All that is so obvious in restrospect that my mind keeps trying to convince me that I always knew it. (Despite evidence to the contrary).
In a competitive environment there can not be a widely known path for success, because the competitors will adapt untill the path changes to something unknown. Thus, if you know about a model, it's probably not valid anymore, and you simply can't study it for long enough to discover the non-linearities.
I like the article, but the author suggests (or at least creates the impression) that "nonlinearity" causes "path dependence". This is not necessarily true.
Path-independence or -dependence is more about history than the nature of linearity or nonlinearity in variables itself.
From Wikipedia: "A nonholonomic (a path-independent) system in physics and mathematics is a system whose state depends on the path taken to achieve it."
Here is counter-example where non-linearity does not cause path-dependence. Fields that obey an inverse-square law (such as electromagnetism) are called conservative (path-independent) because the work needed to move an object from one point to the other is not dependent on the path.
Your reasoning about the electromagnetic field is right. I think the element of confusion here is the definition of nonlinear. Nonlinear in system theory means that the superposition principle is not valid.
From Wikipedia: "The superposition principle states that the net 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".
The electromagnetic field in vacuum is linear, because it obeys Maxwell's equations and it is a solution of a set of linear equations. The output is always proportional to the input: you put a bigger charge and you get a bigger field. A linear combination of electromagnetic fields, with constant, real coefficients, is a new field which obeys Maxwell's equations, thus it is still linear and the superposition principle is still valid.
The traditional name for systems that are deterministic but unpredictable because they can be radically changed by differences in starting conditions that are below the threshold of measurement is "chaos". Or, in human affairs, "luck". We like to draw parallels and deduce conclusions, but a lot of it is down to survivor bias and confirmation bias. We think of some hypothesis of success; soon we see examples everywhere.
It doesn't quite matter: inhuman complexity renders all of them intractable for ordinary comprehension. History is, as Barzun observed, "above all concrete and particular, not general and abstract".
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[ 2.2 ms ] story [ 39.7 ms ] threadIn a competitive environment there can not be a widely known path for success, because the competitors will adapt untill the path changes to something unknown. Thus, if you know about a model, it's probably not valid anymore, and you simply can't study it for long enough to discover the non-linearities.
Path-independence or -dependence is more about history than the nature of linearity or nonlinearity in variables itself.
From Wikipedia: "A nonholonomic (a path-independent) system in physics and mathematics is a system whose state depends on the path taken to achieve it."
Here is counter-example where non-linearity does not cause path-dependence. Fields that obey an inverse-square law (such as electromagnetism) are called conservative (path-independent) because the work needed to move an object from one point to the other is not dependent on the path.
But, I could be missing something. Let me know!
It doesn't quite matter: inhuman complexity renders all of them intractable for ordinary comprehension. History is, as Barzun observed, "above all concrete and particular, not general and abstract".