Monday 11 September 2017

Chaos Theory: And Why It Matters

Posted by Keith Tidman

Computer-generated image demonstrating that the behaviour of dynamical systems is highly sensitive to initial conditions

Future events in a complex, dynamical, nonlinear system are determined by their initial conditions. In such cases, the dependence of events on initial conditions is highly sensitive. That exquisite sensitivity is capable of resulting in dramatically large differences in future outcomes and behaviours, depending on the actual initial conditions and their trajectory over time — how follow-on events nonlinearly cascade and unpredictably branch out along potentially myriad paths. The idea is at the heart of so-called ‘Chaos Theory’.

The effect may show up in a wide range of disciplines, including the natural, environmental, social, medical, and computer sciences (including artificial intelligence), mathematics and modeling, engineering — and philosophy — among others. The implication of sensitivity to initial conditions is that eventual, longer-term outcomes or events are largely unpredictable; however, that is not to say they are random — there’s an important difference. Chaos is not randomness; nor is it disorder*. There is no contradiction or inconsistency between chaos and determinism. Rather, there remains a cause-and-effect — that is, deterministic — relationship between those initial conditions and later events, even after the widening passage of time during which large nonlinear instabilities and disturbances expand exponentially. Effect becomes cause, cause becomes effect, which becomes cause . . . ad infinitum. As Chrysippus, a third-century BC Stoic philosopher, presciently remarked:
‘Everything that happens is followed by something else which depends on it by causal necessity. Likewise, everything that happens is preceded by something with which it is causally connected’.
Accordingly, the dynamical, nonlinear system’s future behaviour is completely determined by its initial conditions, even though the paths of the relationship — which quickly get massively complex via factors such as divergence, repetition, and feedback — may not be traceable. A corollary is that not just the future is unpredictable, but the past — history — also defies complete understanding and reconstruction, given the mind-boggling branching of events occurring over decades, centuries, and millennia. Our lives routinely demonstrate these principles: the long-term effects of initial conditions on complex, dynamical social, economic, ecologic, and pedagogic systems, to cite just a few examples, are likewise subject to chaos and unpredictability.

Chaos theory thus describes the behaviour of systems that are impossible to predict or control. These processes and phenomena have been described by the unique qualities of fractal patterns like the one above — graphically demonstrated, for example, by nerve pathways, sea shells, ferns, crystals, trees, stalagmites, rivers, snow flakes, canyons, lightning, peacocks, clouds, shorelines, and myriad other natural things. Fractal patterns, through their branching and recursive shape (repeated over and over), offer us a graphical, geometric image of chaos. They capture the infinite complexity of not just nature but of complex, nonlinear systems in general — including manmade ones, such as expanding cities and traffic patterns. Even tiny errors in measuring the state of a complex system get mega-amplified, making prediction unreliable, even impossible, in the longer term. In the words of the 20th-century physicist Richard Feynman:
‘Trying to understand the way nature works involves . . . beautiful tightropes of logic on which one has to walk in order not to make a mistake in predicting what will happen’.
The exquisite sensitivity to initial conditions is metaphorically described as the ‘butterfly effect’. The term was made famous by the mathematician and meteorologist Edward Lorenz in a 1972 paper in which he questioned whether the flapping of a butterfly’s wings in Brazil — an ostensibly miniscule change in initial conditions in space-time — might trigger a tornado in Texas — a massive consequential result stemming from the complexly intervening (unpredictable) sequence of events. As Aristotle foreshadowed, ‘The least initial deviation . . . is multiplied later a thousandfold’.

Lorenz’s work that accidentally led to this understanding and demonstration of chaos theory dated back to the preceding decade. In 1961 (in an era of limited computer power) he was performing a study of weather prediction, employing a computer model for his simulations. In wanting to run his simulation again, he rounded the variables from six to three digits, assuming that such an ever-so-tiny change couldn’t matter to the results — a commonsense expectation at the time. However, to the astonishment of Lorenz, the computer model resulted in weather predictions that radically differed from the first run — all the more so the longer the model ran using the slightly truncated initial conditions. This serendipitous event, though initially garnering little attention among Lorenz's academic peers, eventually ended up setting the stage for chaos theory.

