Complicated Simplicity

Rationales and Irrationals

Scientists have been asking the wrong question. They have focused upon complexity as the thing that requires explanation, and they have taken simplicity for granted. The answer to complexity turns out to be fairly obvious and not, in itself, especially interesting: if you have a lot of simple interactors, and let them interact, then the result can be rather complicated. The interesting question is precisely the opposite, the question that most scientists never thought to ask because they did not see that there was a question to ask. Where does the simplicity come from?

Lets begin by making it clear what sort of simplicities we are thinking of. Simplicities like the numerology of atoms – whole numbers of electrons, protons, and neutrons. Magic numbers like 8 and 18 that , by way of electron shells, determine the atoms’ chemical properties. Simplicities like the gas laws, which relate temperature, pressure, and volume in gases, which we know are really extraordinarily complicated systems of vast numbers of atoms, bouncing off one another at tremendous speed. Simplicities like Newton’s law of gravitation, asserting that the attractive force of gravity falls off according to the inverse square of the distance. The deep biological simplicities, such as hemoglobin, chlorophll, the double helix of DNA, and homeotic genes. And purely mathematical laws, such as the Pythagorean theorem and the fact that the circumference of any circle is Pi times its diameter.

Integer ratios in chemistry led to the acceptance of atomic theory long before anybody could see an atom. Does God love whole numbers? We humans seem to think so. “God made the integers,” said the mathematician Leopold Kronecker; “all else is the work of man”. Kronecker was a wholistic fundamentalist. We were baffled by chlorine’s atomic weight of 35.46, and sighed with relief when it turned out to be a mixture of three and a bit parts of 35 to one of 37. We’ve gotten used to Pi, mostly because it shows up in so many places. But still find it peculiar that the universe should employ such a curious number when plain 3, or maybe 22/7, would be so much tidier. We are always impressed by integers whole numbers. We would be less impressed by Newton’s law of gravity if it involved the inverse 2.167th power. An inverse 2.167th law looks like an impirical fudge; an inverse square law looks like a universal truth.
We like “clean” geometry too: circles, ellipses, the regular solids. Buckminsterfullerence attracts scientists’ attention because it is a new form of carbon, but even more because it is a truncated icosahedron. What is it that attracts us to whole numbers and regular geometric shapes?
A mathematical image captures what we are up against. A so called real number is anything that can be represented by a decimal, possibly nonterminating. Buried in the real numbers, like diamonds in clay, are the integers: the whole numbers 0, 1, 2, 3, …and their negatives. From the integers you can build so called rational n umbers; these are fractions such as 2/3 or 22/7. the ancient Pythagoreans discovered that some real numbers are irrational: they cannot be represented as exact fractions, however big the numerator and denominator may be. Their first example was radical 2, the square root of two. Later on it turned out that Pi is irrational. There are infinitely many irrational numbers, and infinitely many rational numbers. But Georg Cantor showed that, in a reasonable sense, there are more irrationals than rationals. Overwhelmingly more.
Despite the prevalence of irrationals, they and the rationals are intimately mixed together. Given any irrational number, you can find a rational number – indeed, infinitely many – as close to it as you wish; correct, say, to a billion decimal places. Its not hard to do this: write the irrational as a decimal, and throw away all decimal places after the billionth. Terminating decimals are always rational. It is also true (and only marginally harder to prove) that given any rational number you can find in irrational number – indeed, infinitely many – as close to it as you wish. Rationals and irrationals are mixed together like sugar and flour in a cake and remain mixed even if you look at them through a microscope, however powerful it might be.

Lets phrase this in terms of simplicity and complexity. Rational numbers are simple in comparison to irrationals: they are built from simple ingredients (integers) by simple rules (make a fraction). Irrationals are more complicated; their description is generally more involved. For example, you cant talk about Pi without first sorting out circles and lengths and things like that. So, in the world of real numbers, the simple ones and the complicated ones are mixed together so finely that if all you do is look at the first billion decimal places, you cant tell which kind you have got. The mathematics at the bottom of the reductionist funnel may seem to offer infinite precision, but every simple nugget of mathematics is surrounded by clouds of far more complicated – but almost indistinguishable – pieces.

