Scaling to infinity
What does a rabbit, a well and a spaceship have in common?
Before you start exercising your grey cells, let me quickly re-tell you the apocryphal story of Achilles and the tortoise.
So, the Greek Philosopher Zeno of Elea (c. 490 - 430 BC) in his quest to explore the concept of Monism, devised a series of paradoxes. Aristotle and others propagated these paradoxes orally through their famous discourses, and one of them was about a race between the great warrior Achilles and a humble tortoise.
Achilles being an excellent runner, decided to give the tortoise a head-start of 100 meters. The race began in due time, and by the time Achilles had cruised through the gap, the tortoise had steadily walked ahead a bit further. Now Achilles has to catch up, and he runs. But as he covers the gap, the tortoise walks ahead a bit more. He runs again to bridge the gap. And this keep happening ad infinitum. So, the question is:
** Does Achilles win the race? **
This is a classic example of a logical knot. Douglas Hofstadter has dealt with this extensively in his amazing magnus opus, in the context of Bach's three-part invention. You need to read the book, trust me. No amount of summary can do justice to the "golden braid" that Douglas lays open in his tome. However, what can be contextualized is the concept of infinity in an action as simple as a race.
As Achilles tries to catch up to the Tortoise, the motion of the latter continues in the background. The goalpost is not limited to the visible, but is a sum of the aggregate of the goalpost, field, grass, players and audience. As our quest for accuracy proceeds, the number and density of variables in the function space grows rapidly. An interesting digression could be a correlation between the nature of the space and its influence on the variables. Intuition tells us that the nature of the game should, in fact, influence the nature and dynamics of the variables involved.
Reeling our thread back to the drawing board, think of the tortoise as a perpetually running automaton. In case of societies, it could be modelled by Conway's Game of Life. The rules are set, and the tortoise sets into motion. It doesn't care about obstacles or any fast warriors trying to catch up. In reality, however, it does care. Society does compare itself through economical connections to other societies. There is an exchange of ideas, capital and debt with each active connection. Free markets are the best example of a system driven almost entirely by these dynamic rules, and it is fascinating how macroeconomic variables emerge from the granular microeconomics at lowest possible level. The tortoise is, in fact, built up of an infinite tower of tortoises.
The rabbit in the classic story takes rest in between, but the analogy with Achilles is direct. The ignorance of the nature of the tortoise's walk impacts the outcome of the race. There we have our first symptom of an infinity - ignorance.
Obviously, if you're walking down a dense forest and are ignorant of traps hiding in the undergrowth, you shall fall into a well. This well is an instance of an infinity lurking under the garb of an ignored variable or an overlooked instance of truth. The well represents a singularity, an inevitable result of the nature of the function. If space dictates how variables behave, they should indeed indicate the structure of infinity {^1}.
And so, in case of emergent systems like societies and bee swarms, these infinities show up as nontrivial behavior outside the purview of predictability. What do we do then, if society reaches a deadlock? A well, deep enough to lock into stagnation, economic or social or both? Is there a saving grace for a society staring at imminent collapse?
The answer is a resounding YES.
This is not an optimistic denial of truth. In fact, optimists end up ignoring more about the system than pessimists, but that's a story for another day.
Why I say there's a ladder out of the well, is due to the simple fact that the system is greater than the sum of its aggregates and hence has ample room for disruptive behavior. Introduce a little variable that nudges the economy out of a local minimum and watch how the system wiggles into a different state. Often what happens with systems in transition, is that they are stuck in a metastable equilibrium and all it takes is a nontrivial nudge. An actual spaceship out of the well.
Now this is not a promised solution, but it does work! In case of thermodynamics, every state function does have at least one inflection point, where the system "decides" if it wants to remain in the original state or transition to a new one. This point of decision is a metastable equilibrium, a window of opportunity for driving change. This when the iron is hot, and the hammer must yield a decisive blow. That's the only escape out of the well.
Achilles needs to realize that the tortoise shall keep walking at its pace come rain or sunshine. So, he must NOT lock into the tortoise as a goal, rather focus on the infinities ahead and build up a steady pace for the sake of his run. The only way he can ever win the race, is to ignore the tiny infinity ahead and focus on his own feet and their pace. The old adage holds true, that comparison is the death of originality.
The spaceship must cruise into deep space regardless of any gravitational wells that try to rope it in. That is how exploration happens.