Constructivism without verificationism

• Main Author: Hansen, Casper Storm
• Published: University of Aberdeen 2014
• Subjects: 100; Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.629405

This dissertation lays the philosophical groundwork for a mathematics that combines some of the virtues of constructivist mathematics and classical mathematics; from the former, an austere ontology and independency of actual in nity, and from the latter, objectivity of truth values and a strong logic. The rst half of the dissertation is concerned with classical mathematics and Brouwer's intuitionism. Classical mathematics is criticized for relying on a notion of in nity that may not even be possibly instantiable. Brouwer is criticized for relying on an incoherent notion of free choice sequences and based on a novel interpretation of intuitionism in terms of a strong and a weak notion of truth for unnecessarily mixing mentalism about mathematical objects with veri cationism. The second half develops a so-called non-veri cationist constructivist philos- ophy of mathematics that accepts all and only possible constructions as truth makers, independently of what is and can be veri ed about them. Classical arithmetic is vindicated on this basis. Classical set theory is, of course, not. Rather, it is argued that a mentalistic set theory has to be non-well-founded and that this forces a revision of logic. However, the result is not intuitionis- tic logic but something akin to the logic that comes out of Kripke's theory of truth: bivalence fails only in the absence of groundedness, not in the absence of proof and decidability. The akin to reservation is due to the fact that there are serious problems with the speci c formal theory given by Kripke. His theory is extended and modi ed with the aim of solving these problems while staying within the bounds of non-veri cationist constructivism.