 The neo-Fregean movement seeks to reveal that a significant amount of mathematics is analytic. This is a stronger claim than that it is a priori and derives no part of its justification from empirical science, or even from successful applications within the empirical sciences. For that would hold of mathematics (or indeed any other branch of knowledge) conceived of as synthetic a priori. The neo-Fregean maintains in addition that significant parts of mathematics flow logically from principles that are analytic of (or definitional of) their central concepts or predicates, such as ‘natural number’ or ‘real number’. That is, they flow from the very meanings of those central predicates. (We opt here for the linguistic version of the analyticity claim). Note the stress here on ‘significant  parts’.[35]  We know from Gödel’s second incompleteness theorem that any consistent and sufficiently strong theory of arithmetic is unable to prove or refute (the formalized statement of) its own consistency. The latter statement is true, but unprovable. In light of the incompleteness phenomena, one would be hard pressed to make good on the claim that all mathematical truths are true by virtue only of such logical considerations as can be captured in systems of formal  proof.[36]  When the first principles of a mathematical theory, such as arithmetic, form an essentially incomplete axiomatization, the logicist will have to maintain that the justification of any new first principle can be furnished in some strictly logical sense. 