  Peirce began with a philosophical statement of a different kind about mathematics which has become his best remembered single statement “Mathematics is the science that draws necessary conclusions” (Peirce 1870, p. 1). What does ‘necessary’ denote? Perhaps he was following a tradition in algebra, upheld especially by Britons such as George Peacock and Augustus De Morgan (a recipient of the lithograph), of distinguishing the ‘form’ of an algebra from its ‘matter’ (that is, an interpretation or application to a given mathematical and/or physical situation) and claiming that its form alone would deliver the consequences from the premises. In his first draft of his text he wrote the rather more comprehensible “Mathematics is the science that draws inferences”, and in the second draft “Mathematics is the science that draws consequences”, though the last word was altered to yield the enigmatic form involving ‘necessary’ used in the book. The change is not just verbal; he must have realised that the earlier forms were not sufficient (they are satisfied by other sciences, for example), and so added the crucial adjective. Certainly no whiff of modal logic was in his air. His statement appears in the mathematical literature fairly often, but usually without explanation. One feature is clear, but often is not stressed. In all versions Peirce always used the active verb ‘draws’: mathematics was concerned with the act of drawing conclusions, not with the theory of so acting, which belonged in disciplines such as logic. He continued: 