Lorenz’s contributions came to qualify the classical laws of Nature represented by Isaac Newton, whose Mathematical Principles of Natural Philosophy three hundred-plus years earlier famously laid out a well-ordered, mechanical system — epically reducing the universe to ‘clockwork’ precision and predictability. It provided us, and still does, with a sufficiently workable approximation of the world we live in.

No allowance, in the preceding descriptions, for indeterminacy and unpredictability. That said, an important exception to determinism would require venturing beyond the macroscopic systems of the classical world into the microscopic systems of the quantum mechanical world — where indeterminism (probability) prevails. Today, some people construe the classical string of causes and effects and clockwork-like precision as perhaps pointing to an original cause in the form of some ultimate designer of the universe, or more simply a god — predetermining how the universe’s history is to unfold.

It is not the case, as has been thought too ambitiously by some, that all that humankind needs to do is get cleverer at acquiring deeper understanding, and dismiss any notion of limitations, in order to render everything predictable. Conforming to this reasoning, the 18th century Dutch thinker, Baruch Spinoza, asserted,
‘Nothing in Nature is random. . . . A thing appears random only through the incompleteness of our knowledge’.

*Another example of chaos is brain activity, where a thought and the originating firing of neurons — among the staggering ninety billion neurons, one hundred trillion synapses, and unimaginable alternative pathways — results in the unpredictable, near-infinite sequence of electromechanical transmissions. Such exquisite goings-on may well have implications for consciousness and free will. Since consciousness is the root of self-identity — our own identity, and that of others — it matters that consciousness is simultaneously the product of, and subject to, the nonlinear complexity and unpredictability associated with chaos. The connections are embedded in realism. The saving grace is that cause-and-effect and determinism are, however, still in play in all possible permutations of how individual consciousness and the universe subtly connect.


Thomas O. Scarborough said...

Newton’s view of the world seems to be remarkably oblivious of its vast complexity, and Spinoza’s seems to be remarkably insightful, if one may interpret him as saying that ‘incompleteness’ cannot, in principle, be overcome.

There is a constellation of related terms, of which you use several. A term which is suggested here, though you do not use it, is ‘uncertainty’. Could you reflect on the distinction between chaos and uncertainty?

Out of curiosity, do you think that a truly omniscient mind, supposing that there were such a thing, could predict everything? or not?

docmartincohen said...

Thank you, Keith, for another interesting and stimulating post. I've long had a soft spot for the theory of chaos - surely it explains amny things. But here's a slightly different way to look at it, which you might enjoy.

"A young man boarded a train bound for Odessa and finds a carriage empty except for a prosperous looking passenger reading a newspaper. After some time, the young man leans forward and asks politely: 'Excuse me , sir, could you tell me the time?' The other passenger, angrily puts down his paper and glares at him angrily. 'Absolutely not, and I resent your impudence! Get your own watch, wastrel!'
'What the devil is wrong with you!', the young man replies indignantly, 'I only asked you for the time!'
The other passenger makes as though to continue reading his paper and then sighs and says: 'All right, I'll explain. First, you would like me to tell you the time, yes? But then there will be some small talk and soon you will discover that we are both going to Odessa. Then it will turn out that we are both going to the International Arts Festival there, but that you do not have anywhere to stay and all the hotels will be booked up. Since I live in Odessa in a large house, i will invit e you to stay there and you will meet Sophie, my beautiful daughter. The two of you will get on very well and after a few more visits you will both fall in love. At that point you will then expect my approval so that you and Sophie can get married. So now you wonder why I am angry? Because I refuse to let my daughter marry anyone who cannot even afford a watch!'

What's this got to do with chaos theory? Well, the joke is making fun of conditional necessity - the idea that everyday life is governed by predictable laws of cause and effect. If I heat the kettle, to use the example of Agnes Heller in The Immortal Comedy: The Comic Phenomenon in Art, Literature, and Life, the water will boil. If I forget that I am heating the kettle, when I return in half an hour the water will have disappeared and the kettle will be ruined too.

