibly, a set of linguistically based rules adopted by, and activities carried out by, the mathematicians.
The set of mathematicians is partially ordered by the relations of power and status.
The informational content of the artefacts are the creations of the set of mathematicians, as they carry out the linguistically based activities constrained by the rules. This informational content becomes recognised as part of the concepts, methods and results that make up mathematics, when it is accepted by members of the set of mathematicians with power and status.
All the time these sets are changing, and thus mathematics is continuously evolving.
Versions of some of the concepts, methods and results that make up mathematics are represented in the minds of the mathematicians, and are symbolised in the information technology artefacts. Each of these representations includes a structure of relationships connecting the constituent objects and entities (the concepts). There is a powerful myth that these conceptual structures are all substructures of a single idealised structure of mathematics, or some past version of it. In fact, the various structures make up a family, with family resemblances between them (to use Wittgenstein’s metaphor). In all likelihood, the intersection of all of the structures would be empty, even if we were to exclude instances of substructures that disagree with the idealised structure, and are considered incorrect. (However this operation of intersection is of course impossible, since the structures are intangible.)
Membership of the set of mathematicians results from the interaction of a person with other mathematicians, and with information artefacts. Over a period of time this results in a personal internalised mathematical structure and set of rules, derived through personal negotiation with other persons, until a ‘fit’ between structures is achieved (using the term in the sense of Glasersfeld, 1984). The set of mathematicians is a fuzzy set with different strengths of membership (which could be quantified from 0 to 1). This includes ‘strong’ members (active research mathematicians) and ‘weak’ members (teachers of mathematics). The ‘weakest’ members would simply be numerate citizens.
The set of rules and the idealised structure of mathematics are so significant that accounts of mathematics acceptable to many persons refer only to (some of) the rules and the structure.
The concepts of mathematics are derived by abstraction from direct experience of the physical world, from the generalisation and abstraction of previously constructed concepts, by negotiating meanings with others during discourse, or by some combination of these means. Many of the concepts generalised in mathematics come from the physical and other sciences (which likewise derive many of their concepts from mathematics).
Mathematics rests on spoken (and thought and read) natural language, and mathematical symbolism is a refinement and extension of written natural language. Mathematical concepts refine and abstract natural language concepts. Mathematical truths arise from the definitional truths of natural language, which is acquired by persons through social interaction. The truths of mathematics are defined by implicit social agreement on what constitute acceptable mathematical concepts, relationships between them, and methods of deriving new truths from old.
Mathematics is a branch of knowledge which is indissolubly connected with other knowledge, through the web of language. Language functions by facilitating the formulation of theories about social situations and physical reality. Dialogue with other persons and interactions with the physical world play a key role in refining these theories, which consequently are continually being revised to improve the ‘fit’. As a part of the web of language, mathematics thus maintains contact with the theories describing social and physical reality. Of these, physical reality is the most obdurate, since there is an enduring physical world, even if our theories of it change. Through the physical sciences mathematics plays a key role in theories providing well-fitting descriptions of aspects of physical reality. Thus the ‘fir of mathematical structures in areas beyond mathematics is continuously being tested. Indeed, new mathematical structures invented to ‘fit’ other areas have been a source of renewal for mathematics throughout its history.
A key feature of mathematics is its perceived impersonality, its objectification. Thus the concepts, methods and other creations of mathematics are ruthlessly reformulated and altered by mathematicians, in contrast, say, to literary creations. Such changes are subject to very strict and general mathematical rules and values. This objectification of the rules in mathematics also has the result of offsetting some of the effect of sectional interests exercised by those with power in the community of mathematicians.
Mathematicians form a community with a mathematical culture, that is a more or less shared set of concepts and methods, a set of values and rules (which is often understood implicitly), within the contexts of social institutions and power relations.
This is a tentative and greatly simplified formulation of a social constructivist view of mathematics. It combines sociological, psychological, as well as philosophical perspectives. It also stands in need of justification. As an alternative, I anticipate same of the objections to this view and answer them.
