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Monday, March 13, 2006

The history of quaternions

If this paper of mine is to be believed, we should expect the best of the action in the philosophical study of mathematics to take place between what I call there genealogical and tradition-constituted accounts of mathematical rationality. An example of a paper I'd classify as genealogical is Andrew Pickering's 'Concepts and the Mangle of Practice: Constructing Quarternions' (in Mathematics, Science, and Postclassical Theory, Barbara Herrnstein Smith and Arkady Plotnitsky (eds.), 1997, pp. 40-82), which tells the story of Hamilton's work on quaternions. Genealogical histories of a practice tend to delight in bringing contingency to centre stage - things could have been so very different. What is very noticeable in such histories is that often the very early days of a practice are treated. This gives the genealogist the advantage of only needing to study a handful of people with all their idiosyncracies. The underlying thought is that if so much could have been so different while the course of a practice was being set, how different things could be decades later. And, if we can find a sharp change of direction away from the original pioneer's intentions late on our story, so much the better. Most of the original thinking guiding the practice will be revealed to be just a story. Any number of stories might have governed at that time, leading mathematics in very different directions.

So, in Pickering's paper, with the pace of research so slow, we can dwell on Hamilton's idiosyncratic metaphysical views, and we can tell the story of the quaternions as having "mutated over time into the vector analysis central to modern physics." (p. 45). Hamilton had failed to reach his original goals, only achieving "a local association of calculation with geometry rather than a global one. He had contructed a one-to-one correspondence between a particular algebraic system and a particular geometric system, not an all purpose link between algebra and geometry, considered as abstract, all-encompassing entities." (p. 59). The quaternions could not form the required calculus for reasoning about entities in three-dimensional space. Even after Hamilton had considered multiplication on just the imaginary part, where the product of two lines could be an ordinary number or another imaginary, "...the association of algebra with geometry remained local. No contemporary physical theories, for example, spoke of entities in three-dimensional space obeying Hamilton's rules. (p. 60). "It was only in the 1880s, after Hamilton's death, that Josiah Willard Gibbs and Oliver Heaviside laid out the fundamentals of vector analysis, dismembering the quaternion system into more useful parts in the process. This key moment in the delocalization of quaternions was also the moment of their dintegration." (p. 60). From this an innocent reader might take it that, by and large, that was that as far as the quaternions were concerned.

So what do we make, then, of this paper, which documents the use of the quaternions and allied algebras in physics to recent times in the form of an analytic bibliography of 1300 references? With so many man-hours devoted to the extraction of whatever can be found to be useful aboout quaternions, and their relationships with other mathematical entities, do the first few decades of their lives tell us very much? Although it makes for engaging history, do we learn so much about the ways in which mathematics operates at its highest level of organisation from the quixotic quests of individuals, rather than from an account of droves of workers, most of whom must necessarily remain largely faceless?

Responding to a journalist's account of the 'Mathematics and Narrative' conference, in his letter to the Independent (British national daily) - Mathematicians struggle for truth - Ronnie Brown stresses that mathematics is not the work of a handful of individuals, but rather "a world-wide collaborative effort involving tens of thousands". But, to raise again the kind of question of an earlier post, how can a history of the application of the quaternions, the work of hundreds of people, be written?


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