Dialectica Dialectica Vol

dialectica dialectica Vol. 63, N° 4 (2009), pp. 381–395 DOI: 10.1111/j.1746-8361.2009.01214.x Vectors and Beyond: Geometric Algebra and its Philosophical Significancedltc_1214 381..396 Peter Simons† In mathematics, like in everything else, it is the Darwinian struggle for life of ideas that leads to the survival of the concepts which we actually use, often believed to have come to us fully armed with goodness from some mysterious Platonic repository of truths. Simon Altmann 1. Introduction The purpose of this paper is to draw the attention of philosophers and others interested in the applicability of mathematics to a quiet revolution that is taking place around the theory of vectors and their application. It is not that new mathematics is being invented – on the contrary, the basic concepts are over a century old – but rather that this old theory, having languished for many decades as a quaint backwater, is being rediscovered and properly applied for the first time. The philosophical importance of this quiet revolution is not that new applications for old mathematics are being found. That presumably happens much of the time. Rather it is that this new range of applications affords us a novel insight into the reasons why vectors and their mathematical kin find application at all in the real world. Indirectly, it tells us something general but interesting about the nature of the spatiotemporal world we all inhabit, and that is of philosophical significance. Quite what this significance amounts to is not yet clear. I should stress that nothing here is original: the history is quite accessible from several sources, and the mathematics is commonplace to those who know it. However, for philosophers who are not themselves mathematicians, I hope this may be of some interest. 2. Excerpts from the history of vector theory Dramatis personae Benjamin Olinde Rodrigues (1795–1851) Sir William Rowan Hamilton (1805–1865) Hermann Günther Grassmann (1808–1877) Peter Guthrie Tait (1831–1901) † Department of Philosophy, Trinity College Dublin, Dublin 2, Ireland; Email: [email protected] © 2010 The Author. Journal compilation © 2010 Editorial Board of dialectica. Published by Blackwell Publishing Ltd., 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA 382 Peter Simons Josiah Willard Gibbs (1839–1903) William Kingdon Clifford (1845–1879) Oliver Heaviside (1850–1925) David Hestenes (b. 1933) There is a classic history of vector theory, a marvellous book by Michael Crowe (1967). The subsequent history since Crowe’s book appeared shows that while he is quite right that vectors won out over quaternions in “the Great Quaternionic War” of roughly 1890–1910, it is no longer the case, as it seemed to him then, that vector analysis plus other things will remain the undisputed formalism of choice in modern applied mathematics. Although the idea of directed quantities was present in earlier mathematics, it emerged most obviously in the Wessel–Argand geometric interpretation of complex numbers. It was after thirteen years of trying to generalize this to three dimensions that in a famous episode Hamilton got the insight that three rather than two extra imaginary units were needed, and this was the key to the quaternions. The discovery, which can be located in space and time to a very small region near Broome Bridge in Dublin on the morning of 16 October 1843, is one of the best-known stories of mathematics. Quaternions (sometimes called ‘hypercomplex numbers’) are quantities: qw=+++ijk x y z where w, x, y, w are real, addition is defined termwise, and the imaginary units ijksatisfy: i22== j k 2 = ijk =−=−=1;; ij ji k jk =−= kj i ; ki =−= ik j Quaternions form the largest (non-commutative) division algebra: every non-zero quaternion has an inverse. From 1846 Hamilton called w the scalar part and ix + jy + kz the pure or vector part of a quaternion, and if q, q′ are two pure quaternions (vectors): qq′ =−() xx′ + yy′ + zz′ + ijk() yz′ − zy′ + () zx′ − xz′ + () xy′ − yx′ So quaternion multiplication embodies both (the negative of) the scalar product, and the vector product of later theory. Just as complex numbers handle rotations in the plane, so quaternions are meant to handle rotations in three-dimensional space. But, misled by the analogy with the two-dimensional case, where multiplication by a single unit complex number produces a rotation, Hamilton assumed such rotations could be represented by a single multiplication qv, where v is any vector and q is a unit quaternion. This one-sided multiplication works only for a special case: rotations orthogonal to the © 2010 The Author. Journal compilation © 2010 Editorial Board of dialectica. Vectors and Beyond 383 vector, about its axis. Thus from the very evening of 16 October 1843 Hamilton bungled the quaternion account of rotation, which should be dealt with by pre- and postmultiplication by a quaternion rvr-1, with r defined