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. The examples given here are heterogeneous in their origins and functions. Cassirer and Grattan-Guinness find their principles and notions (respectively) in applied mathematics and empirical science. Lautman sought the same ideas (in his Platonic sense of ‘idea’) in mathematics and physics, though he looked longer and harder in pure mathematics than in physics (this doctrine, that mathematics and physics have a common root, is part of his Platonism). The heuristic patterns in Lakatos and Polya are more closely specific to pure mathematics, though this may be an artefact of their contingent mathematical interests. Regarding function, heuristic patterns are not necessarily the deepest of these shapes, precisely because they are evident before (or at least during) mathematical investigation, whereas Lautman’s Platonic ideas typically come into view late in the day, when they are instantiated in several different mathematical theories. Cassirer’s principles and Grattan-Guinness’ notions seem to be intermediate, having both heuristic and ontological significance (reading ‘ontological’ here in something like Lautman’s Platonic-Heideggerian sense). The aim of this paper is not to undertake the large task of classifying and comparing these various kinds of enabling structures. Rather it is to