Objections to this view
(1) If the most certain of all truths, namely mathematical truths, are but social constructs, then they are conceivably false. A fortiori there are no truths. Consequently the above view is not a true account, as there are none. Hence there is no need to accept it.
This is correct. The social constructivist view of mathematics is offered as an explanatory hypothesis, not as a truth. If made sufficiently precise, insofar as it is descriptive it could, in principle, be refuted. Failing that, its retention must depend on its explanatory utility.
(2) The view presented above conflates philosophical, psychological and sociological explanations and concepts, and thus fails to offer a coherent account of mathematics from a philosophical perspective.
To insist that these fields must be kept separate is to prejudge the issue. For the above account makes the assumption that all human knowledge rests on human language, and hence is part of a connected web. There is a growing train of thought in philosophy which admits into epistemology considerations of human activity or pragmatics (Dewey, Wittgenstein, Polanyi, Toulmin), psychology (Piaget, Lorenz, Bruner, von Glasersfeld) or sociology (Kuhn, Lakatos, Feyerabend, Barnes, Bloor, Restivo). Thus a boundary-crossing approach has a good provenance.
Ultimately, the aim of this social constructivist account of mathematics, when fully developed, is to show that shared explanations can explain:
the psychology of individuals learning mathematics;the historical development of mathematics;mathematics as a living social institution;the philosophy of mathematics.
This may be ambitious, but is not illegitimate. Currently modern physics is seeking to unify its various theories. The history of mathematics likewise provides plenty of examples of theoretical unification. This is also a worthwhile goal for the philosophy of mathematics.
These remarks notwithstanding, the account is not proposed as a purely philosophical account. For these, the reader is referred to Ernest (1990, 1991).
(3) A social constructivist account of mathematics rests the ‘certainty’ of mathematical knowledge on shared language use, social agreement, and arbitrary definitions and conventions. It therefore cannot account for two central aspects of mathematics: the conviction held by many that it is objective and certain, and its “unreasonable effectiveness” (Wigner 1960) in providing models of physical reality.
These are indeed two of the challenges facing this account. The certainty of mathematics is something that emerges gradually in learners of mathematics as they develop the mathematical part of language, and internalise the meanings of mathematical concepts and the relationships between them. Throughout this development there is interchange with others which leads to the agreement or ‘fit’ between individuals’ constructions. (Although psychologists and mathematics education researchers well know that for many the ‘fit’ is poor.) Certainty only emerges as the end product of this constructive process. Thus the certainty of the equivalence between ‘not-A or B’and ‘A (materially) implies Bç is unquestioned by most mathematicians and logicians. For many who have not been through the same learning experiences this will not be so. For they will not have internalised the refined mathematical meanings of the logical connectives, and their interrelationships. Thus the mathematicians’ intuition of the certainty of this basic fact depends on their experience. Similarly ‘1 + 1 = 1’ is only known to be false, with certainty, because of our learned definitions, rules and assumed interpretation. In fact it is true, with certainty, in Boolean algebra, when we routinely make different assumptions. The difference is that ‘1 + 1 = 2’ is a truth of the mathematics embedded in natural language usage, whereas Boolean algebra is a more artificial creation (according to Russell, the first ‘pure’ mathematics).
It is the traditional philosophies of (pure) mathematics which have so much trouble with its ‘Unreasonable effectiveness’. The social constructivist account, by admitting the connection of knowledge, the experiential origins of mathematical concepts and structures, and the hypothetical nature of all knowledge, finds more than coincidence in the applicability of mathematical structures to the world. However there is an important point here. What is proposed is not ‘the social construction of reality’, but the social construction of our knowledge of reality. And the process of this construction involves ‘fitting’ our conjectures as closely to reality as we can, with the possibility of falsification, as Popper (1959) suggests. Thus although our linguistic and scientific definitions and conventions are arbitrary, in that they have been freely created within the constraints of implicit or explicit rules, they have to withstand the gruelling tests of consistency not only within language, but also in ‘fitting’ of hypotheses to reality. (See Ernest, 1990, 1991, for a fuller development of these theses.)
