Word: theorems
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...fondness for logic and reason and a mother who believed in starting her son's education early. By age 10, Godel was studying math, religion and several languages. By 25 he had produced what many consider the most important result of 20th century mathematics: his famous "incompleteness theorem." Godel's astonishing and disorienting discovery, published in 1931, proved that nearly a century of effort by the world's greatest mathematicians was doomed to failure...
...appreciate Godel's theorem, it is crucial to understand how mathematics was perceived at the time. After many centuries of being a typically sloppy human mishmash in which vague intuitions and precise logic coexisted on equal terms, mathematics at the end of the 19th century was finally being shaped up. So-called formal systems were devised (the prime example being Russell and Whitehead's Principia Mathematica) in which theorems, following strict rules of inference, sprout from axioms like limbs from a tree. This process of theorem sprouting had to start somewhere, and that is where the axioms came in: they...
...thinking of theorems as patterns of symbols, Godel discovered that it is possible for a statement in a formal system not only to talk about itself, but also to deny its own theoremhood. The consequences of this unexpected tangle lurking inside mathematics were rich, mind-boggling and--rather oddly--very sad for the Martians. Why sad? Because the Martians--like Russell and Whitehead--had hoped with all their hearts that their formal system would capture all true statements of mathematics. If Godel's statement is true, it is not a theorem in their textbooks and will never, ever show...
...Were the above 100 percent readings three days later just a fluke, or were they the output of a normally functioning system? What's truly going on? Perhaps inquiries should be forwarded to British mathematician Andrew Wiles, who gained celebrity for his complicated proof of Fermat's Last Theorem, to see if he can provide a solution. Of course, I myself have discovered a remarkable little explanation, but unfortunately this column is too short to contain...
Mazur is well-known for his work on topology, and his accomplishments in the area of number theory are widely seen as a factor in the 1996 proof of Fermat's Last Theorem by Andrew Wiles...