Pure mathematics will remain more reliable than most other forms of knowledge, but its claim to a unique status will no longer be sustainable. So predicts Brian Davies, author of the article "Whither Mathematics?", which appears in the December 2005 issue of Notices of the AMS.
For centuries mathematics has been seen as the one area of human endeavor in which it is possible to discover irrefutable, timeless truths. Indeed, theorems proved by Euclid are just as true today as they were when first written down more than 2000 years ago. That the sun will rise tomorrow is less certain than that two plus two will remain equal to four.
However, the 20th century witnessed at least three crises that shook the foundations on which the certainty of mathematics seemed to rest. The first was the work of Kurt Gödel, who proved in the 1930s that any sufficiently rich axiom system is guaranteed to possess statements that cannot be proved or disproved within the system. The second crisis concerned the Four-Color Theorem, whose statement is so simple a child could grasp it but whose proof necessitated lengthy and intensive computer calculations. A conceptual proof that could be understood by a human without such computing power has never been found.
The third crisis deals with the Classification of Finite Simple Groups, a grand scheme for organizing and understanding basic objects called finite simple groups. Finite simple groups are absolutely fundamental across all of mathematics, something like the basic elements of matter, and their classification can be thought of as analogous to the periodic table of the elements. Indeed, the classification plays as fundamental a role in mathematics as the periodic table does in chemistry and physics.
And yet, to this day, no one knows for sure whether the classification is complete and correct. Mathematicians have come up with a general scheme, which can be summarized in a few sentences, for what the classification should look like. However, it has been an enormous challenge to try to prove rigorously that this scheme really captures every possible finite simple group. Scores of mathematicians have written hundreds of research papers, totaling thousands of pages, trying to prove various parts of the classification. No one knows for certain whether this body of work constitutes a complete and correct proof.
4. Which of the following statements about the Classification of Finite Simple Groups is correct?
When you are finished answering the questions, go on to the next part of the exercise here.