 The arguments from productivity and systematicity are persuasive. Many linguists and philosophers have suggested that the explanation of these phenomena that presupposes compositionality is not only the best, but also the only one imaginable. So, before I survey some of the putative counterexamples to compositionality from the semantic literature, to bolster the imagination I will discuss a simple non-linguistic case where our understanding is productive and systematic despite apparent lack of compositionality in the system of representations.