Engineering - Molecular Theory of the Living Cell - page 120

mechanisms for a great many physical phenomena that have previously seemed completely
mysterious [p.8]. (5.4)

... traditional mathematical models have never seemed to come even close to capturing the

kind of complexity we see in biology. But the discoveries in this topic show that simple
programs can produce a high level of complexity. And in fact it turns out that such
programs can reproduce many features of biological systems - and seem to capture some
of the essential mechanisms through which genetic programs manage to generate the actual
biological forms we see. [p. 9]. (5.5)
Over and over again we will see the same kind of thing: that even though the underlying
rules for a system are simple, and even though the system is started from simple initial
conditions, the behavior that the system shows can nevertheless be highly complex. And I
will argue that it is this basic phenomenon that is ultimately responsible for most of the
complexity that we see in nature. [p. 28]. (5.6)
... intuitions from traditional science and mathematics have always tended to suggest that
unless one adds all sorts of complications, most systems will never be able to exhibit any
very relevant behavior. But the results so far in this topic have shown that such intuition is
far from correct, and that in reality even systems with extremely simple rules can give rise
to behaviors of great complexity. [p. 110]. (5.7)

It may be asserted here that Statements (5.6) and (5.7) apply to biology. If so, these
statements would represent the most important contributions that Wolfram has
made to biology. Based on his numerous computer experiments, often involving
millions of iterations of a set of simple rules applied to simple initial conditions,
Wolfram concluded that complex structures can arise from simple programs. For
example, he was able to simulate complex structures and processes such as the
shapes of shells (Fig. 5.1 ) and trees and turbulence, using simple rules governing
the behavior of cellular automata, from which he inferred that all complex
structures and phenomena in nature can originate from recursive operations of
sets of simple rules. The similarity between the computer-generated shell shapes
and the real ones shown in Fig. 5.1 is striking and seems to provide credibility to
Wolfram's assertions, i.e., Statements (5.1)-(5.7).
Although I do agree with Wolfram that his NKS does have the ability to
represent or simulate certain complex phenomena in nature that could not even
be approached using traditional mathematical tools, I suggest that both the tradi-
tional mathematics and NKS may still be subject to the constraints of the cookie-
cutter paradigm described in Sect. 2.3.9 . That is, no matter which model one
adopts, either traditional mathematical or NKS, models always cut out only those
aspects of reality that fit the model (i.e., the cookie cutter) and leave behind “holes”
and the rest of in the dough that are beyond the capability of the models employed.
We may depict this idea as shown in Fig. 5.2 .

5.2.2 Complexity, Emergence, and Information

Ricard (2006) defines a complex system as a composite system whose properties or
degrees of freedom cannot be predicted from those of its components. In other