This paper poses two problems for von Glasersfeld’s Radical Constructivism. The first problem concerns the rejection of the idea that it is possible to share meanings. The second problem is that Radical Constructivism rejects the notion of an objectively existing reality of which we can have objective knowledge. Yet with respect to mathematics, von Glasersfeld seems to claim that it is possible to obtain objective knowledge. We propose an alternative position – Constructive Realism – that gives a description of what mathematical objects are and gives an account of why knowledge in mathematics is objective. Furthermore, we argue that some of the assumptions, used in von Glasersfeld’s description of how numbers are formed, support the claim that some meanings are objective and that communication is possible. Finally, we consider some of the implications this position has for mathematics education.
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