The spectrum of practises of mathematical reconstruction is explored on the basis of a case study on a partly successful mathematical reconstruction of the Chinese Remainder Theorem. In the 19th century L. Matthiessen reconstructed two versions of this theorem on the basis of a corrupted secondary source concerning ancient Chinese mathematics. He identified the more restricted version of the theorem with a Gaussian approach, whereas the other more general one was described as something new surpassing contemporary mathematical European achievements. I identify and compare two different types of mathematical reconstructions in Matthiessen's contributions, and explore their historiographic functions. To capture the relation between mathematical reconstruction and anachronism, the time scheme in the case study is analyzed and linked to the concept of pluritemporality according to Landwehr. This more complex perspective on the category of time in historical research suggests that anachronism should be re-conceptualized. It allows for a discussion of some of the conditions under which mathematical reconstructions can be used in a historiographically sensitive way in a different setting. I argue that this kind of historiographically sensitive mathematical reconstruction can be regarded as a productive historiographical method

Historians are constantly confronted with the twin problems of translating texts and interpreting their meanings. When mathematicians like Georg Cantor or Abraham Robinson demonstrate the consistency of concepts that, since the paradoxes of Zeno and Democritus, have been assumed to be paradoxical notions like infinitesimals or the actual infinite, how should the works of earlier mathematicians be regarded, who either used such concepts or believed they had proven their impossibility? Is it anachronistic to use nonstandard analysis or transfinite numbers to “rehabilitate” or explain the works of Leibniz, Euler, Cauchy, or Peirce, for example, as recent mathematicians, historians, and philosophers of mathematics have attempted? At the other extreme, chronologically, how may ideas readily accepted in the West – like incommensurable numbers, parallel lines, and similar triangles – but foreign to traditional Chinese mathematics have adversely affected the interpretations of ancient Chinese mathematical works?

This chapter is devoted to what I call textual anachronism. By this expression, I refer to forms of anachronism that lead to interpreting ancient texts on the basis of anachronistic assumptions with respect to how these texts made sense for the ancient actors. This is, for instance, the case when historians take the textual components that they find in ancient documents (like a mathematical problem, an algorithm, a proof, and a diagram), as they would take what they consider to be modern counterparts, and interpret these components on that basis. I argue that this form of anachronism has caused misinterpretations of several kinds, which I identify and analyze, using examples from four historical contexts. I further discuss the historiographical implications of this type of anachronism. I conclude with the thesis that one can limit the effects of this form of anachronism by using a historical approach to forms of text. Moreover, a historical approach of this kind sheds light on the fact that anachronism has a history that might also be interesting to consider.