Strengths of the social constructivist account
The social constructivist account of mathematics is proposed because it appears fruitful in being able to provide satisfactory accounts of more aspects of mathematics than many other views. As is suggested above, it accounts for:
mathematics descriptively instead of prescriptively,the growth and changing nature of mathematics,mathematical activity as well as mathematical knowledge,the links between the psychology, sociology and philosophy of mathematics, andthe link between pure mathematics and the world.
In addition, the view can explain the apparent objective existence of mathematical objects, and the mechanism behind the creation and growth of mathematics.
(4) How can the success of the Platonist view, that mathematicalobjects and structures have an objective existence, be explained?
If the constructivist view that our knowledge of reality is a mental construction – albeit mediated by human interaction – is accepted, then not surprisingly, mentally constructed mathematical realities can be as potent as physical realities. Certainly other fictions have a powerful impact upon our lives. Consider only the concept of ‘money’. We cannot deny the existence and power of this denotationless symbol. But it is clearly a social construct.
The uniformity of mathematical meanings amongst mathematicians, and a shared view of the structure of mathematics, result from an extended period of training in which students are indoctrinated with the ‘standard’ structure. This is achieved through common learning experiences and the use of key texts, such as Euclid, Van der Waerden, Bourbaki, Birkhoff and MacLane, Rudin, etc. (or their more recent equivalents). Most students fall away during this process. Those that remain have successfully negotiated and internalised a sub-structure that fits with a subset of the official one – at least in part. These mathematicians will have called up their mental mathematical schemas and used and reinforced them through use so much that they seem to exist objectively. Thus the objects of mathematics are mental constructions which have been given so much solidity, that they seem to have a life of their own. This accounts for why some of the greatest mathematicians are Platonists. They simply have made their mental worlds of mathematics more real than the rest of us by constantly revisiting and extending them.
(5) What mechanisms account for the development of mathematical ideas?
Here Piaget’s genetic epistemology has a great deal to offer. The mental constructions of individuals, through the processes of communication and negotiation, within the strict constraints imposed by the rules of mathematics (also mediated by the exercise of power by mathematicians) are added to the official conceptual structure of mathematics. Thus there is a social process by means of which the ideas of individuals spread out through enlarging groups of persons, and then may be taken up by ‘official’ mathematics. The actual genesis of mathematical ideas within individual minds involves vertical and horizontal processes (by analogy with inductive and deductive processes, respectively). The vertical processes involve generalisation, abstraction and reification, and are akin to concept formation. Typically, this process involves the transformation of properties, constructions, or collections of constructions into objects. Thus, for example, we can reconstruct the creation of the number concept, beginning with ordination. 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 generalisation ‘5’, is applied as an adjective to the whole collection of 5 objects. The adjective ‘5’ (applicable to a set), is refied into an object, ‘5’, which is a noun, the name of a thing in itself. Later, the collection of such numbers is refied into the set ‘number’. Psychologically, it is plausible that the objects of mathematics are ‘refied constructions’. But this has also been offered as a philosophical account of the nature of mathematics by Machover (1983). Such a genetic account holds promise as a way of linking the psychology, history and philosophy of mathematics.
The 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 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’.
This account, which suggests the mechanism which objectifies the concepts of mathematics, may also help to explain why mathematics as a whole is seen to be objectified. To project idealised structures into a Platonic realm may be a natural application of this process of reification. It may be the case that mathematics would have been severely handicapped in its evolution without the myth of its objectivity, and the tendency to objectify it. For this myth permits a focus on the objects and processes of mathematics as shared things-in-themselves, subject only to the rules of the game.
The above sketch begins to show that a social account of mathematics is needed to do it justice, if the reconceptualization of the philosophy of mathematics offered above is accepted. Social constructivism is one of a number of possible accounts, and is further elaborated elsewhere (Ernest, 1991).
Some implications for education
It is appropriate to indicate, however briefly, some of the educational implications of the social constructivist account of mathematics.
