complex bakeoffs and error bounds

[Vaughan Pratt]

	Interval arithmetic is one approach to dealing with uncertainty.

Indeed.  However the definition of "interval" as a pair [a,b], which
works fine for the reals, makes no sense in the complex plane.  The
right definition of "interval" is "convex set", and at dimensions
higher than one this is still too weak to be useful.

A more robust and generally useful definition would I think be "ball"
(in the sense of "interior of sphere").  Thus an "interval" would
consist of a single quantity together with a radius bounding the
error.  This works not just for the complex numbers but for any Banach
space if just adding and subtracting, and for any Hilbert space if also
multiplying.  The passage to these more general spaces should permit
the integration of interval arithmetic with more traditional error
analysis for matrix packages.

What *doesn't* change in this passage is the real value of real-valued
interval arithmetic.  The trouble with interval arithmetic is that it
is even more paranoid than the Simpson jury, who acquitted OJ because
(as I read it) of the reasonable likelihood of police tampering with
the evidence in the light of the Fuhrman tapes enthusiastically
admitting to such practices.  (Jury reasoning: If as the LAPD has
stated quite clearly Mr. Simpson is indeed guilty, what harm could
there be in a little creative tampering to turn the mushy
circumstantial evidence that fate dealt the LAPD into a slam-dunk
case?  If you are at most once shy of the LAPD, that reasoning might
not work as well for you as for the jury.)  In practice computer
arithmetic randomly walks around the right answer and does not deviate
anywhere near as far from it as interval arithmetic conservatively
calculates to be possible.

The trouble with interval arithmetic is that every answer is innocent
(= possibly the true answer) until proven guilty, in the *mathematical*
sense of "proven".  With long computations, as with long trials, that
gets to be an impractically tall order.