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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

=

(

) =

∑

=0

. 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

she

pleases.

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

a

of opinion, whereas (0, 1, 2, 3) has no cycle, so the violator does worse in

senses

mentioned by the ancients in the Katha Upanishad!

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