## Linear

Of central importance in context of complexity is the notion of *linearity*, which should not be confused with the colloquial meaning but rather understood from systems science.

A problem is linear if it can be broken into a sum of mutually independent sub-problems. When, to the contrary, the various components/aspects of a problem interact with each other so as to render impossible their separation for solving the problem step by step or in blocks, then the situation is non-linear.

Another way to express the same concept is to use the systems theory definition: a system is linear if it responds with direct proportionality to inputs. This is a system that obeys the **superposition principle**: the response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually[1].

Linearity is a logical artefact, as the “systems” and the “problems” to be encountered in nature are all essentially nonlinear.

However, to simplify the studies or for application purposes, one often resorts to linearity as a first-degree approximation: if the effects of non-linearity can be considered negligible, a mathematical **model** can be built that represents the system as if it were linear [2].

A linear mathematical model consists in representing the system as a polynomial function whose coefficients are independent from one another, or so weakly dependent that their mutual interactions can be neglected without losing too much information.

This approach is fecund in many situations. As an example: every audio amplifier is intrinsically nonlinear but, within certain frequency limits, it will behave in a linear fashion and be useful for *hi-fi*.

Linear models are useful because subject to the hypothesis of linearity many natural systems resemble to one another: their behavior can be described with the **same** mathematical equations even if the contexts are very different, such as mechanics, electronics, chemistry, biology, economics, and so on.

Enormous scientific and technological advances have been made using simplifying linearity assumptions, before computers started allowing to venture into nonlinear territory.

[1] R.P. Feynman, R.Leighton, M. Sands, *The Feynman Lectures on Physics*, Addison-Wesley 1964, Vol. 1, pag 25-3

[2] *Op.cit.*, Vol. 1, pag. 25-9

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