What can our understanding of analogical reasoning in mathematics tell us about the nature of metaphor? I wish to propose that it suggests the possibility of distinguishing between profound and shallow metaphors along the lines of a mathematical distinction between ‘important’ or ‘fruitful’ results and trivial ones. First, I will offer examples of parallels between what mathematicians say about analogy and what poets say about metaphor, then discuss and, I hope, resolve, apparent points of non-correspondence. I will conclude by drawing out the connexion of this issue to the debate between constructivists and realists in mathematics, and its importance for our evaluation of poetry as a way of thinking about deep metaphysical issues.