Saturday, December 5, 2015

Common Misconceptions – No Asymptotics

There are a great many misconceptions around about what an alien civilization might look like, and some of the characteristics of them. One of the most glaring is based on the same misconception as much other thinking here on Earth in many different fields. It is common among popularizers.

This misconception centers about which curve to use for trend extrapolation. If something increased by 1% over the last time period, day, week, month, year, decade, then by multiplying it by some length of time will provide an extrapolation of what the something might be after that length of time. This is known as arithmetic extrapolation. Like most methods of trend extrapolation, it is good for a short length of time, which is measured by the length of data. If you have 100 years of data, running out another year using linear extrapolation isn’t a bad idea. Running out ten years might be okay as well, but a bit more chancy. Running out a hundred years is absurd, and running out a thousand years worse than absurd. There is simply not enough data in the series to provide a good indication that a straight line is a good trend line to use.

Other people prefer to use geometric extrapolation. They substitute multiplication for addition. So, in the previous example, instead of saying that each extrapolated period increases by 1% of the last value with data, they multiply each successive period by 1.01. Not much difference by the second period, which is 1.0201 times the last data point instead of 1.02. At twenty time periods beyond the last data point, changes begin to be significant, being 1.22 of the last data point instead of 1.2. The curve generated by this process is called exponential, and the growth is called exponential growth, instead of linear growth.

Both of these trend extrapolations run into the same problem, known as infinity. They are based on the idea that infinity is a reasonable value for whatever it was that was being extrapolated. Almost nothing has no limits. That means these two methods must fail at some point, and without knowing where, they are surely going to lead to errors in conclusions. These methods are solely mathematical, and can be done by anyone without the slightest idea of why the growth might be limited and how that could happen. The alternative to these is trend projection based on the idea of finite limits, and everything pretty much has them. To know where they are and how growth tapers off when they are approached requires some specialist knowledge of the thing that is being extrapolated. This can be erroneous, of course, and extrapolation based on faulty conceptions of limits can make the opposite error, of predicting too little growth instead of too much, which is what the first two methods are guaranteed to do, sooner or later.

The growth of something, anything at all, that has a maximum value is called asymptotic extrapolation. The simplest way, as to the mathematics that are used, is to use a logistics curve for the extrapolation instead of a straight line or a exponential curve. The mathematical term logistics curve is use, sometimes specifically for a particular formula, or sometimes generically for any curve that looks like a squashed S. There are dozens of these curves that have been used in various trend extrapolations by people who understand that something has a limit.

The shape of the curve is a smooth climb between two parallel horizontal lines. One is the lower limit, perhaps zero, and the other is the upper limit. The curve slowly leaves the lower horizontal line, starts rising faster and faster until it reaches somewhere around halfway between the two limiting lines, at an inflection point where it has the steepest slope, and then it reduces its slope as it comes closer to the upper limit, eventually trailing off just below it, getting closer more and more slowly.

This is the shape of the logistics curve.

Population of bacteria in a situation where there is a finite food supply is a typical example. The bacteria count can start off at one, and the growth rate is proportional to the number of bacteria, initially, as long as the number of bacteria is small relative to the food supply. When the number begins to become a sizable fraction of the food supply, like 10%, growth stops as the bacteria soon find difficulty in finding food at a fast enough rate to promote growth at the maximum biological rate. As they grow even more, the numbers will approach the amount tolerated by the food supply rate. They reach an asymptotic value governed exactly by the food supply rate.

The areas where this type of extrapolation commonly occur in discussing alien civilizations concern intelligence, technology, energy sources and resources, to name the most important. When someone writes about alien civilization being a million years old, they extrapolate our improvements in computer intelligence by a hundred thousand times and discuss the fanciful ability of an intelligent machine of that era. But intelligence is a finite quantity, no matter how you define it. It can’t grow a million times in areas that are useful. Processing power can increase greatly, but that is not intelligence. One of the faults of extrapolating is to do it on something which is not accurately defined, and then make projections that sound like they are beyond all comprehension. They are beyond comprehension because they make no sense.

Technology is another poorly extrapolated concept. However you define it, the generation of new technology, theories, ideas, inventions, discoveries, will only go on a short time before it reaches the state of knowing everything important. People who do this sometimes have a very poor image of what science is, and confuse knowing how many grains of sand are on a planet in another galaxy with knowing the fundamental constants of nature. One is useless data, the others are critical parameters.

Energy and resources are also finite. A large population of aliens cannot live forever on a planet, even with a high degree of recycling, as the losses eventually use up the supplies. There are finite supplies, such as the minerals available at affordable costs in a solar system, and finite sources, like the energy falling on a planet’s surface from its sun. These are limits which cause linear and exponential extrapolations to fail.

Conclusions drawn from non-asymptotic extrapolation simply need to be re-thought. By using reasonable methods of extrapolation, many more insights into what alien civilizations must be and must do become available. These conclusions may be less exciting to write about, but they make more sense.

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