We are magical beings in a scientific age. Notwithstanding all the remarkable achievements of our species in terms of understanding and harnessing nature, we are born to magical thought and not to reason.
From the introduction paragraph to "Propensity to Believe," James E. Alcock, in The Flight from Science and Reason, New Your Academy of Sciences,1996.
This sentence caught my eye after reading a similarly placed introduction sentence in a "next generation" agile management book...
For centuries, mathematicians have preferred to work with linear (ordered) systems, and they considered nonlinear (complex) systems to be a special group.
My least favorite reference source is Wikipedia, but that source says simply:
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
Going upstairs to the library on a quest for a mathematics history book - which I couldn't find - I spotted instead the Principia and several "guides" to the Principia. The Principia (the replica of the original I have is from 1932) is a geometric text, since calculus was banned by the church. The "guides" - Florian Cajori's revised translation of Motte's translation from Latin to English (my version), Cohen, Whitman, and Cohen's translation and "a Guide to Newton's Principia," and my favorite, Chandrasekhar's Newton's Principia for the Common Reader (that's me) all start with Newton "inventing" his Calculus using non-linear relationships between the independent variables of is version of gravity.
Of course Wikipedia's definition of nonlinear is more accessible:
A nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables.
Lo and behold those orbital mechanics diagrams, with the second order differential equations (all geometrically drawn from the calculus) in Principia and the equations that supported them - but were not published for fear of punishment - are of course Non-Linear integral and differential equations, since they involve variables with exponents.
So back in those past centuries I guess those mathematicians were actually using non-linear math to work out the orbits of planets, making attempts to explain gravity of the orbits, solve those elliptical (not elliptical in the number theory definition) equations of motion to predict where the moon and planets would be some time in the future.
But again care is needed with terminology. The non-linear aspacts of say large amplitude pendulums, forget the double pendulum for the moment, can be approximated with a series expansion. And the indeterminate aspects of these system does not mean the are not solvable or simulated. Just that the behavior cannot be determined in closed form and a progressive solution is needed to determine what happens at a future time.
The propensity to believe that extensions can be established to baseline principles, where established language, and definitions are agreed to, can be used as the basis one's message, while ignoring those principles is sporty business at best and flawed at worse. Such an approach, no matter how well intentioned, lays the resulting message on a foundation of sand. The difficulty of course is that approach - redefining established principles for ones own needs - is actually not necessary. Either use the right analogies or find better ways to explain the message without drawing attention to the fact you're making this stuff up as you go.
