ABSTRACT. An analysis of traditional mathematical proof is undertaken, with an implicit contrast to formal derivations. The semantic interpretation of mathematical terms plays a role in the former that doesn't appear in the latter. This semantic interpretation - with an accompanying role for intuition - is explained in terms of inference packages, which are psychologically-bundled ways of phenomenologically exploring the effect of several assumptions at once without explicit recognition of what those assumptions are. Although its correspondence with a derivation is the (ultimate) justification for the success of a traditional proof, the certainty that mathematicians experience when they study successful traditional proofs is not due to that correspondence, but rather - for the most part - to the role of inference packages in their reasoning.