In this story, known as the 'case of The Man on the Train to Odessa', the joke is that the supposed sequence of events is not fully determined - at any point there are too many other factors and indeed imponderables to make the assumption: 'the young man wants to have a big house and beautiful wife therefore will try to marry my daughter' reliable. Yet once the absurd logic is accepted, the punchline itself does seem somehow reasonable - and this reasonable quality is also amusing in the contrast between the parochial nature of the concern and the grand events supposed to be linked. It is 'chaos theory' in reverse. Is a hurricane in the United States considered undesirable as it implies a butterfly having to beat its wing in Brazil?

Keith said...

Thank you, Thomas, for posing critical questions.

As to your first question — the ‘distinction’ between chaos and uncertainty — definitions matter. However, I would suggest, using my essay’s theme as a springboard, that uncertainty is another way of saying ‘unpredictability’. Just as with unpredictability, uncertainty emerges from some mix of these key elements: When the initial conditions are overwhelmingly many, or abstruse, or poorly known, or unknown, or even unknowable. When the unspooling of the string of causes and effects is hard or impossible to trace (which becomes exponentially more difficult as time passes). And when the recursive branching in rapidly increasing numbers of paths, fractal-like, overwhelms by its sheer scale the ability to understand. To these extents, uncertainty and unpredictability are interchangeable.

Now, of course, the preceding paragraph describes the macro world. However, your word ‘uncertainty’, if applied to the micro (particle) world of quantum theory, has a different meaning. One interpretation is what’s referred to as the Heisenberg uncertainty principle (or indeterminacy principle). It states that the more precisely one property of a particular pair is known about a particle, the less precisely the other paired property may be determined. A common example gives this point a degree of concreteness: the more precisely the ‘momentum’ of a particle is known, the less precisely can the particle’s ‘position’ be determined — and vice versa. Sticking with quantum theory for one more moment, uncertainty (and probability) arises, too, from the effect of observation, in particular measurement. It’s when observation of a quantum phenomenon causes the collapse of a so-called wavefunction that was formerly in superposition, probabilistically resulting in one of alternative realities. So, as to ‘uncertainty’, it matters whether we’re venturing into the macro or micro world.

As to the last topic you raise, I suggest that, by definition, an ‘omniscient’ mind could, as you ask, predict everything. So, if, hypothetically, a creature were to emerge from the swamp with (unqualified) omniscience, its omniscience would include capably controlling for the three conditions I outlined in the first paragraph above: ability to know and understand every initial condition (no matter how many or how complex); ability to trace the subsequent string of causes and effects (no matter for how long, or how fast, or how many); and ability to follow the innumerably branching paths along which events travel (no matter the scale, or how exponentially multiplying the branches). To expand this discussion out to a cosmic scale, if there were an ‘omniscient’ mind behind the universe’s existence, that mind could likewise predict everything — assuming, importantly, one agrees that the ability to predict everything is fundamental to omniscience. However, the preceding situation stirs a hornet’s nest, raising questions as to what an omniscient — thereby all-predicting — mind means for how all future events in the universe unfold: deterministically/mechanistically or indeterministically/probabilistically. And, by extension, what the answer to that either-or conundrum means for the purpose of human consciousness and, of course, for the presence or absence of free will.

Keith said...

An amusing tale, Martin — and a curious spin on chaos theory.

I can envision an additional step (an additional ‘effect’) to the train tale: Poor Sophie, who has exasperatingly endured living with such an insufferably idiosyncratic and overbearing dad, is desperate to leave home, prompting her to flirt with the young man whom she happens by chance to meet at Odessa’s International Arts Festival, leading her eventually to marry the young man and bid adieu to her patriarchal family, to live happily ever after. (Hmm; could we could turn chaos theory into a children’s book?)

One last point: I especially liked your closing comment, flipping the conventional butterfly => hurricane construct. That is, the hurricane in the United States counterintuitively causing the butterfly in Brazil to flap its wings.

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