Syntactic analogies and impossible extensions

Mathematicians study shapes, structures and patterns. However, there are shapes, structures and patterns within the body and practice of mathematics that are not the direct objects of mathematical study. Rather, they are part of the explanation of how mathematical study is possible, and thus demand the attention of epistemologists and phenomenologists as well as mathematicians. Partial philosophical accounts of these enabling structures include heuristic in the senses of Polya and Lakatos; principles in the sense of Cassirer; ideas in the sense of Lautman and notions in the sense of Grattan-Guinness (Polya, 1954; Lakatos, 1976; Cassirer, 1956; Lautman, 2006; Grattan-Guinness, 2008). The study of these structures lies in the intersection of mathematics and philosophy because some of these shapes, structures and patterns may eventually submit to mathematical treatment, but others may have a `Protean' quality that will always escape formal treatment. [opening paragraph]