Mind, Body, World- Foundations of Cognitive Science, 2013a

applied recursively to inscribe smaller triangles into any of the triangles that were
added to the figure in a previous stage of drawing. Figure 3-5 shows the result when
this rule is applied four times to Figure 3-4.
Figure 3-5. The Sierpinski triangle that results when the recursive rule is applied
four times to Figure 3-4.
The Sierpinski triangle, and all other fractals that are created by recursion, are
intrinsically self-similar. That is, if one were to take one of the smaller triangles
from which Figure 3-4 is constructed and magnify it, one would see still see the
hierarchical structure that is illustrated above. The structure of the whole is identical
to the (smaller) structure of the parts. In the next section, we see that the recursive
nature of human language reveals itself in the same way.
3.3 Phrase Markers and Fractals
Consider a finite set of elements (e.g., words, phonemes, morphemes) that can, by
applying certain rules, be combined to create a sentence or expression that is finite
in length. A language can be defined as the set of all of the possible expressions that
can generated in this way from the same set of building blocks and the same set of
rules (Chomsky, 1957). From this perspective, one can define a grammar as a device
that can distinguish the set of grammatical expressions from all other expressions,
including those that are generated from the same elements but which violate the
rules that define the language. In modern linguistics, a basic issue to investigate is
the nature of the grammar that defines a natural human language.
Chomsky (1957) noted that one characteristic of a natural language such as
English is that a sentence can be lengthened by inserting a clause into its midst. As
we see in the following section, this means that the grammar of natural languages
is complicated enough that simple machines, such as finite state automata, are not
powerful enough to serve as grammars for them.
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