🚀 go-pugleaf

     Being one of the earlier posters who appealed to it as a
philosophic criterion for comparing scientific theories, I now
feel obliged to put forth my defense of Ockham's Razor. (The
spelling "Occam" is also correct.)

     The first order of business is describing precisely what (in
this article) is an application of Ockham's razor. From reading
the many postings on the subject, it seems there is no concensus
on this. In presenting the definition below, no claim is pressed
that this is closest to what William originally intended. It is
simply the version that I wish to defend.

  o  The term "theory" refers to a set of statements. The set,
     though probably infinite in cardinality, must be finitely
     describable to qualify as a usable scientific theory. But in
     discussing the relative merits of theories, it is the entire
     set that is important. This point is expanded below.
     
  o  I assume some noncontroversial method for identifying which
     statements are testable by observation. These might include
     such statements as "grass is green" or "if A is performed, B
     will be observed". (This is a very controversial assumption,
     but one that is not the subject of this article.)
     
  o  A theory X can now be divided into an "observable component"
     C and its non-observable component X - C. Note that X - C
     includes mixed statements. (If the integration path is
     closed, conservation of flux is observed.) Only statements
     that are completely testable are in C. In general, both C
     and its complement are infinite.

     A theory Y is stronger than (equal to) a theory X if every
statement in X is also in Y (and vice versa). A theory Y with
observational component D has greater (equal) explanatory power
than a theory X with observational component C if every statement
in C is also in D (and vice versa). 

     The version of Ockham's Razor that this article defends can
now be stated. If X and Y are two theories and Y is properly
stronger than X but has the same explanatory power, then X is
preferred. (If C and D are the observational components of X and
Y respectively, then C = D but X [ Y.)

     The justification for this is not difficult. Scientific
theories are not deductively derived. The only reason for putting
any stock in them at all is their explanatory power.  A
scientific theory is assumed true because of the observable
statements it predicts (contains). In particular, the only reason
for believing any statement y in Y that is not in D is because it
helps explain D. But in the case above this is not true for the
statements in Y that are not in X. They can all be dropped with
no weakening of the explanatory power of the theory. 

     By talking about entire theories rather than their finite
descriptions via sets of axioms, the difficulties that arise
because of different equivalent axiomatizations are avoided.
Sometimes two axiomatizations can be directly compared. One set
of axioms may properly contain the other. But this is rare, and
can almost always be ducked by the defenders of a theory by
changing the axiom set. In science, it is an entire theory that
is assumed true. Discussing a theory only in terms of its axioms
is to focus on a description of the theory, usually one of many,
rather than the theory itself. 

     At this point, a potential criticism of the above view must
be presented and dismissed. It might be argued that in light of
the strong argument above for weak theories, the best theory is
simply the concerned set of testable statements, C. But C
(probably) does not have a finite axiomatization. This puts it
out of the running as a usable theory.  While in comparing two
theories one should not worry about particular finite
descriptions, it is undoubtedly true that the motivation behind
posing theories in the first place is to find finite
(understandable) descriptions of the infinite range of observable
phenomena. 

     A second rejoinder that might be made is that taking such a
strong stance for weak theories completely disallows assuming the
existence of entities that cannot be directly observed. One
cannot assume that electrons exist, only that observations will
be in accord with QED. The first reply to this complaint is that
this is not so bad. The second reply is that definitions come
for free. One can define an electron as the locus of a set of
behavior. Adding a definition to a theory, for simplifying
discussion, should not be viewed as changing its content.
Similarly, mathematical content is not important in comparing
physical theories.

Russell