496 Paul Thompson
assertion because it not specified by the theory itself.^18
Justifying the assertion that an isomorphism exists is a complex task which
requires: the employment of a range of other scientific theories, the adoption of
theories of methodology (e.g., theories of experimental design, goodness of fit, etc.),
the employment of a variety of domains of mathematics (e.g., probability, statistics,
linear algebra), etc. If a theory and the phenomena within its intended scope are
shown to be isomorphic, then explaining and predicting outcomes within the model
constitutes explaining and predicting outcomes in the empirical world. Advocates
of the semantic view have seen the separation of the theory and the methods of
its application and confirmation as a major logical, heuristic and methodological
advantage of the conception.
An isomorphism or homeomorphism is a one-to-one correspondence between
the elements of one or more sets resulting from a bijective mapping. A bijective
mapping is a ‘one-to-one’ (injective) and ‘onto’ (surjective) mapping. A function
fis a one-to-one mapping function if and only if:
(x 1 =x 2 )⇔(f(x 1 )=f(x 2 )).A functionfis an onto mapping function if:
y∈Y⇒∃x∈X:f(x)=y.The essential feature of an isomorphism in the context of scientific theories is the
assertion of the “sameness” of the structure and behaviour of the model and the
empirical world (i.e., if there is a one-to-one, and onto, mapping which preserves
relations, functions and constants). As a consequence of characterising the relation
between a theory and the empirical world as an isomorphism or homeomorphism,
(^18) In the syntactic conception, on the other hand, the semantics is provided by correspondence
rules whicharepart of the theory and directly link the formal system to the phenomenal world.
In effect the correspondence rules define an empirical model of the formal system. That empirical
model is understood as logically equivalent to the phenomenal system to which the theory applies.
It is for this reason that actual phenomena can be deduced from the statements of the theory.
That is, the statements of the theory are laws that describe theactualbehaviourofobjectsinthe
world. Hence, any behaviour deduced from the statements of the theory is either a prediction
about whatactuallywill happen under the specified circumstance or an explanation of what
actuallydid happen under the specified circumstances. The interpreted formal system directly
describes the behaviour of entities in the world. In a semantic conception, a theory is defined
directly by specifying in mathematical English the behaviour of a system. Most importantly, laws
do not describe the behaviour of objects in the world, they specify the nature and behaviour
of an abstract system. This abstract system is, independently of its specification, claimed to
be isomorphic to a particular empirical system. Establishing this isomorphism, as I shall argue
below, requires the employment of a range of other scientific theories and the adoption of theories
of methodology (e.g., theories of experimental design, goodness of fit, etc.). In essence, the
substance of the difference between the two conceptions is that the semantic conception calls
into question the possibility of providing an adequate semantics for a scientific theory by means
of correspondence rules. And, calls into question the need for any reference to a formal system
since the semantics can be provided directly by defining a mathematical model. Advocates of
the semantic view have seen the separation of the theory and the method of its application as a
major logical, heuristic and methodological advantage of the conception.