Determinacy and Non-Classicality in Arithmetic

Wider research context / theoretical framework: The mathematician’s self-image is often portrayed as that of a theoretician studying a domain of immutable truths, a domain devoid of ambiguity or indeterminacy. But is this picture sustainable in light of twentieth-century mathematics and philosophy?
It seems not. The work of Kurt Gödel and Paul Cohen and many others has shown just how pervasive independent mathematical sentences are. Standard logical techniques easily establish the existence of different mathematical structures relative to which independent sentences are true or false, respectively. It seems that independent sentences are true relative to some intended ways
of interpreting our mathematical theories and false relative to others, yielding a case of indeterminacy. Faced with independence results, philosophers of mathematics divide themselves into two broad camps: some double down on the mathematician’s self-image and argue that independence only reveals that our theories do not describe mathematical truth with enough precision, whilst others renounce orthodoxy and accept that some mathematical sentences simply lack determinate truth-values. The debate is immediately complicated by the fact that some authors privilege (in)determinacy only within particular sub-fields.