Synthetic Fracterm Calculus

Previously, in [Bergstra and Tucker 2023], we provided a systematic description of elementary arithmetic concerning addition, multiplication, subtraction and division as it is practiced. Called the naive fracterm calculus, it captured a consensus on what ideas and options were widely accepted, rejected or varied according to taste. We contrasted this state of the practical art with a plurality of its formal algebraic and logical axiomatisations, some of which were motivated by computer arithmetic. We identified a significant gap between the wide embrace of the naive fracterm calculus and the narrow precisely defined formalisations. In this paper, we introduce a new intermediate and informal axiomatisation of elementary arithmetic to bridge that gap; it is called the synthetic fracterm calculus. Compared with naive fracterm calculus, the synthetic fracterm calculus is more systematic, resolves several ambiguities and prepares for reasoning underpinned by logic; indeed, it admits direct formalisations, which the naive fracterm calculus does not. The methods of these papers may have wider application, wherever formalisations are needed to analyse and standardise practices.