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so methods that reverse their opinions needlessly are less reliable than those that are guaranteed to
keep one on the straightest possible path to the truth.
Let us see how that idea applies to a standard theory choice problem―curve fitting.
Assume that the truth is some polynomial law
) =
. Suppose that the question is
what the polynomial degree of the true relation is, where the degree of
is the greatest
such that
is non-zero. The data are increasingly precise open rectangles in the plane, through which the
graph of
passes. The answers to the question concerning polynomial degree are epistemically
, in the following way. Think of inquiry as an endless game between nature and the scientist.
Nature wins the game if, in the limit, she presents ever more, ever tighter open rectangles for some
polynomial relation
on which the scientist does not converge to the true degree of
. The scientist
wins if she converges to the true degree of
, or if nature fails to present ever more data true of
some polynomial. Nature can start by presenting more and more data for some degree zero curve
(i.e., a flat line). If nature continues to do so and the scientist never converges to the answer “degree
0,” then the scientist loses. So the scientist has to plump for “degree 0” sooner or later. By then, at
most finitely many open rectangles have been presented. Since the rectangles are all open, it
remains possible to tilt the line a little to obtain a curve of degree 1 that is compatible with the
rectangles presented already. Now nature can continue presenting more rectangles for the tilted
line until, on pain of losing the game, the scientist plumps for the conclusion “degree 1.” At that
point, nature can inflect the tilted line slightly into a parabola that still fits all the open rectangles
presented so far, and so on. So nature has a winning strategy to force an arbitrary, convergent
scientist to reverse opinion from degree 0 to degree 1 to degree 2, and so on, to any degree
Here is the analogy to the freeway example. The freeway to the truth is the sequence of
theories “degree 0,” “degree 1,” …, “degree
” through which nature can force an arbitrary,
convergent method. Since nature has a winning strategy to do so, no method can guarantee better
worst-case performance than that, just as no route home can be better than the freeway. So the
freeway to the truth is the sequence “degree 0,” “degree 1,” …, “degree
,” …. Think of each stage
along the freeway to the truth as an exit to a possible destination city.
Now consider an Ockham violator. Suppose that she even gets lucky, and jumps
immediately to degree 3, which happens to be
. Then, the anticipated cubic effects (two
inflection points in the graph of
) do not come in. Time passes. They still do not come in. She
realizes that if they never come in, she is going to lose the truth-finding game (think of the driver,
meandering through the Pennsylvania mountains). So, eventually, she has to reverse her opinion
from degree 3 to degree 0. Now, nature
has a winning strategy to force the scientist back up
through degrees 1, 2, and 3, so her overall path to the truth is (3, 0, 1, 2, 3) rather than the optimal,
Ockham path (0, 1, 2, 3). Her reward for getting lucky at hitting on the complex truth a priori is
just an extra reversal of opinion compared to the Ockham method. Note that (3, 0, 1, 2, 3) is also
of opinion, whereas (0, 1, 2, 3) has no cycle, so the violator does worse in
mentioned by the ancients in the Katha Upanishad!