Affine logic for constructive mathematics
Abstract
We show that numerous distinctive concepts of constructive mathematics arise automatically from an "antithesis" translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically "constructivize" classical definitions, handling the resulting bookkeeping automatically.
Keywords
Cite
@article{arxiv.1805.07518,
title = {Affine logic for constructive mathematics},
author = {Michael Shulman},
journal= {arXiv preprint arXiv:1805.07518},
year = {2022}
}
Comments
51 pages. v2: More general and precise treatment of semantics, using categories instead of posets, and hyperdoctrines and comprehension instead of triposes. Changed name to "antithesis translation" and more properly emphasized "affine" over "linear". v3-4: minor fixes; final version to appear in BSL