 Is there any limit to this process of “concept-stretching”, or any distinction to be drawn between interesting and frivolous concept-stretching? Can this process yield, not fallible conjectures and proofs, but certainty? Lakatos’s editors distinguish the certainty of proofs from the certainty of the axioms from which all proofs must proceed. They claim that rigorous proof-procedures have been attained, and that “There is no serious sense in which such proofs are fallible” (P&amp;R: 57). Quite so. But only because we have decided not to “stretch” the logical concepts that lie behind those rigorous and formalizable proof-procedures. A rigorous proof in classical logic may not be valid in intuitionistic or paraconsistent logics. And the key point is that a proof, however rigorous, only establishes that if the axioms are true, then so is the theorem. If the axioms themselves remain fallible, then so do the theorems rigorously derived from them. Providing foundations for mathematics requires the axioms to be made certain, by deriving them from logic or set theory or something else. Lakatos claimed that this foundational project had collapsed (see below,  §3.2).