Page 151 - MODES of EXPLANATION

Basic HTML Version

That is really the crux for any attempt to provide a non-circular, non-metaphysical
justification of Ockham’s Razor. It is
easy
to see that you do worse at finding the truth in
simple
worlds if you assume that the truth is complex. But if you also do
better
in complex worlds by
inferring a complex theory, then the argument for simplicity requires a circular or question-
begging assumption that the truth is simple, which prompts metaphysical speculations. To break
out of the circle, it is necessary to show something apparently paradoxical―that presuming a
complex theory makes your truth-finding performance worse, even in worlds in which the
complex
theory is true. But it really is no more paradoxical than driving north to get on to a
south-bound interstate.
To examine the same idea from a slightly different angle, consider the worst-case reversal
bounds for Ockham and for the violator. In general, over degree
n
, Ockham reverses opinion at
most
n
times, but the violator reverses her opinion at least
n
+ 1 times.
So Ockham
dominates
the
violator in terms of worst-case reversal bounds over answers to the question. Therefore, Ockham’s
razor is, in a sense, necessary for optimal worst-case reversal performance in the polynomial
degree problem. It is also sufficient: every convergent method for the polynomial degree problem
that never rules out the simplest possibility compatible with information is both reversal efficient,
in terms of worst-case bounds over answers. It is not hard to see that the result extends to any
problem in which the alternative theories are totally ordered by simplicity as T
1
, T
2
, T
3
, …. Call
back-and-forth results of that kind
Ockham efficiency theorems
.
The preceding argument can be generalized. First, the above Ockham efficiency theorem
holds only in the fairly simple setting of the polynomial degree problem, in which answers are
sequentially ordered by simplicity in an obvious way. One can extend the argument to a very
general class of theory choice problems. A theory choice problem involves three components: (1)
a set of possible worlds, or ways the world could be; (2) the possible information that would be
received, eventually, in each world; and (3) the question, or the range of possible theories among
which one is choosing. Simplicity reflects
iterations
of the traditional problem of induction that
one faces in such a problem. You face the problem of induction from one answer to another in a
theory choice problem if, regardless of how the first answer is true, you would never receive
information ruling out the second answer. The problem of induction from one answer to another
defines an order on answers, and you face an iterated problem of induction along paths in that
order
The polynomial degree problem has just one such path “degree 0,” “degree 1,” …, “degree
n
,” …, but that is a very special case. In general, simplicity orders, like freeway systems, can
branch
. For example, suppose that the question is not merely to find the polynomial degree of
f
,
but the
form
of
f
, which is defined as the set of all
i
such that is non-zero in the normal form
polynomial expression of
f
.
Second, scientists don’t usually plump for a unique theory at every stage of inquiry. They
typically suspend judgment for a while by disjoining all answers compatible with experience until
a sufficiently long run of simple data
confirms
the simplest answer. So Ockham’s razor should be
stated generally enough to deal with
disjunctive
beliefs over branching simplicity orders. That
5