Clouds of imitators

The world of possible scientific theories (not just the true ones, but every thing you could imagine and a lot more besides) is just the same. When thinking about the dynamics of a system, we found it useful to consider a mathematical fiction: the space for all possible states. Even though the system does not occupy all of those states, but selects them according to the dynamic, we cannot understand the selection mechanism unless we contemplate the possibility that the system might have done something different. In the same way, when considering scientific theories, we have to work in theory space, the space of all possible theories. Not just the theories that have been agreed upon as a result of lengthy series of experiments, but the alternative theories that might have been adopted had the evidence been different, or indeed had the evidence been the same. The reasons for accepting a particular theory tacitly involve the reasons for rejecting its competitors; so we have to consider the competitors, even if they eventually turn out to be “wrong”.
In this mental space of possible theories, every theory is surrounded by a cloud of different but almost indistinguishable others. As close as you like to a simple theory – such as Newton’s law of gravity – there is a halo of nearby theories that are as complicated as you wish, theories like “actually it should be the inverse 2.000000000000000000000000001st power law.” The halo is not that straightforward, though; it includes theories such as “the inverse-square law holds universally, except for one atom in the Andromeda galaxy at 7:15 pm on Christmas Day, 1842. that is not a very appealing theory, we admit; but if it is true, and Newton’s law is false, its too late to find out.

Newton’s halo also includes general relativity, a geometric description of gravitation. Relativity says that mass is really the curvature of space time, which means that the kinds of masses we encounter in everyday life behave pretty much as Newton said, but enormous masses don’t. this is why Newton’s law looked pretty good for a long time and why we still use it to put communication satellites into orbit. It also means that you could state the theory of relativity as the inverse square law, provided masses aren’t too big. But we would not be impressed b a theory that was stated that way, because its uncomfortably close to the inverse square law holds universally, except for one atom in the Andromeda galaxy at 7:15 pm on Christmass Day, 1842. What impresses us about relativity is its mathematical simplicity and elegance. However as science develops theories that started simple tend to get more complicated. Overtime we revise our views of what is on is not simple. The number Pi started out looking pretty complicated, but by now we’ve become so used to it that it seems almost as simple as 2. Without chaos theory to explain why Mandelbrot set, and a new one called the Feigenbaum number are really simple, both would have seemed fiendishly complicated.

Emergence

The sudden discovery of new simplicities amid some highly complex, apparently unstructured muddle is a fairly rare event – a lot rarer than the discovery of some new detail of the complexity. At least 999 out of a thousand scientific papers are about complex details , but the one that we treasure and for which we award a Noble prize is the one that reveals a new simplicity. It is as if simplicities are all around us, but scattered rather thinly. Some scientists are rather good at longing hands on them they must have the right kind of mind, seeing the world with unusual clarity. why do such simplicities exist, and how do we discover them?

Emergent phenomena, as philosophers call regularities of behavior that somehow seem to transcend their own ingredients. Life is an emergent phenomenon – emerging from chemistry by way of DNA.statistical regularities are certainly one important and widespread mechanism for emergence. Curiously, the existence of statistical regularities probably traces back to the deterministic nature of chaos. Determinism implies underlying laws, and the regularities of statistics are traces for those laws on the macroscopic level. It may sound a very strange thing to say, but really random systems would not possess statistical regularities. No averages, no standard deviations, no correlations. However many emergent features do not come from statistics. There is nothing statistical about Pi, the Mandelbrot set or chlorophyll. Any large scale simplicity can be exploited by a system that is suitably attuned to it. This is especially clear in the workings of the stock market. But in the stock market, this leads to disruption and instability, but in physics or biology it leads to recognizable and exploitable features that can be used as building for higher-level functions and structures. For example, flowers can exploit accidental differences of the emergent feature “color” of chemicals by creating upon them a sophisticated advertising campaign to attract bees. The success of the campaign depends on the bee’s ability to see the colors, though – so as far as a flower’s public-relations department is concerned, “color” just means what bees can see. Statistics is just one way for a system to collapse the chaos of its fine structure and develop a reliable large scale feature. Other kinds of feature can crystallize out from underlying chaos – numbers, shapes, patterns of repetitive behavior, many of those features have their own intricate internal structure which is quite different from the underlying rules that generated the feature in the first place. The intricacy of the Mandelbrot set bears no obvious relation to the simplicity of the process that produces it. The rules for making a Mandelbrot set are dynamic; the internal intricacies are geometric. That is why it is describe as a complicated simplicity, there is no contradiction.

Extracted from;
Editors: Jack Cohn and Ian Stewart, The collapse of Chaos, Discovering simplicity in a complex world