November 30, 2009
The Mathematics of Global Warming
The forecasts of global warming are based on mathematical solutions for equations of weather models. 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 scientists' understanding. Often the mathematicians can describe the evolution through time of the scientists' 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 three hundred years since Newton. Engineers constantly use models and mathematics to gain insight into the physics of their field.
However, for many situations of interest, the mathematics become too difficult to be helpful. 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 example 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 amounts of time and small n-values. 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 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. This means that a small variation in initial conditions will 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 weather models 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 thirty models being used around the world to predict the weather is just a different inaccurate approximation of the weather equations. (Of course, this is an issue only 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, one cannot logically conclude that any of the global warming 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.
Peter Landesman (firstname.lastname@example.org) is the author of the 3D-maze book Spacemazes, with which children can have fun while learning mathematics.