One aspect of this view is that mathematics is seen as embedded in a cultural context. It leads to the conclusion that the view that mathematics somehow exists apart from everyday human affairs is a dangerous myth. It is dangerous, not only because it is philosophically unsound, but also it has damaging results in education. Thus, if mathematics is viewed as a body of infallible, objective knowledge, then mathematics bears no social responsibility. The underachievement of sectors of the population, such as women; the sense of cultural alienation from mathematics felt by many groups of students; the relationship of mathematics to human affairs such as the transmission of social and political values; its role in the distribution of wealth and power; the mathematical practices of the shops, streets, homes, and so on – all of this is irrelevant to mathematics.
On the other hand, if mathematics is viewed as a social construct, then the aims of teaching mathematics need to include the empowerment of learners to create their own mathematical knowledge; mathematics can be reshaped, at least in school, to give all groups more access to its concepts, and to the wealth and power its knowledge brings; the social contexts of the uses and practices of mathematics can no longer be legitimately pushed aside, the uses and implicit values of mathematics need to be squarely faced, and so on.
This second view of mathematics as a dynamically organised structure located in a social and cultural context, identifies it as a problem posing and solving activity. It is viewed as a process of inquiry and coming to know, a continually expanding field of human creation and invention, not a finished product. Such a dynamic problem solving view of mathematics embodied in the mathematics curriculum, and enacted by the teacher, has powerful classroom consequences. In terms of the aims of teaching mathematics the most radical of these consequences are to facilitate confident problem posing and solving; the active construction of understanding built on learners’ own knowledge; and the exploration and autonomous pursuit of the learners’ own interests.
If mathematics is understood to be a dynamic, living, cultural product, then this should also be reflected in the school curriculum. Thus mathematics needs to be studied in living contexts which are meaningful and relevant to the learners. Such contexts include the languages and cultures of the learners, their everyday lives, as well as their school based experiences. If mathematics is to empower learners to become active and confident problem solvers, they need to experience a human mathematics which they can make their own. The social constructivist view places a great deal of emphasis on the social negotiation of meaning. Clearly this has very strong implications for discussion in the mathematics classroom.
The social constructivist view also raises the importance of the study of the history of mathematics, not just as a token of the contribution of many cultures, but as a record of humankind’s struggle – throughout time – to problematise situations and solve them mathematically – and to revise and improve previous solution attempts. By legitimating the social origins of mathematics, this view provides a rationale, as well as a foundation for a multicultural approach to mathematics.
Conclusion
This paper has attempted to sketch a social constructivist approach to the philosophy of mathematics. The sketch is greatly simplified, and reference has been made to fuller accounts. The need for a social view of mathematics arises from the inadequacy of the traditional foundationist accounts. As I have indicated, those of us in education have an additional reason for wanting a more human account of the nature of mathematics. Anything else alienates and disempowers learners.
References
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Ernest P. (1990) Social Constructivism as a Philosophy of Mathematics: Radical Constructivism Rehabilitated? Presented at PME-14, Mexico, July 15–20, 1990.
Ernest P. (1991) The Philosophy of Mathematics Education, Falmer Press, London.
Glasersfeld E. von (1984) An Introduction to Radical Constructivism. In: Watzlawick, ed., The Invented Reality, Norton, New York: 17–40. http://cepa.info/1279
Kitcher P. (1983) The Nature of Mathematical Knowledge, Oxford University Press, Oxford.
Kuhn T. S. (1970) The Structure of Scientific Revolutions, Chicago University Press, Chicago.
Lakatos I. (1976) Proofs and Refutations, Cambridge University Press, Cambridge.
Lakatos I. (1978) Philosophical Papers (Volume 2) Cambridge University Press, Cam-bridge.
Machover M. (1983) Towards a New Philosophy of Mathematics. British Journal for the Philosophy of Science 34: 1–11.
Popper K. (1959) The Logic of Scientific Discovery, Hutchinson, London.
Tymoczko T. (ed.) (1986) New Directions in the Philosophy of Mathematics, Birkhauser, Boston.
Wigner E. P. (1960) The Unreasonable Effectiveness of Mathematics in the Physical Sciences. Reprinted in T. L. Saaty and F. J. Weyl (eds.) The Spirit and Uses of the Mathematical Sciences, McGraw-Hill, New York. (1969) pp. 123–140.
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