Vaughan Pratt writes:
Why not model d as the matrix
0 1 0 0 ?
This is a perfectly good quantity, adding and scaling just like any real, e.g.
2d = 0 2 0 0.
And obviously d^2 = 0.
Part of this idea is implicit in the usual algebraic geometry treatment of infinitesimals as nilpotents. In addition to the usual "point", such that complex functions on this space form the commutative ring C, algebraic geometers like to think about the "point with nth-order nilpotent fuzz", such that complex functions on this space form the commutative ring C[d]/<d^n = 0>. They visualize this as a space slightly bigger than a point: just big enough to tell the difference between the function 0 and the function whose first n-1 derivatives equal zero! To deal with this sort of "space" in a precise way, someone like Grothendieck invented the category of affine schemes, which is just the opposite of the category of commutative rings. But affine schemes are happier as part of a larger category of schemes... and thus topos theory was brought kicking and screaming into the world. To see how this led to a really nice treatment of infinitesimals, see: F. William Lawvere, Outline of synthetic differential geometry, available at http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html or Anders Kock, Synthetic Differential Geometry, Cambridge U. Press, Cambridge, 1981. But, it's also tempting to embed the commutative ring C[d]/<d^n = 0> into the noncommutative ring of nxn complex matrices, by letting d be a slightly off-diagonal matrix, like this: 0 1 0 0 0 0 1 0 (in the case n = 4) 0 0 0 1 0 0 0 0 (Vaughan is considering the case n = 2.) And this is more like how Alain Connes thinks of infinitesimals: as part of the bigger world of noncommutative geometry! Best, jb