Re: Modeling infinitesimals with 2x2 matrices

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

From: Steve Vickers <s.j.vickers@cs.bham.ac.uk>
>I thought that was the whole reason for contemplating infinitesimals in
>the first place - 0/0 cannot be meaningful, but d/d is 1, ed/d is e etc.

Well, you're not alone in thinking that way.  This was the basis for
Robinson's invention of nonstandard analysis: the belief that a/b has to
be defined for all nonzero b in order to make infinitesimals nonparadoxical.

Instead of formulating division a/b as an operation, go back to its motivating
formulation as a system in search of a solution, in this case the system
consisting of the one linear equation a = bx in one unknown.

Had the system been one of ordinary or partial differential equations,
there would be no argument that the solution space could turn out quite
oddly shaped.

Now when a and b are reals the solution space is a rectangle: only the
column indexed by b=0 is undefined.  This remains true when a and b are
extended to the complex numbers, or even to the quaternions.

But if you extend the domain to the algebra R(2) of 2x2 real matrices,
the columns indexed by singular matrices now lose some of their entries.
But not all, and so the solution space ceases to be rectangular.

Robinson believed that the way to make infinitesimals safe for analysis was
to make the solution space for a = bx rectangular.  Today's logicians are
magicians with logic: if logic indicates the impossibility of a rectangular
solution space, no need to abandon that goal, just bend logic until the
solution space does become rectangular.  The students will bend with you,
at least those who've approached nonstandard analysis with the proper
upbeat spirit about how much simpler analysis becomes when infinitesimals
can be objectified.  Power tools are wonderful.

To answer your question (or comment), in the system of refined numbers I
described, if b is infinitesimal and nonzero, a = bx is solvable if and only
if a is infinitesimal.  The division table is no longer rectangular.  So what?

One might grumble that a = bx can't have an infinitesimal part when a and
b are both infinitesimals, but in the simple cases that's a plus.  In more
complicated cases, 2x2 matrices aren't enough, you need nxn matrices,
with distance of nonzero entries from the diagonal measuring the degree
of their infinitesimality (if that's a word).  In this case d^n = 0 only
for higher n's.

After thinking along those lines for a bit more the other day, I decided
that even though I liked this approach better than throwing ultrafilters at
it, it still wasn't as good as doing analysis in Boole's finite difference
calculus with h remaining unbound throughout, the approach I'd used since the
early 1970's.  That approach has the great advantage of being able to use the
same analysis in classical and quantum physics by setting h=0 to interpret a
result classically and setting it to Planck's constant to interpret the same
result quantumly.  As a case in point, the same integration formulas can
deliver areas under smooth curves and discrete summations of e.g. n^3, the
latter with h=1. (I already wrote a bit about that two or three messages ago.)

The right power tools are even more wonderful.

Vaughan Pratt