“I have not yet any clear view as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties”

declared John Graves in reaction to his mathematician friend’s invention of the quaternions (Hamilton 1843). Henri Poincaré held it as:

“a revolution in arithmetic which is entirely similar to the one which Lobachevsky effected in geometry”

Yet what exactly were the prevailing conceptions so radically violated thereby ? For more background to this cf the book by Israel Kleiner A History of Abstract Algebra (Birkhäuser 2007), from which also the quotations above are taken (chapter “History of Ring Theory”). The main aspect regards, as far as and in terms which I can understand, the decoupling, for certain algebras, of multiplication from is arithmetical understanding as the repetition of addition. Multiplication is not associative for such algebras, as it is called. This means that the “products” are no longer “determinded” by the uniformity and linearity of one operation applied to the involved quantities, but rather that they can be “formed” by modulating the differential impacts of various operations in a many dimensional space (where the dimensionality is qualified positioning-value system). These spaces must be conceived as modeling spaces constituted by certain precepts (instantiated in the choice of the dimensionality installed), not merely a representational space. They are representational spaces of what we imagine, or something like this.

Hamilton’s own motivation in introducing the quaternions was to extend the algebra of vectors in the plane to an algebra of vectors in 3-space. The quaternions acted as a catalyst for the exploration of diverse “number systems” with properties which departed in various ways from those of the real and complex numbers – especially because the extended the “jump” performed already by the complex numbers from 1 dimension to 2 dimensions. The quaternions continued this exploration into 3 dimensions, but eventually number systems were invented exploring higher dimensionalities, culminating in Ernst Grassmann’s so called “exterior geometry”, or in German Ausdehnungslehre (1844), a vector algebra in n-dimensional space. For Grassmann this held the promise of continuing to work out the Leibnizian idea of a geometria speciosa, in combination with a mathesis universalis. The continuous magnitudes of geometry could become objects of specification, generalization, classification, he dreamt – and this without the threat of arbitrariness and non-rigorousness which lead Leibniz to his neo-pythagorean assumption of the pre-established harmony (which Kant opposed to much against). Giving primacy to quantities (numbers) before qualities (forms) does not anymore leave us, necessarily, freely afloat in the numerical infinities without possibility for orientation in and architectonics of reasoning. Algebraic method, once it follows conceptual and symbolic laws within the transparency of their constitution (categories and axioms) is universal without totalizing what it allows to comprehend.

To explore the immune reaction against this direction, it is worth reading Bertrand Russell’s treaties On the Foundations of Geometry (1888) which reads as a radical plea to disregard the emerging inversion inversion, and instead return to the Kantian view on intuition and “institutionalize” a fundamental(ist) rôle for it.

cf for an interesting and recent paper on Grassmann’s fascinosum:

Applications of Geometric Algebra and the Geometric Product to solve Geometric Problems, by Ramon Gonzales Calvet (Proceedings to Applied Geometric Algebras in Computer Science and Engineering 2010)

cf  for a diagrammatic map of the people involved in this field, and their philosophical proximities, stakes, sympathies, legacies, have a look at Gerhard Dirmoser’s map of Grassmann’s and Riemann’s environment here.