In the simplest possible second-order differential equation,

d2y/dx2 == ky

the solution depends on the sign and value of the parameter k. Because it is so important, I call k “the key”.

At positive values ofk, the solution is an exponential rise curve

y == eX

Here y increases slowly at first, and then very rapidly, the rate of growth itself increasing. “The bigger it is, the faster it grows” is the basic rule for both y itself and its rate of increase and the acceleration of increase and so on. This describes an avalanche or an explosion or an unrestricted population increase of bacteria on a Petri dish or people in the world – anything that reproduces along the pattern 2 gives 4, 4 gives 8, 8 gives 16 etc. in geometric series. Raoul Naroll calls these processes “snowballs”, because these grow to avalanches as they roll down the hillside. Physicists and chemists call them “chain reactions”, which lead to explosions, chemical or nuclear. (One neutron splits a heavy nucleus, giving two or more neutrons, which split more heavy atoms…)

Naroll compares Richardson arms races to “snowballs”, in which arms expenditures of two opponents grow in mutual reaction to each other. (We have two equations similar to the first one given here, but with cross-reference of each nation to the other; plus a damping term and a constant term added.) Maruyama calls snowballs “second cybernetics”; a result of positive feedback, while normal (“first”) cybernetics results from negative feedback. But if cybernetics means “steering” or “regulation”, only “first cybernetics” qualifies; so-:ealled “second cybernetics” is a misnomer, because it describes run- away processes, out of control.

Environmentalists wony about exponential growth. because it creates problems which seem small at first. but suddenly confront us with an irretrievable catastrophe, past the point of no return. The classic example here is the Parable of the Lily Pond. Water lilies growing on a pond are threatening eventually to cover the whole surface and choke off the oxygen supply to aquatic life. People on the shore leisurely decide that they will intervene when the pond is half-covered. (Nothing to worry about until then.) If the pond will be completely covered in 30 days, when is the time of half-coverage when remedial action is planned for? The stunning answer is the 29th day, just on the eve of destruction.

Diagram of the 3 cases.

If the parameter k is negative and small, the solution function is a negative exponential, a curve that approaches the base value rapidly at first, but then asymptotically, i.e. ever more slowly and never actually gets there. It is exemplified by radioactive decay, or any other process which can be characterized by a half-life (the time when it gets half-way to the base-line)’-The next half-life (same time duration as the first) gets it to quarter of the distance from the base-line, the next half-life to 1/8, and so on.

If k is negative and large, the solution function becomes a sine or cosine wave, i.e. an oscillation about the base-line, and probably a damped wave, i.e. with progressively decreasing amplitude. (The “envelope” of the wave is a decreasing exponential on both sides of the base-line.) If negative k is regarded as the measure of a “restorative force”, then with a small negative k it never quite restores the original state (base-line), but with a large negative k it “overshoots” and starts back, overshoots again (but a little less), and so on; it oscillates like a swing or a pendulum or a stretched spring or a vibrating string on a musical instrument. The base-line is an “attractor” to the iterated overshoots; it is also an attractor to the decreasing exponential, except that the latter does not overshoot and so need not return. Some iterated functions approach their attractor from one side only (like the exponential), some bracket it from two sides (like the wave). (Alternatively, the oscillation can be considered to have two attractors. See essay “How Things Come Together” in Section XII.)

Why is the solution function with negative k either a decay-type exponential or a sine/cosine wave? This goes back to the. deep and beautiful mathematical relationship called Euler’s theorem: e1X = cos x + i sin x derived from the Maclaurin series describing these functions. (i is the imaginary unit, the square root of – 1.) Mathematically as well as intuitively, exponentials and waves are deeply connected. The second-order differential equation with negative k represents Maruyama’s “first cybernetics.. or just plain cybernetics, describing processes with negative feedback. It describes devices like thermostats, or control devices in chemical industry which keep the pH constant or the reactant concentration constant. It reminds one-of biological homeostasis, although, as is explained below, this operates by a more complex mechanism; however, it does keep e.g. the glucose concentration in the blood constant, or our body temperature constant. The rule for thermostats or homeostasis alike is: “The bigger the deviation from the base-line, the faster the deviation decreases.” This applies to both decay exponentials and waves. It is the opposite of the rule for increasing exponentials for unrestricted growth. as described above.

A note about homeostasis. There are three ways that cybernetic control can operate to keep a variable close to a chosen base-line:

1. Equilibrium systems typified by the thermostat.
2. Steady-state systems which are open to matter and energy flows and operate, away from equilibrium, like a bathtub in which both the faucet and the drain are open; by adjusting these, the level of water in the bathtub can be kept constant.
3. True homeostasis, where a steady state away from equilibrium in open systems is maintained by means of multiple interlocking cycles.

Next, imagine a surface with a small amount of roughness, i.e. hills and valleys at the molecular scale in the crystal lattice, immersed in a solution from which molecules are deposited on that rough surface. We would normally expect a faster deposition in the valleys than on the hills, and therefore a smoothing effect. This does happen in slow growth of crystals from saturated solutions or in deposition by electrolysis. We can do “electro-polishing” that way. Such processes are reminiscent of negative k processes, which also smooth out deviations.

But we can also have a process in which deposition takes place preferentially on the hills of surface roughness, perhaps because they are more exposed (they stick out). This process accentuates roughness, and is called fractal growth. This goes on at ever smaller dimensions: any bump on a hill grows faster than the rest of the hill, any spike on each bump grows faster than the rest of the bump, and so on ad infinitum, at least in theory. (In practice we are stopped by the indivisibility of atoms.) Thus coastlines and clouds and many natural structures are not Euclidean solids with smooth borders, but are rough at any size level we care to look at. This type of growth process is reminiscent of one with a positive value of k: any original deviation is accentuated, the more so the bigger it is. But fractal growth is not really “run-away” like an explosion, because the successive structures get ever smaller in scale. The process does not produce thin threads extending to infinity, but only very bumpy (infinitely rough) partial filling in of a given space which is its limiting envelope. This is well illustrated by the fractal (infinitely branched) Koch snowflake formed from an equilateral triangle by a process that keeps sprouting more equilateral triangles from the middles of each side, iterating forever. The fractal snowflake has an infinitely long perimeter, but a finite volume. (perhaps we should expect snowflakes to be related to snowballs.)

Life makes use of both positive feedback and negative feedback, i.e. positive and negative k. Positive feedback is involved in flipping to a higher pattern through an accumulation of fluctuations, as in the creation of a new species. Negative feedback is involved in homeostasis and general pattern maintenance. Negative feedback is much more common. “Conservation” predominates on the runs of the developmental staircase. (See the essay “Rise and Run” in this Section.) “Radicalism” prevails only on the rises and is dangerous, creating a crisis; but of course, a crisis is also an opportunity for innovation.

Social and political revolutions (e.g. the French and the Russian Revolution) are self-accelerating processes (positive k), in that the more radical factions (the Jacobins or the Bolsheviks) drive out the more moderate factions (the Girondins or the Mensheviks). This is opposite to what happens in “normal politics” (a phrase coined to be parallel to Kuhn’s “normal science”), where all parties and voters tend to be pushed to the political centre – a thermostat-like cybernetic process called “consensus-building” – until there is hardly any difference between conservative, liberal, and social democrat. When or why does the revolutionary dynamic set in? It is like Kuhn’s paradigm shift, or a climatic flip to an ice age, or a change from laminar to turbulent flow.

Thus the original equation and its parameter k relate to a wide range of phenomena in the physical and social world. It is truly the Key.