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Math Trek

A mathematician puts Fermat's Last Theorem on an axiomatic diet

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Fermat’s Last Theorem is so simple to state, but so hard to prove. Though the 350-year-old claim is a straightforward one about integers, the proof that University of Oxford mathematician Andrew Wiles finally created for it nearly two decades ago required almost unimaginably complex theoretical machinery. The proof was a dazzling demonstration of that machinery’s value, but one aspect of it troubled mathematicians: It relied on stronger axioms than mathematics normally requires, and ones far more complex than are needed to state the problem. Surely, many mathematicians thought, it was possible to prove Fermat’s Last Theorem while assuming less.

Proof — the demonstration of logical consequences arising from a set of axioms — is at the heart of mathematics. But the particular axioms that underlie mathematics aren’t universally agreed upon. The most commonly used axioms are called set theory. But for some theorems, mathematicians assume additional axioms as well. Fewer axioms suffice for others, because set theory involves concepts like infinity that aren’t always needed.

Fermat’s Last Theorem seems too simple to require the full apparatus of set theory, much less even more axioms. Around 1630, Pierre de Fermat noted in the margin of a book that he had discovered a “truly marvelous demonstration” that there are no integers a, b and c that make the equation an + bn = cn true if n is a whole number greater than 2. Unfortunately, he said, the margin was too small for his proof.

When Wiles finally proved the theorem in 1994, he used a deep connection between Fermat’s Last Theorem and algebraic geometry, a field in its infancy in Fermat’s time. Modern algebraic geometry was built using extraordinarily powerful tools developed in the mid-20th century by the mathematician Alexander Grothendieck that rely on an extra axiom in addition to those of standard mathematics — so Wiles’ proof did too.

Grothendieck introduced the axiom to deal with a pesky logical problem. His tools got much of their power through being extremely abstract.  For example, instead of just considering the set of all whole numbers, Grothendieck also included the spaces defined by equations of whole numbers. (Think, for example, of the circles and parabolas and ellipses defined by equations in two variables). Then he threw in the set of all functions between such spaces, and so on, with one set building on another.

Set theory (that is, the standard axioms of mathematics) puts limits on how far this process can go. For example, there is no set of all sets, because a set of all sets would lead to logical difficulties like the famous Russell paradox: Imagine the set whose elements are precisely those sets that don’t contain themselves. Does it contain itself? Prepare yourself for some brain-twisting here: The answer can’t be yes, because then by its own definition it’s a set that doesn’t contain itself. But the answer can’t be no, either: If the set doesn’t contain itself, then it satisfies its own definition, so it does contain itself. Damned if you do, damned if you don’t!

With his sets built on sets built on sets, Grothendieck wanted to go into territory perilously close to this. So he created a new axiom that would allow him to use these very large sets of sets of sets while still keeping clear of paradox. This axiom posited the existence of a larger type of set called a “universe.” Grothendieck’s approach was fabulously successful, allowing him to prove one major theorem after another and laying the groundwork for modern algebraic geometry, among other fields — all leading to the remarkable success of the proof of Fermat’s Last Theorem. But all this work relies on the axiom of universes. “Everyone sees that this is a quick and dirty fix,” says Colin McLarty of Case Western Reserve University in Cleveland. “But it works.”

Intuitively, many mathematicians thought Grothendieck didn’t need his fancy universes. Ordinary set theory seemed like it should suffice — or even less. In particular, set theory involves axioms about infinity that seem unnecessary for something like Fermat’s Last Theorem, which involves only finite whole numbers. But no one had proved it until now. McLarty recently showed that Fermat’s Last Theorem can be proven using the ordinary axioms of mathematics, and even a bit less. He presented his results at the Joint Mathematics Meetings in San Diego in January.

“This justifies a feeling that lots of people had and I had too,” McLarty says. McLarty showed that a subset of set theory called finite-order arithmetic, which assumes less about the concept of infinity, is sufficient to undergird Grothendieck’s work.

“This is a well-done, major, clarifying first step,” says Harvey Friedman of Ohio State University. Friedman believes the work could be extended to show that Fermat’s Last Theorem requires only axioms relevant to arithmetic, with no use of infinity at all.

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