November 30, 2009

The Mathematics of Global Warming

The forecasts of global warming are based on the mathematical solutions of equations in models of the weather. But all of these solutions are inaccurate. Therefore no valid scientific conclusions can be made concerning global warming. The false claim for the effectiveness of mathematics is an unreported scandal at least as important as the recent climate data fraud. Why is the math important? And why don't the climatologists use it correctly?

Mathematics has a fundamental role in the development of all physical sciences. First the researchers strive to understand the laws of nature determining the behavior of what they are studying. Then they build a model and express these laws in the mathematics of differential and difference equations. Next the mathematicians analyze the solutions to these equations to improve the understanding of the scientist. Often the mathematicians can describe the evolution through time of the scientist's model.

The most famous successful use of mathematics in this way was Isaac Newton's demonstration that the planets travel in elliptical paths around the sun. He formulated the law of gravity (that the rate of change of the velocity between two masses is inversely proportional to the square of the distance between them) and then developed the mathematics of differential calculus to demonstrate his result.

Every college physics student studies many of the simple models and their successful solutions that have been found over the 300 years after Newton. Engineers constantly use models and mathematics to gain insight into the physics of their field.

However, for many situations of interest, the mathematics may become too difficult. The mathematicians are unable to answer the scientist's important questions because a complete understanding of the differential equations is beyond human knowledge. A famous longstanding such unsolved problem is the n-body problem: if more than two planets are revolving around one another, according to the law of gravity, will the planets ram each other or will they drift out to infinity?

Fortunately, in the last fifty years computers have been able to help mathematicians solve complex models over short time periods. Numerical analysts have developed techniques to graph solutions to differential equations and thus to yield new information about the model under consideration. All college calculus students use calculators to find solutions to simple differential equations called integrals. Space-travel is possible because computers can solve the n-body problem for short times and small n. The design of the stealth jet fighter could not have been accomplished without the computing speed of parallel processors. These successes have unrealistically raised the expectations for the application of mathematics to scientific problems.
Unfortunately, even assuming the model of the physics is correct, computers and mathematicians cannot solve more difficult problems such as the weather equations for several reasons. First, the solution may require more computations than computers can make. Faster and faster computers push back the speed barrier every year. Second, it may be too difficult to collect enough data to accurately determine the initial conditions of the model. Third, the equations of the model may be non-linear. This means that no simplification of the equations can accurately predict the properties of the solutions of the differential equations. The solutions are often unstable. That is a small variation in initial conditions lead to large variations some time later. This property makes it impossible to compute solutions over long time periods.

As an expert in the solutions of non-linear differential equations, I can attest to the fact that the more than two-dozen non-linear differential equations in the models of the weather are too difficult for humans to have any idea how to solve accurately. No approximation over long time periods has any chance of accurately predicting global warming. Yet approximation is exactly what the global warming advocates are doing. Each of the more than 30 models being used around the world to predict the weather is just a different inaccurate approximation of the weather equations. (Of course this is only an issue if the model of the weather is correct. It is probably not because the climatologists probably do not understand all of the physical processes determining the weather.)

Therefore, logically one cannot conclude that any of the predictions are correct. To base economic policy on the wishful thinking of these so-called scientists is just foolhardy from a mathematical point of view. The leaders of the mathematical community, ensconced in universities flush with global warming dollars, have not adequately explained to the public the above facts.

President Obama should appoint a Mathematics Czar to consult before he goes to Copenhagen.

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