Linearity.

How do we cope with what we do not know? It is impossible for us to have perfect knowledge of the world. Even in situations where it is possible to gain more knowledge we have tendency to make assumptions and produce decisions without waiting for all of the available information. This does make sense: there is an optimization that has to take place, which mediates the need to act in time, the conclusions the brain is able to extract with the information it has, and how long it would take to obtain more information.

We have to habitually induct to draw the best conclusions with the information we have. This requires assumptions, and the purpose of this post is to discuss one typical assumption people make and how.

Linearity.

The weight of an object is a linear function of the mass. If I put an object on a scale, and then put a second object on the scale, the weight is twice as much. For a pendulum, if it is perturbed only slightly from the equilibrium position to start the motion, the fact that the sine of the angle is about equal to the angle itself means the force of gravity pulling it back will be linear in the angle (1). In contexts less obviously related to physics, try these questions:

If twice as many people enter a room, does the volume level of their conversation about double, or is the increase greater than that?

If a city suffers a net 2% out-migration of the population, will the effects be about 1/3rd that of a 6% out-migration or more severe? Which effects would be linear, and which not?

Is it better to pick a Basketball team with nothing but the very best players, or is it better to pick a team that works together better than any other? Can we expect teams selected through both principles to be the same, or sometimes radically different?

If everyone on the highway wants to get where they’re going sooner, will they if they all decide to drive faster?

Many of the processes of change that people experience are “smooth”, meaning they change gradually, or changes occur continuously. People have to extrapolate, and it seems their intuition is reproducing a kind of linear approximation, where they project out from the point of their present experience and expect change in the future at the same present rate. This is the assumption of linearity in a temporal situation. You may notice that interaction among the parts of the system is important in each of the questions I asked above. In the time-oriented case, nonlinearity can sometimes result in correlations of later states to prior states – a significant dependence on history so that the system is sort of interacting with itself across time.

I think most people don’t even take a single course in basic physics. Those that do have trouble even with acceleration, velocity, and position. It seems curvature is not an easy thing to intuit about, and indeed I have always been surprised at how completely wrong my own intuitions have been about many physics problems. In simple cases, though, you can find a variable that is linear (velocity for a free-falling object) and apply your intuition with that, combine it with some math and manage pretty well.

Some problems are inescapably nonlinear, however. Some problems involve interactions between system components, and maybe even a response of components to global properties (people and politics, mass media). Some problems exist for which it can be demonstrated no equation for the solution exists – I can’t write down an answer that can be computed by hand in a few steps. Some kinds of problems require a computer to literally walk the state of the system forward one instant at a time. There exist problems for which no known algorithm can reach a solution in a finite span of time. It also can be shown there exists a class of problems for which no such algorithm could ever exist, assuming classical logical operations.

One thing that should be disturbing is how often climate scientist’s predictions fall short – I have noted a pattern where the predictions made several years ago seem to have underestimated how dire the situation would become. They have been uncovering unexpected effects, and they are typically bad. These problems, we hope, and have good reason to hope, are of the ‘walk the system forward’ kind, but there is the problem of chaos at short timescales and it does not seem to be clear under what circumstances the larger scale cyclic behaviors could be guaranteed to continue suppressing this (2).

I think people have developed a very solid track record of not accurately predicting the future. Now I’m referring not just to changes in the climate, but also about the impact of technology and other changes (3). I should probably not have as much confidence in my own expectations about the future as I do, but I have this idea in my head that things are about to change very profoundly. It is so implausible that the earth could possibly continue as it has, we must be approaching a singularity of a kind – a sharp, discontinuous change in the rate of change, and a “phase transition” into a new regime of global system behavior. It could be amazing or it could be cataclysmic. I expect the worst, but this could be my own pessimism in play. I hope everyday that I am just some fool.

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  1. I think it’s funny that “clockwork” universe is used to describe a deterministic one, given that a pendulum clock with a sufficient driving force, and perturbed sufficiently from equilibrium can behave chaotically.
  2. I mean mostly that it’s not clear to me. I do not study the climate professionally.
  3. Arthur C. Clarke aside, https://www.youtube.com/watch?v=aajlLeTgrEg. WaPo: http://www.washingtonpost.com/news/wonkblog/wp/2015/10/04/what-people-in-1900-thought-the-year-2000-would-look-like/?tid=pm_business_pop_b
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