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Outside of purely mathematical considerations, perhaps the greatest value of functional decomposition is the insight it provides into the structure of the world. When a functional decomposition can be achieved, this provides ontological information about what structures actually exist in the world, and how they can be predicted and manipulated. For example, in the illustration above, if it is learned that <math>{x_1}</math> depends directly only on <math>{x_2}</math>, this means that for purposes of prediction of <math>{x_1}</math>, it suffices to know only <math>{x_2}</math>. Moreover, interventions to influence <math>{x_1}</math> can be taken directly on <math>{x_2}</math>, and nothing additional can be gained by intervening on variables <math>\{x_3,x_4,x_5\}</math>, since these only act through <math>{x_2}</math> in any case.
==Philosophical considerations==
{{Importance section|date=November 2021}}
The philosophical antecedents and ramifications of functional decomposition are quite broad, as functional decomposition in one guise or another underlies all of modern science. Here we review just a few of these philosophical considerations.
===Reductionist tradition===
One of the major distinctions that is often drawn between [[Eastern philosophy]] and [[Western Philosophy]] is that the Eastern philosophers tended to espouse ideas favoring [[holism]] while the Western thinkers tended to espouse ideas favoring [[reductionism]]. This distinction between East and West is akin to other philosophical distinctions (such as [[Philosophical realism|realism]] vs. [[anti-realism]]). Some examples of the Eastern holistic spirit:
* "Open your mouth, increase your activities, start making distinctions between things, and you'll toil forever without hope." <small>— The [[Tao Te Ching]] of [[Lao Tzu]] (Brian Browne Walker, translator)</small>
* "It's a hard job for [people] to see the meaning of the fact that everything, including ourselves, depends on everything else and has no permanent self-existence." <small>— [[Majjhima Nikaya]] (Anne Bankroft, translator)</small>
*"A name is imposed on what is thought to be a thing or a state and this divides it from other things and other states. But when you pursue what lies behind the name, you find a greater and greater subtlety that has no divisions..." <small>— [[Visuddhi Magga]] (Anne Bankroft, translator)</small>
The Western tradition, from its origins among the [[Greek philosophers]], preferred a position in which drawing correct distinctions, divisions, and contrasts was considered the very pinnacle of insight. In the [[Aristotelianism|Aristotelian]]/[[Porphyry (philosopher)|Porphyrian]] worldview, to be able to distinguish (via strict proof) which qualities of a thing represent its [[essence]] vs. [[Property (philosophy)|property]] vs. [[accident (philosophy)|accident]] vs. [[Intensional definition|definition]], and by virtue of this formal description to segregate that entity into its proper place in the taxonomy of nature — this was to achieve the very height of wisdom.
===Characteristics of hierarchy and modularity===
In natural or artificial systems that require components to be integrated in some fashion, but where the number of components exceeds what could reasonably be fully interconnected (due to square wise growth in number of connections (= n over two or = n * (n - 1) / 2)), one often finds that some degree of hierarchicality must be employed in the solution. The general advantages of sparse hierarchical systems over densely connected systems—and quantitative estimates of these advantage—are presented by {{Harvtxt|Resnikoff|1989}}. In prosaic terms, a hierarchy is "a collection of elements that combine lawfully into complex wholes which depend for their properties upon those of their constituent parts," and wherein novelty is "fundamentally combinatorial, iterative, and transparent" {{Harv|McGinn|1994}}.
An important notion that always arises in connection with hierarchies is modularity, which is effectively implied by the sparseness of connections in hierarchical topologies. In physical systems, a module is generally a set of interacting components that relates to the external world via a very limited interface, thus concealing most aspects of its internal structure. As a result, modifications that are made to the internals of a module (to improve efficiency for example) do not necessarily create a ripple effect through the rest of the system {{Harv|Fodor|1983}}. This feature makes the effective use of modularity a centerpiece of all good software and hardware engineering.
===Inevitability of hierarchy and modularity===
There are many compelling arguments regarding the prevalence and necessity of hierarchy/modularity in nature {{Harv|Koestler|1973}}. {{Harvtxt|Simon|1996}} points out that among evolving systems, only those that can manage to obtain and then reuse stable subassemblies (modules) are likely to be able to search through the fitness landscape with a reasonably quick pace; thus, Simon submits that "among possible complex forms, hierarchies are the ones that have the time to evolve." This line of thinking has led to the even stronger claim that although "we do not know what forms of life have evolved on other planets in the universe, ... we can safely assume that 'wherever there is life, it must be hierarchically organized'" {{Harv|Koestler|1967}}. This would be a fortunate state of affairs since the existence of simple and isolable subsystems is thought to be a precondition for successful science {{Harv|Fodor|1983}}. In any case, experience certainly seems to indicate that much of the world possesses hierarchical structure.
It has been proposed that perception itself is a process of hierarchical decomposition {{Harv|Leyton|1992}}, and that phenomena which are not essentially hierarchical in nature may not even be "theoretically intelligible" to the human mind ({{Harvnb|McGinn|1994}},{{Harvnb|Simon|1996}}). In Simon's words,
{{quotation|The fact then that many complex systems have a nearly decomposable, hierarchic structure is a major facilitating factor enabling us to understand, describe, and even "see" such systems and their parts. Or perhaps the proposition should be put the other way round. If there are important systems in the world that are complex without being hierarchic, they may to a considerable extent escape our observation and understanding. Analysis of their behavior would involve such detailed knowledge and calculations of the interactions of their elementary parts that it would be beyond our capacities of memory or computation.}}
==Applications==
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