During the 19th century, however, there emerged another cluster of ways of regarding axioms, which we shall refer to as formalist. What they had in common was a rejection of the idea just mentioned that the axioms can be regarded simply as true statements about a subject matter external to them. One part of the motivation for the emergence of formalism lay in the different axiom systems for geometry — Euclidean, hyperbolic, projective, spherical — which mathematicians began to study. The words ‘point’ and ‘line’ occur in all, but the claims made using these words conﬂict. So they cannot all be true, at any rate not unconditionally. One view, then, is that axioms should be thought of as assumptions which we suppose in order to demonstrate the properties of those structures that exemplify them. The expositor of an ax- iomatic theory is thus as much concerned with truth on this view as on the realist one, but the truths asserted are conditional: if any structure satisﬁes the axioms, then it satisﬁes the theorem. This view has gone under various names in the literature — implicationism, deductivism, if-thenism, eliminat- ive structuralism. Here we shall call it implicationism

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