More thoughts on whether the cosmos is mathematics ____________________________________________ It's been my conviction for as long as I can remember ["How long's that?" Uh... how long's what?] that a firm distinction must be made between pure mathematical modelling --- a cosy activity within a comforting, disciplined world whose confines I greatly appreciate --- and applied engineering --- a nasty, unpredictable business demanding unquantifiable intuition based on considerable experience. I see no convincing reason why cosmology should not also be expected to observe this distinction. But the second stage of the process may involve a considerable psychological hurdle: for the whole point of designing a model is for its user to be able to identify model with application; yet this confusion is precisely what its designer must resist! I can't make up my mind just now whether or not the following anecdote really does support rejection of Tegmark's thesis. But the point it makes seems worthwhile --- and anyway it gives me a pretext to go boring on again about Geometric Algebra. Clifford algebras are a more general concept embracing complex numbers, quaternions, and some other more exotic species of "number": the basic idea is to start with the real numbers, then attach a fixed set of "imaginary" generators. Some of these square to 0, some to -1, some to +1; any two distinct generators anti-commute; otherwise they observe the usual laws of algebra. Several theoretical physicists have utilised CA's for building cosmological models. My own aspirations are more humble: I just want to do ordinary geometry in Euclidean 3-space, hopefully in a more systematic fashion than it is usually hacked about in present-day computer graphics etc. For this I employ an algebra I call DCQ (dual complex quaternion), with 4 generators o,x,y,z such that o^2 = 0, x^2 = y^2 = z^2 = 1, and o x = - x o, x y = - y x, etc. [Several alternatives have been proposed by other authors, notably David Hestenes.] Having chosen a particular "pure model" --- the algebra, a matter of mathematics --- I am faced with an entirely separate decision regarding its application to the "real world" --- projective geometry. In this case the decision is so apparently straightforward that it seems usually to have been made unconsciously: obviously, the point with real Cartesian coordinates (a,b,c) will be represented (homogeneously) by a x + b y + c z + 1 o --- yes? The coordinate system which results from this naive identification is quite useful for some things --- for instance, subspaces and their meet and join operations are representable --- but frankly it's barely worth the effort. So, actually --- no! In fact, an immensely superior application identifies instead the "number" a x + b y + c z + d o with the _plane_ a x + b y + c z + d = 0 (in the usual notation). The unexpected bonuses of this alternative include metrical quantities and proper and improper isometries, the combination of which permit the design of algorithms in robotics which do not at present appear to be possible to motivate or formulate by conventional means, however determined. Finally, the point of this little saga is to illustrate that --- even within this very restricted universe --- it is not possible to ditch the "baggage": the pure model may be the same in each case; but lacking any identification, the real worlds may nonetheless be distinct. OK, I'll shut up and go away now. Fred Lunnon