Re: Modeling infinitesimals with 2x2 matrices

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

>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.

On reflection this is not so simple in the case when b in a/b is
infinitesimal.  First, noting that R(2) is noncommutative, the requirement
should be phrased as two equations, a = bx and a = xb to prevent multiple
solutions whose diagonal is not constant.

But while this duplication then determines a unique real part (the diagonal),
the two equations fail to pin down the infinitesimal part (the upper right
entry).  That's the sort of thing that happens with matrices of less than
full rank.

When there is no solution, certainly a/b should be considered undefined.
But when there are multiple solutions, the question arises as to whether to
punt completely (as with 0/0) or do something creative such as setting the
undefined infinitesimal part to 0 (as with the ratio of two infinitesimals).

One test is whether the "dominant" term is fixed, but this breaks down
for 0/b.

A better test would be to use the rank of b to decide how much of the
quotient a/b to ignore---if b has rank 1 (a nonzero infinitesimal) then
ignore the infinitesimal part of a/b.

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One virtue of Robinson's approach is its universality with respect to all
first-order definable functions; this however is not sufficient to compensate
for its more counterintuitive aspects.  Now that I'm starting to see that
the zero-divisor approach is less easily managed than I'd first thought,
I'm not so sold on it either at this point (but maybe all its difficulties
have been overcome somewhere...?).

Meanwhile I remain convinced that Boole's finite difference approach to
handling infinitesimals is superior (recall the trick here: setting h=0
instead of some positive quantity like 1 or Planck's constant introduces no
artificial singularities with Boole's method).  His 1860 *A Treatise on the
Calculus of Finite Differences," substantially revised by J.F. Moulton for
the 1872 edition after Boole's death, is 336 pages of inspired analysis.
(You can get second hand copies for $10 from Amazon; my very second hand
copy has "F.S. Curry, Trin. Coll., Feb. 1881" written on the inside cover.)

The preface to the first edition starts out,

"In the following exposition of the Calculus of Finite Differences,
particular attention has been paid to the connexion of its methods with
those of the Differential Calculus---a connexion which in some instances
involves far more than a merely formal analogy.

Indeed the work is in some measure designed as a sequel to my *Treatise
on Differential Equations*.  And it has been composed on the same plan."

An updated version of this book incorporating the greatly matured perspective
on linear algebra since then could be a worthwhile project for someone
interested in improving on the existing explications of infinitesimals as
real objects.  While Boole's system beats the current crop hands down in
principle (in my view anyway), in outlook it is showing its age.

Category theory creatively applied might also help.  I confess to having no
idea how intuitionistic logic could be brought to bear effectively though.
I can see that not cancelling certain double negations might preserve certain
nuances that convey certain constructively motivated notions, but to my
untrained eye these come across as nuances with a capital N when their
contribution is assessed in the larger picture of alternative approaches
to constructivizing infinitesimals.  That makes me either a beer guzzler
at a wine tasting or the owner of a screwdriver in a room full of hammer
owners depending on one's outlook.  :)

YBMV (Your biases may vary.)

Vaughan Pratt
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