not (as in Hamilton) by the angle of the rotation but by its half-angle. The geometry of composing rotations and the need to deal in half-angles had already been properly done in 1840 by a French mathematician who was also a socialist banker, Benjamin Olinde Rodrigues, but had (not unsurprisingly) escaped Hamilton’s notice. Particularly insightful and detailed accounts of Hamilton’s mistake and Rodrigues’s success are given by Altmann (1986, 1989, 1992), who deserves much credit for showing that the neglect of quaternions was for a long time excessive, as well as for investigating the hitherto highly obscure Rodrigues (Altmann and Ortiz, 2005).1 Hamilton was convinced that quaternions were the key to the universe. For a while, spurred by his deserved fame and his enthusiastic advocacy, quaternions were all the rage. His 1853 Lectures on Quaternions ran to nearly 900 pages, and James Clerk Maxwell used them though without total endorsement in his Treatise on Electricity and Magnetism. In the propagation of the quaternionic gospel, Hamilton was ably seconded by his bulldog, the Scottish mathematician Peter Guthrie Tait, but gradually the drawbacks of quaternions became apparent, not least through the work of Maxwell, who eventually dropped them. Quaternions came under fire from two thinkers who independently came up with the more flexible mathematics of vectors, the American mathematician–physicist–chemist Josiah Willard Gibbs and the English engineer–physicist Oliver Heaviside. Gibbs’s Yale lectures on vector analysis began to be circulated around 1881, while Heaviside’s Electromagnetic Theory came out in 1891. After the publication in 1901 by Edwin Bidwell Wilson of Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics and Founded upon the Lectures of J. Willard Gibbs, the war was decided. That book reads like any modern textbook of vector theory, and indeed it was the first such. For example the definitions of scalar and vector are wholly modern: Definition:Avector is a quantity which is considered as possessing direction as well as magnitude. Definition:Ascalar is a quantity which is considered as possessing magnitude but no direction. Gibbs and Heaviside arrived at their ideas independently and in a remarkably similar way: by separating the scalar and vector parts of the quaternion product as two distinct products, and changing the sign of the scalar part. After embarking on vector theory, Gibbs discovered some convergence with ideas that had already 1 Quarternions have, sinceAltmann published his book in 1986, and in part because of the book, had about as much of a revival as they merit: they are now regularly used by space engineers and computer games developers for efficient and singularity-free representations of 3-dimensional spatial rotations, as of space vehicles and virtual game perspectives. See Kuipers (2002). © 2010 The Author. Journal compilation © 2010 Editorial Board of dialectica. 384 Peter Simons been proposed by Hermann Günther Grassmann.2 Grassmann was, in his day, and even to some extent now, one of the most unjustly neglected mathematicians of all time. His revolutionary Ausdehnungslehre [Theory of Extension] was first published in 1844 (Grassmann 1844), with a second revised and cut-down edition in 1862 (Grassmann 1862) and a third, which restored some of the first edition’s cuts, appearing in 1878 only after his death. Grassmann was led to his algebra of geometry initially by a problem in applied mathematics: that of describing the flow of water in tides (Grassmann 1840). For this purpose he introduced two notions which later became known as the scalar (inner) product and exterior (or outer) product of two vectors. Grassmann’s work, which was conceived for any finite number of dimensions, not just two or three, was more revolutionary than Hamil- ton’s and at a level of abstraction which even professional mathematicians such as Möbius and Kummer found offputting. The Gibbs–Heaviside vector theory is as much a cut-down version of Grassmann’s work as of Hamilton’s, though because of Grassmann’s obscurity they discovered that only subsequently. Vectors in the Gibbs–Heaviside vein caught on rapidly in the early twentieth century, being both adequate to many applications and easy enough for non-mathematicians to grasp and manipulate. From then on, vector algebra and vector analysis have held sway throughout most of the twentieth century. The quaternionists did not however give up without a massive struggle. In 1888 Gibbs wrote in a letter: a Kampf ums Dasein is just commencing between the different methods and notations of multiple algebra, especially between the ideas of Grassmann & of Hamilton. The rhetoric in the Great Quaternionic War became quite heated. So Tait wrote: Even Professor Gibbs must be ranked as one of the retarders of Quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann.

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