Re: Modeling infinitesimals with 2x2 matrices

[Vaughan Pratt <pratt@CS.Stanford.EDU>]

I'm told that Bell's "microlinear calculus" in his 1998 book on infinitesimals
is equivalent to the matrix approach I suggested, so that was not new after
all, other than perhaps its formulation in terms of 2x2 matrices.

On the other hand it is apparently mixed in with Bell's strongly
intuitionistic outlook, whereas it would seem intuitively that something
so simple as a model of d^2 = 0 should transcend whether one is working
intuitionistically or classically.  A more classical version of Bell's account
might be of interest (perhaps to relatively few people on the categories
mailing list though, which seems to have a strongly intuitionistic slant).

Meanwhile I received the April issue of Mathematics Magazine just now, and it
has an article on pp. 118-129 on "Geometry of Generalized Complex Numbers"
by Anthony and Joseph Harkin.  The microlinear calculus, under the names
"Study product" and "parabolic complex numbers," apparently dates back
to Study's 1903 book Geometrie der Dynamen.  The Harkins associate i^2 =
-1,0,1 with respectively Ordinary (i.e. complex) product, Study product,
and Clifford product (though Clifford algebras include ordinary product as
well, the quaternions being a Clifford algebra).

The article makes no mention of infinitesimals, and it would be interesting to
try to find the appropriate infinitesimal interpretations of the geometric
properties of the parabolic complex plane.

One approach I very much like to infinitesimals that I haven't seen in the
nonstandard analysis/infinitesimal literature (but would certainly appreciate
pointers) is one that does all the work with what one might call finitesimals.
A finitesimal h is just a positive real that you plan one day to reduce to
zero, and thus organize everything around it to that end.

Polynomials in R[x] of degree d form a (d+1)-dimensional vector space.
The usual basis for this space is the d+1 monomials x^i for i in 0..d.
However if one fixes h > 0 and takes the basis to be 1, x, x(x-h),
x(x-h)(x-2h),... then Boole's difference calculus works essentially
identically to the infinitesimal calculus for polynomials represented in
the monomial basis.  Since h is a free variable throughout the development,
one can do all the work first and then drive h to 0 uniformly everywhere
at the end.  Expressions such as x^i (Knuth writes an underbar under the i
and calls it "x to the falling i") mention h only implicitly and hence don't
change (as symbolic expressions) as h changes, though their numerical values
at any given x change.  The Stirling numbers of the first and second kind,
organized as matrices, constitute linear transformations from the bases
for h=1 to h=0 and back again, respectively.

I've looked from time to time at how one might extend this to exponentials
and logarithms, but have never been satisfied with the results.  It would be
nice to know how to deal exactly with exp(it) for nonzero h.  If this were
possible it might give an even nicer constructive treatment of infinitesimals
than the others, and one that didn't care at all whether one was classically
or intuitionistically inclined.

Vaughan Pratt