Discovering mappings between concept hierarchies is widely regarded as one of the hardest and most urgent problems facing the Semantic Web. The problem is even harder in domains where concepts are inherently vague and ill-defined, and cannot be given a crisp definition. A notion of approximate concept mapping is required in such domains, but until now, no such notion is available. The first contribution of this paper is a definition for approximate mappings between concepts. Roughly, a mapping between two concepts is decomposed into a number of submappings, and a sloppiness value determines the fraction of these submappings that can be ignored when establishing the mapping. A potential problem of such a definition is that with an increasing sloppiness value, it will gradually allow mappings between any two arbitrary concepts. To improve on this trivial behaviour, we need to design a heuristic weighting which minimises the sloppiness required to conclude desirable matches, but at the s...