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new genes that are formed through novel sequences. On higher levels of biological
organisation, emergent structures and functions can similarly arise from novel com-
binations of previously existing molecular, cellular, and organismic constituents.
In psychology, associationist theories hold that emergent mental states arise from
novel combinations of pre-existing primitive sensations and ideas. Whether cast in
terms of platonic forms, material atoms, or mental states, combinatoric emergence is
compatible with reductionist programs for explaining macroscopic structure through
microscopic interactions (Holland
1998
).
This strategy for generating structural and functional variety from a relatively
small set of primitive parts is a powerful one that is firmly embedded in many of our
most advanced informational systems. In the analytic-deductive mode of exploration
and understanding, one first adopts some set of axiomatic, primitive assumptions,
and then explores the manifold, logically-necessary consequences of those assump-
tions. In the realm of logic and mathematics, the primitives are axioms and their
consequences are deduced by means of logical operations on the axioms. Digital
computers are ideally suited for this task of generating combinations of symbol-
primitives and logical operations on them that can then be evaluated for useful,
interesting, and/or unforeseen formal properties. In the field of symbolic artificial
intelligence (AI) these kinds of symbolic search strategies have been refined to a
high degree. Correspondingly, in the realm of adaptive, trainable machines, directed
searches use evaluative feedback to improve mappings between features and clas-
sification decisions. Ultimately these decisions specify appropriate physical actions
that are taken. In virtually all trainable classifiers, the feature primitives are fixed and
pre-specified by the designer, contingent on the nature of the classification prob-
lem at hand. What formally distinguishes different kinds of trainable machines is
the structure of the combination-space being traversed, the nature of the evaluative
feedback, and the rules that steer the search processes.
15.2.4 Limits on Computations on Existing Primitives
Combinatoric novelty is a dynamic, creative strategy insofar as it constantly brings
into being new combinations of elements. However, such combinatoric realms are
inherently limited by their fixed, closed sets of primitive elements. Consider the set
of the digits 0-9 vs. a set of 10 arbitrarily distinguished objects. The first set is well-
defined, has 10 actual members, and is closed, while the latter set is ill-defined, has
an indefinite number of potential members, and is open.
1
All that can happen within well-defined universes are recombinations of existing,
pre-specified symbols—there is no means by which new primitive symbols can be
created by simply recombining existing ones. It seems obvious enough that one
1
A Platonist could claim that all sets are open because they can include null sets and sets of sets
ad infinitum
, but we are only considering here sets whose members are collections of concrete
individual elements, much in the same spirit as Goodman (
1972
).
