9.4: Final word

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Finding proofs is an art. In some ways, it’s like programming: you have a set of building blocks, each one defined very precisely, and your goal is to figure out how to assemble those blocks into a structure that starts with only axioms and ends with your conclusion. It takes skill, patience, practice, and sometimes a little bit of luck.

Many mathematicians spend years pursuing one doggedly difficult proof, like Appel and Haken who finally cracked the infamous four-color map problem in 1976, or Andrew Wiles who solved Fermat’s Last Theorem in 1994. Some famous mathematical properties may never have proofs, such as Christian Goldbach’s 1742 conjecture that every even integer is the sum of two primes, or the most elusive and important question in computing theory: does P=NP? (Put very simply: does the class of problems where it’s easy to verify a solution once you have it but crazy hard to find one actually have an easy algorithm for finding them we just haven’t figured out yet? Most computer scientists think “no," but despite a mind-boggling number of hours invested by the brightest minds in the world, no one has ever been able to prove it one way or the other.)

Most practicing computer scientists spend time taking advantage of the known results about mathematical objects and structures, and rarely if ever have to construct a water-tight proof about them. For the more theoretically-minded student, however, who enjoys probing the basis behind the tools and speculating about additional properties that might exist, devising proofs is an essential skill that can also be very rewarding.