1. Social constructivism which is an influential theory in education is construed in a wide variety of ways as Sally Haslanger, "Objective Reality, Male Reality, and Social Construction" in Women, Knowledge, and Reality: Exploration of Feminist Philosophy, 2d ed. (New York: Routledge,1996), 84-107 notes: "[T]here is striking diversity in how the term 'social construction' (and its cognates) is used, and consequently diversity in what revisions to the old models are proposed. In addition to the claims that race, gender, and sexuality are socially constructed, it is also claimed, for example, that the 'subject,' 'identity,' 'knowledge,' 'truth,' 'nature,' and 'reality' are each socially constructed. On occasion it is possible to find the claim that 'everything' is socially constructed, or that it is socially constructed 'all the way down.'" Despite this diversity it seems that a common thread running through social constructivist theories is a rejection of the epistemologies of both rationalism and empiricism. Instead, it is argued that the social relations both "construct" our knowledge and somehow overturn it.

2. See, for example, Paul Ernest, The Philosophy of Mathematics Education (London, U.K.: Macmillian Co., 1991) and "Social Constructivism and the Psychology of Mathematics Education," in Constructing Mathematical Knowledge: Epistemology and Mathematics Education, ed. Paul Ernest (London U.K.: Falmer Press, 1994). Ernest's personal website <www.ex.ac.uk/~PErnest/> gives a good indication of his involvement in mathematics education and the philosophy of mathematics education.

3. Paul Ernest, Social Constructivism as a Philosophy of Mathematics (New York: SUNY Press, 1998). This book will be cited as SCPM in the text for all subsequent references.

4. G.H. Hardy, "Mathematical Proof," in Mind 38, no. 149 (January 1929): 5.

5. Kurt Gödel, "What is Cantor's Continuum Problem?" (1982) in Philosophy of Mathematics, ed. P. Benacerraf and J. Putnam (1964; reprint, Cambridge: Cambridge University Press, 1993), 483-84.

6. Gödel, "What is Cantor's Continuum Problem?" 484 and Charles Parsons, "Mathematical Intuition," in Proceedings of the Aristotelian Society 53 (1980): 145-68.

7. James Brown, Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures (London: Routledge, 1999), 13.

8. This contention seems to be similar to a claim about the character of the language used to describe spatial relations. In a survey paper, Michel Denis reports on findings of psychologists that demonstrate "the importance for speakers of sharing identical (or sufficiently similar) representations of the domain of space. In addition, it is crucial for felicitous social interactions that speakers share knowledge of specific lexical meanings of spatial expressions and have also capacities for setting up common deictic space. These factors are especially important in collaborative dialogue about spatial objects or configurations"; Michel Denis, "Imagery and the Description of Spatial Configurations," in Models of Visuospatial Cognition, ed. Manual de Vega et al. (New York: Oxford University Press, 1996), 190. This somewhat unclear passage seems to contend that the language employed to describe spatial relations cannot be arbitrarily changed because the precision of this language is a necessary condition for social discourse about spatial relations.

9. Meno, 82b.

10. Bernard Lonergan, Insight: A Study of Human Understanding (Toronto: University of Toronto Press, 1957/1992), 31-32. (I have added the emphasis in this passage.)

11. The "roughly Aristotelian approach" which I have presented has a kinship with Charles Parsons's approach in "Mathematical Intuition."