On partita doppia

In 1494 Fra Luca Pacioli published in Venice one of the first printed mathematical books [P1494]. One section, Computis e Scripturis, is the first published description of partita doppia or double-entry bookkeeping, the foundation of accounting. Double-entry bookkeeping had been developed over a period of 200 years by Italian merchants and bankers. The aim of accounting is the measurement of a distributed concurrent system, and it is our contention that it is one of the earliest and most successful mathematical theories of concurrency. It is interesting that at that time negative numbers were not accepted; in the following century mathematicians repaired this deficiency though there seems to have been no attempt to explicate the distributed algebra underlying partita doppia. For further information about early accounting see [BJ84], [DeR56]. In this paper we give a precise mathematical account of partita doppia in terms of an algebraic structure on Span(RGraph) the bicategory of spans of reflexive graphs. Some interesting new mathematical considerations arise in the study of this example. The fact that accounting concerns the measurement of