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Here’s an example. It’s a population ecology example, but not because population ecology is the only bit of ecology that’s about dynamical systems. Purely for the sake of simplicity (because it doesn’t affect my argument at all), let’s say it’s a closed, deterministic, well-mixed system with no population structure or evolution or anything like that, so we can describe the dynamics with just two coupled equations, What is billiards one for prey dynamics and one for predator dynamics. You cannot think about this dynamical system in terms of sequences of causal events. You cannot think about equilibria in terms of sequences of causal events, it’s like trying to think about smells in terms of their colors, or bricks in terms of their love of Mozart. "The prey go up, which causes the predators to go up, which causes the prey to crash, which causes the predators to crash." In lecture, even I’ve been known to slip and fall back on talking this way, and when I do the students’ eyes light up because it "clicks" with them, they feel like they "get" it, they find it natural to think that way.

Now, you could try to drill down even further, down to the underlying physiological (or whatever) causes of individual births and deaths, and the underlying mechanisms linking per-capita birth and death probabilities to species’ abundances. A prey individual was born, which caused prey abundance to increase by one, which caused… In the real world one could in principle write down, in temporal order of occurrence, all the individual birth and death events in both species. People with all sorts of table billiards experience from everywhere in the world believe that they have the real pool secrets of winning. Now I can hear some of you saying, ok, that’s true of the math we use to describe the world, but it’s not literally true of the real world. They are, but that’s not my point here. That’s why colliding billiard balls are a paradigmatic example of causality in philosophy. Why do the predators and prey cycle? Or, let’s say the predators and prey exhibit cyclic dynamics. For concreteness, let’s say it’s a limit cycle in the Rosenzweig-MacArthur model. No. What that increase in prey abundance did was slightly change the expected time until the next birth or death event, by increasing prey abundance and (in any reasonable model) feeding back to slightly change the per-capita probabilities per unit time of giving birth and dying.

Billiards dates back to the 15th century while pool which was known as pocket billiards dates back to the 18th century. If there are not enough points remaining on the table for a player to win the frame, that player may offer to concede the frame while at the table (but not while their opponent is still at the table); a frame concession is a common occurrence in professional snooker. You’ve got some prey that reproduce and die, and some of those deaths are due to predators. Births and deaths are happening instantly and continuously. At every instant in time, prey are being born, and prey are dying, and those two rates are precisely equal in magnitude but opposite in sign. And at every instant in time, predators are being born and predators are dying, and those two rates are precisely equal in magnitude but opposite in sign. Some of the more popular ones are carom, snooker, and British style pool. The British balls are however, 51 mm in diameter. They are! I’m just saying that the nature of that causality is such that it’s misleading to think about it as "Event A causes event B which causes event C which causes event D…

The balls used are smaller and the table is pocketed. If you’re setting up a pool table in your bar, pub or even your rec room, then don’t underestimate the power of good pool table lighting. It might start out as 8 Ball, move to 9 Ball, transition to straight pool, and then explore the possibilities of 1 Pocket. In straight pool, one level will be when you can play through two racks of pool. But you truly are able to touch the lowest level of billiard god-hood when you realize that the game is based on a very simple skill. When you consider the question a little more, you realize that being successful at pool requires a very specific pool skill. When you get a little better, you gain a little control with the speed that you apply to the cue ball. Your opponent gains the advantage of ball in hand, entitling them to place the cue ball anywhere on the table before taking their shot. For more information about billiard table removals you can go here.