Learning from Liu Hui? A Different Way to Do Mathematics, Volume 49, Number 7

C ould we have done mathematics differently? At a logical level this question is trivial: research mathematicians spend their time exploring all the ways one can “do it differently” and then doing them. There are no signs that the mathematical enterprise has any artificial barriers round it that stand in the way of this task. But my question refers to something a little less foundational. What if we make the “counterfactual move” of trying to imagine the history of mathematics with one of its great monuments no longer there—say Euclid, for example. Could we imagine a possible history of mathematics without Euclid? By “mathematics without Euclid” I do not of course mean non-Euclidean geometry, but rather a mathematics stripped of the whole axiomatic-deductive scheme for which Euclid’s writing served as the great exemplar and entry point for generations of western mathematicians. At first glance the likely course of development of such a mathematics seems so different from our own that it might deserve a place in one of the more intellectually inclined episodes of Star Trek (Spock: “It’s mathematics, Jim, but not as we know it”). But in fact it is not even necessary to imagine such an alternative history of mathematics, since one already exists. Ancient China developed its own mathematical culture based on a radically different approach to the structuring of mathematical thought. Unfortunately, not only western but also Chinese historians of mathematics have often failed to see this point. As a result, early Chinese mathematicians have been portrayed as if they were doing the same job as Euclid, only—to be blunt—a whole lot worse. In history, as in other disciplines, using the wrong tools for the job often breaks the thing you were trying to fix. But what were ancient Chinese mathematicians up to if they were not playing the Euclidean game? I hope that the answer to this question may be of more than historical interest, since it bears directly on the pressing question of how mathematicians are to be made and made more effectively.