October 01, 2013

The Back Page

Pierre de Fermat was a seventeenth century French lawyer who dabbled in mathematics. I’m not sure if he was a good lawyer, but it is clear that he was a capable mathematician. In 1637, he posited the following simple theorem. If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c. For those of you who remember your junior high algebra, you’ll recognize the Pythagorean Theorem, a2 + b2 = c2 as an example of Fermat’s theorem. For n=2, it works. But for any other values of n greater than two, it does not. For example, a3 + b3 ≠ c3. Fermat claimed to have proved his theorem, but there is no record that he actually pulled it off. How hard could this little theorem be to prove? It turns out, it was really hard. Fermat’s theorem became famous in the math world, garnering its own name: Fermat’s Last Theorem, or FLT for short

The FLT, despite its facial simplicity, was a very tough nut to crack. Many prominent mathematicians after Fermat failed in their attempts at proving the FLT, and it became one of the holy grails of mathematics. In 1993, some 356 years after Fermat proposed the theorem, Andrew Wiles, an English-born Princeton math professor, finally pulled the sword from the stone. His proof was more than one hundred pages long, and it was so complex that only a handful of experts in the world could understand it. (There is an excellent BBC documentary about Wiles’s discovery on YouTube.)

Fermat’s Last Theorem is a great example of a seemingly simple problem that is very hard to solve. Environmental law abounds with such challenges. Take storm water regulation, for example. Water runs downhill. It picks up dirt and other pollutants along the way as it washes over our fields and roads. How hard could it be to regulate it? Extraordinarily difficult, it turns out. It is hard to figure out where it is coming from, and it is harder to corral it in a manner that is cost-effective and still protects the environment. Or take global warming. We know CO2 is the culprit, and we know how to reduce production of the gas, but coming up with a workable international plan to reach that result is a huge undertaking.

It is often easy to spot the problem, but much tougher to solve it. As we move forward to the next phase of environmental protection in this country, we must bear that in mind. The easy problems have been run to ground. The tough ones like global warming and nutrient pollution won’t fall so easily. If the science is not the problem, politics often is. Policy decisions are often no less difficult to resolve than math conundrums. Will it take us 356 years to figure out a way to stop global warming? Let’s hope not.