 The onslaught on formalism continues in a pair of papers “Infinite Regress and the Foundations of Mathematics” (1962) and “A Renaissance of Empiricism in the Recent Philosophy of Mathematics?” (1967a). Here Popper predominates and Hegel recedes. Regress is a critique of both logicism and formalism proper (that is, Hilbert’s programme), concentrating primarily on Russell. Russell sought to rescue mathematics from doubt and uncertainty by deriving the totality of mathematics from self-evident logical axioms via stipulative definitions and water-tight rules of inference. But the discovery of Russell’s Paradox and the felt need to deal with the Liar and related paradoxes blew this ambition sky-high. For some of the axioms that Russell was forced to posit—the Theory of Types which Lakatos sees, in effect, as a monster-barring definition (elevated into an axiom) that avoids the paradoxes by excluding self-referential propositions as meaningless; the Axiom of Reducibility which is needed to relax the unduly restrictive Theory of Types; the Axiom of Infinity which posits an infinity of objects in order to ensure that every natural number has a successor; and the Axiom of Choice (which Russell refers to as the multiplicative axiom)—were either not self-evident, not logical or both. Russell’s fall-back position was to argue that mathematics was not justified by being derivable from his axioms but that his axioms were justified because the truths of mathematics could be derived from them whilst avoiding contradictions: