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If you are familiar with chaos theory, you'll know that the initial conditions in complex systems such as climate or the weather matter a great deal. The so-called "butterfly effect" is an exaggeration, but the effect is very real and is part of the physical system itself. It's not something that gets introduced because the numerical computer model of the physical system is poor.

It seems intuitive that initial conditions "average out". In the simple systems most of us deal with, that's what happens. Flipping a coin is actually one such simple system. It has only two measurable results: heads or tails. Nobody cares about how many times it flipped in the air, how many times it bounced, or even if it landed on a slight angle on the carpet. So you can do statistics and predict how many heads or tails you'll get. If you also had to predict the numbers of flips and bounces and the angle at which it landed on the carpet, you'd find a statistical analysis would be much harder.

Climate or the weather are complex systems where such details matter. It's no good just to say it's getting hotter or it's getting colder. You have to say where it's getting hotter or cooler and by how much. You have to worry about feedback loops such as increased evaporation creating more clouds which might reflect sunlight back into space. Suddenly it's very complex.

Climate scientists can take a model, do a lot of simulated runs with it, and show that it converges to an average. But that's not the same thing as saying it accurately reflects nature. It doesn't justify using that model to predict the future under different conditions such as more CO2. We have only one actual climate history where we have accurate knowledge, namely the last 150 years. Coming up with a model that matches that one brief history doesn't prove that it works with the different climates of past eons. If the climate models were accurate descriptions of the physical system from first principles, you'd have a much better chance of that.