Histories of mathematical practice: reconstruction, genealogy, and the unruly pasts of ruly knowledge

Abstract

Histories of mathematical practice account for mathematical knowledge and action by interpreting presently-available evidence as traces of the events, contexts, and relations that make up the past. Interpretations depend on the assumptions one makes about how mathematical knowledge works, insofar as it is knowledge and insofar as it is mathematical. Though the specific rules and their meanings can differ from context to context, mathematics is a kind of ruly knowledge, expected to follow orderly patterns and principles wherever it is found. The contexts and activities of mathematical practice—how that knowledge is made, shared, applied, and understood—are necessarily less ruly, and different practices leave or occlude different kinds of evidence for historical interpretation. The apparent ruliness of mathematics can be both a resource and an obstacle for understanding its unruly pasts. Historians’ interpretive assumptions and goals have been shaped by centuries of interaction between mathematics research, history, and education.As a guide for mathematics educators and education researchers to historical perspectives on mathematical practice, this article briefly introduces four major interpretive traditions that inform the present discipline of mathematics history. It then illustrates some interpretive approaches and challenges through the history of blackboards in mathematical practice before explaining the two broad kinds of historical interpretation applied to mathematical practices. Reconstruction involves understanding the conditions and contexts of practices in a single historical moment. Genealogy, by contrast, connects elements of the past across time through transmission, interpretation, adaptation, and other kinds of preservation and change.

Historiography of Mathematics from the Mathematician’s Point of View

“…Critiques of the internalist-mathematical approach have naturally focused on the faults of presentist internalist historiography, under the mistaken impression or insinuation that this undermines internalist approaches altogether. The above contextualization suggests instead that, rather than attributing historiographic naiveté to inherent characteristics of the mathematical psyche, a better way to understand the mathematician's relation to history is in terms of the purposes history can serve to the mathematical community (Barany 2020(Barany , 1077.…”