Nominalism and conventionalism in social constructivism

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Paul Ernest

University of Exeter


There are various forms of social constructivism in the social and human sciences, especially in sociology (Berger and Luckmann 1966), discursive psychology (Gergen 1999) and philosophy (Hacking 2001). There are also variants in education and learning theory (Wertsch 1991), in mathematics education (Weinberg and Gavelek 1987), and even in the philosophy of mathematics (Hersh 1997). However, what is meant by the use of the term here is that version of social constructivism as a philosophy of mathematics given its fullest expression to date in my eponymous book (Ernest 1998). To avoid cumbersome circumlocutions, no reference will be made to the range of extant or possible variations or alternatives, although it is not intended to assert the superiority or supremacy of the singular version explicated here.
Social Constructivism is put forward as a philosophy of mathematics with the primary aim of offering an account of mathematical practice, including also the social structuring and the historical development of mathematics. It is a naturalistic philosophy and so it has many elaborated characteristics relating to its descriptive and social aspects. However although ontological and epistemological issues are discussed in Ernest (1998), I wish to focus here on its relationships to traditional ontological and epistemological positions in the philosophy of mathematics. While social constructivism has been contrasted strongly with Platonism and mathematical realism, on the one hand, and with foundationalist and absolutist positions, on the other hand, here I wish to elaborate its positive location in philosophical traditions.
This paper cautiously describes social constructivism as nominalist, with respect to ontology, and conventionalist with respect to epistemology and the foundations of knowledge. The caution is due to the particular variants or interpretations of nominalism and conventionalism that are attributed to social constructivism, for there are dominant traditions within each perspective with which social constructivism is not identified or subsumed under.
By asserting that the objects of mathematics are signs, rather than purely psychological or mental entities (the claim of conceptualism), material entities (the claim of materialism) or objective self-subsistent entities (the claim of realism) what is espoused is claimed to be a variety of nominalism. However, this differs from the most common forms of nominalism, for it is based on a conception of sign that rejects the representational theory of truth. Signs are part of a cultural realm that is intersubjective. It transcends the perceptions and understanding of any one individual but does not transcend the knowledge and practices of humankind as a whole and thus does not belong to any extra-human reality. Elsewhere I have termed this realm ‘objective’, but this requires a new definition of the term, differing from the received, traditional usage and its ontological presuppositions (Ernest 1998).
The related claim that the concepts, terms, theorems, rules of proof and logic, truths and theories of mathematics are socially constructed cultural entities constitutes a form of conventionalism. However, what is not asserted is that the concepts and truths of mathematics are the result of arbitrary, whimsical or even ideologically motivated decisions and choices. Many conventions in mathematics are not conscious decisions but reflections of historical practices laid down for very good reasons. Furthermore, where conscious decisions are made in mathematics they are usually to complete or extend existing rules and practices within mathematics that result in general, simple, elegant, practical and consistent systems. As such, such choices although not actually forced or necessary, for if so they would not be choices within the sense meant here, are nevertheless the choices that come closest to being required by past traditions.1

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