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Reversing the proof

A Penn State mathematician uses reverse mathematics to explore unusual theorems
27 July 2020
Westrick in front of chalkboard

Find the simplest proof. These directions on a geometry assignment provide a puzzle for a student, directing them to use a chain of logical reasoning based on the basic principles of math that we accept to be true—like A + B = B + A. For example, a student might use basic principles about triangles to prove the Pythagorean theorem, which describes the relationship between the length of a right triangle’s sides.

While some students might find these proofs daunting, some relish the challenge. Assistant Professor of Mathematics Linda Westrick has made a career of researching mathematical proofs and the unusual theorems that don’t always play by the rules.

“The kinds of proofs that students do in geometry are not very open ended,” said Westrick. “You have a small list of basic principles, which mathematicians call axioms, and you put them together to solve the puzzle. When I took a discrete math class in high school, I realized that proofs can have so many more pieces and so many different ways that you can put them together. It’s a really different experience. When I took that class, I knew this was what I wanted to study.”

One way that a proof can be simple is if it uses few axioms. Sometimes, in an attempt to find a simpler proof, mathematicians reverse the process: Instead of using axioms to prove a theorem, they assume the theorem is true and work backwards to try to prove an axiom. This process is called reverse mathematics.

“Imagine you have a proof with a handful of axioms,” said Westrick. “In reverse mathematics, we see if the theorem can prove one of those axioms. If not, then you don’t really need that axiom to prove the theorem. So you go back to the original proof and try to work out a simpler version without it. It turns out that five axioms, called the Big Five, frequently appear in reverse mathematics, and most theorems are equivalent to one of them.”

Westrick is applying the principles of reverse mathematics to the field of combinatorics, which revolves around counting and combinations. But there, things start to get unusual. A common theorem in combinatorics called Ramsey's Theorem for Pairs can be proved by one of the Big Five axioms, but the system doesn’t reverse. 

“Ramsey's Theorem for Pairs doesn’t prove this common axiom,” said Westrick. “Some other combinatorial theorems don’t follow the usual patterns either. This opens up a lot of questions about the strength of these combinatorial theorems.”

To explore these questions, Westrick and her collaborators received a grant from the National Science Foundation to establish a Focused Research Group (FRG), which is designed to bring together researchers from multiple institutions to tackle a major research challenge in mathematics. Westrick’s team includes researchers from the University of Connecticut, Notre Dame, the University of Chicago, and the University of California, Berkeley.

“There’s a stereotype of a mathematician sitting alone in a corner working out equations, and we do some of that,” said Westrick. “But we also interact with each other, and those interactions shape the direction that the field goes.”

Westrick also values the opportunities for collaboration at Penn State. She first visited University Park as a graduate student, spending a week working with then Professor of Mathematics Steve Simpson, a noted logician and one of the original founders of reverse mathematics.

“At the time I thought, ‘Hey, I really like it here. I hope I can come back sometime,’ not imagining that I would end up working here,” said Westrick. “The department is really friendly and has a long history in mathematical logic.  I'm enjoying my collaborations here as well as teaching our excellent students.”