When is the algorithm concept pertinent – and when not? Thoughts about algorithms and paradigmatic examples, and about algorithmic and non-algorithmic mathematical cultures

Until some decades ago, it was customary to discuss much pre-Modern mathematics as “algebra”, without agreement between workers about what was to be understood by that word. Then this view came under heavy fire, rarely with more precision.
Now, instead, it has become customary to classify pre-Modern practical arithmetic as “algorithmic mathematics”. In so far as any computation in several steps can be claimed to follow an underlying algorithm (just as it can be explained from an “underlying theorem”, for instance from proportion theory, or from a supposedly underlying algebraic calculation), this can no doubt be justified. Traditionally, however, historians as well as the sources would speak of a rule.
The paper first goes through some of the formative appeals to the algebraic interpretation – Eisenlohr, Zeuthen, Neugebauer – as well as some of the better argued attacks on it (Rodet, Mahoney). Next it asks for the reasons to introduce the algorithmic interpretation, and discusses the adequacy or inadequacy of some uses. Finally, it investigates in which sense various pre-modern mathematical cultures can be characterized globally as “algorithmic”, concluding that this characterization fits ancient Chinese and Sanskrit mathematics but neither early second-millennium
Mediterranean practical arithmetic (including Fibonacci and the Italian abbacus tradition), nor the
Old Babylonian corpus.