Abstract
The paper starts by looking at how ‘practical’ and ‘theoretical’ mathematics and their relation
have been understood from the Greeks to Christian Wolff and by historians of mathematics
from Montucla to recent days. Drawing on earlier work of mine, and on the giants
on whose shoulders I (try to) stand, I then suggest a categorization of the mathematical
knowledge types a historian has to deal with: the ‘sub-scientific’ type, carried by practitioners
taught in an apprenticeship network; the ‘scholasticized’ type, taught supposedly for
practice but in a ‘scribal’ school by masters whose own genuine practice is that of teaching;
and the ‘scientific’ or theory-oriented type. In the end, the utility of this categorization is
tried out on two specific cases.