four color theorem

Computer Models: Questions for Mathematicians and Economists

by Elaine Schwartz    •    Apr 30, 2013    •    1344 Views

Reading about the 4-color theorem for maps, I realized it was about a lot more than math.

My story starts with an recent obituary about one of the mathematicians, Kenneth Appel, who first proved the 4-color theorem. Completing a quest that started in 1852 by Francis Guthrie, Kenneth Appel and his colleague, Wolfgang Haken proved in 1976 that a flat map needs no more than 4 colors.


Until 1976, mathematicians intuitively agreed that maps could be drawn with a 4-color minimum but couldn’t prove it. The reason? It was tough to account for 1,936 potential map configurations like, for example, a nation surrounded by 4 others.

That takes us to the University of Illinois, the 1970s, and a state-of-the-art IBM computer that was as big as a room but had less memory than a cell phone. Still, it provided Appel and Haken the opportunity to prove that the 1,936 configurations needed no more than 4 colors to avoid having 2 adjacent countries with the same color. Concerned with accuracy, though, Dr. Appel had family members checking for typos and accurate print-outs. His sister Laurie found 800 mistakes.

Although Dr. Appel’s obituary described him as having “solved a longstanding problem concerning colors on a map,” not everyone agrees. As one Berkeley mathematician commented, “”Like a landmark Supreme Court case, the proof’s legacy is still felt and hotly debated.” In proofs that depend on computers, transparency is impossible because no one can check for bugs and the unseeable.

A 4 Color US Map

4 color US map

Computer transparency took me to 1998 and the Long-Term Capital bailout. One of the first computer related financial debacles, the demise of Long-Term Capital involved Nobel Prize winning economists. Using programs they developed, the firm claimed to have an infallible model for making money. Essentially, they had programmed the past to determine the future. The problem? The past did not repeat itself. The result? A bailout to avert a financial crisis.

Fast forward to 2013 and Harvard economists Kenneth Rogoff and Carmen Reinhart. Challenged by a University of Massachusetts graduate student, their computations conclude that a relatively large national debt constrains GDP growth. Assuming we are totally neutral on whether they are right or wrong, we can confirm that one of their spread sheets had an incorrect number that skewed computerized results.

You can see where we are going. Whether making maps, investing money or determining economic policy, computer models have a gargantuan impact. I wonder why we are not questioning them more frequently.

Sources and Resources: I recommend reading the NY Times obituary for Kenneth Appel and then more on the four-color theorem in an excellent Economist article from 2005. If you want to continue with more of the math (and my map source), this paper is a possibility. On the economics side of mathematical modeling, here is the Reinhart/Rogoff response to the controversy surrounding their work and a brief BBC overview from the always interesting Tim Harford.

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