Historiography of Mathematics from the Mathematician’s Point of View

Mathematicians used to be highly invested in the study of the history of their own field, but their voice in historiographical discussions has diminished in influence in the past century. One prominent narrative paints this as a justified fall from grace: mathematicians wedded to present mathematical values looked at the past with prejudiced eyes, whereas a new generation of historians were better able to appreciate the past proper, in its own terms. But the best internalist mathematical historiography of old needed no such external corrective. It was already committed to avoiding presentism and anachronism, for reasons that were not in opposition to mathematical values but rather derived directly from a positive vision of the role that history could play in the mathematical community. In this vision, a historical understanding of how a field developed is a proxy for first-hand research experience in that field. It follows that it is essential for historical accounts to thoroughly convey the scope and limitations of alternative conceptions and approaches, including dead-end developments, since this is precisely what sets the critical knowledge gained by first-hand research experience apart from the doctrinal knowledge gained merely from a textbook. Hence, from this point of view, presentist historiography is not the natural outlook of the mathematician, but rather a direct antithesis of the mathematician’s most fundamental reason for studying history. To study past mathematics precisely as it appeared to active researchers at the time is not foreign to the mathematician, but a direct corollary of the mathematician’s core conviction that only those with first-hand research experience in a field of mathematics truly understand it. A sophisticated internalist historiography derived from these ideals was articulated to a greater extent in the past than is commonly recognized today. By going back to its roots, the mathematician’s historiography could revive some of the virtues that have been neglected in recent years.