Mathematics: The loss of certainty

11/01/05

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 will appear 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 Goedel, 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. Many other theorems of a similar type are now known, and more are being discovered every year.

The third crisis seems to show how the uncertainty foreshadowed in the two earlier crises is now having a real impact in mathematics. The Classification of Finite Simple Groups is a grand scheme for organizing and understanding basic objects called finite simple groups (although the objects themselves are finite, there are infinitely many of them). Knowing exactly what finite simple groups are is less important than knowing that they are absolutely fundamental across all of mathematics. They are 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. Many results in mathematics, particularly in the branch known as group theory, depend on the Classification of Finite Simple Groups.

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. What is more, so much time has now passed that the main players who really understand the structure of the classification are dying or retiring, leaving open the possibility that there will never be a definitive answer to the question of whether the classification is true. As Davies puts it:

We have thus arrived at the following situation. A problem that can be formulated in a few sentences has a solution more than ten thousand pages long. The proof has never been written down in its entirety, may never be written down, and as presently envisaged would not be comprehensible to any single individual. The result is important and has been used in a wide variety of other problems in group theory, but it might not be correct.

These three crises could be hinting that the currently dominant Platonic conception of mathematics is inadequate. As Davies remarks:

[These] crises may simply be the analogy of realizing that human beings will never be able to construct buildings a thousand kilometres high and that imagining what such buildings might "really" be like is simply indulging in fantasies.

We are witnessing a profound and irreversible change in mathematics, Davies argues, which will affect decisively its character:

[Mathematics] will be seen as the creation of finite human beings, liable to error in the same way as all other activities in which we indulge. Just as in engineering, mathematicians will have to declare their degree of confidence that certain results are reliable, rather than being able to declare flatly that the proofs are correct.

Davies's article "Whither Mathematics?" (PDF, 448KB) is available at Mathematics: The Loss of Certainty.

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