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Philosophy professor Zeynep Soysal works at the intersection of math and language to answer an age-old question: what makes mathematical expressions meaningful, and what do they reveal about our world?

In the spirit of logical empiricism, Soysal considers mathematical sentences analytical: their truth and meaning are determined by how we use words. She also believes that they are about words.

“There’s something very weird about saying that math is only about words, right?” she admits, “because that’s what it says, it’s only about words, nothing else. The only information [pure math] conveys is linguistic information.”

Possible worlds semantics is a popular way of modeling what humans believe and know. Content, both mental and linguistic, is understood in terms of possible states of the world. When a sentence expresses a proposition, it carves the world into a possible set of worlds. Learning something new rules out possible worlds.

In possible worlds semantics, learning something new involves ruling out possible worlds.

It’s a widely applicable and intuitive theory– but it doesn’t work for math. It’s not uncommon for mathematics to be a problem case, as “a very old and a very big problem in philosophy,” Soysal says.

The issue for possible worlds semantics is that necessary mathematical truths don’t rule out any way the world can be. They just are true, no matter what the world is like. 

So, is it time to abandon possible worlds theory? Soysal doesn’t think so. “My project now is basically to say: actually, the possible worlds account does work just fine for mathematics. Okay, mathematical sentences don’t rule out any way the world can be. But in another sense, they say something about the world.”

What do mathematical sentences say about the world? Something about language, Soysal argues. 

Logical empiricists would argue that the truth of mathematical expressions is determined by language, but not about language. “But actually, it’s both.” Soysal says; “mathematics is about language.”

The content of a mathematical sentence, therefore, is that the sentence itself is true.

Zeynep Soysal is an associate professor in the Department of Philosophy at the University of Rochester and a faculty advisor for the Center of Language Sciences. Find her work at zeynepsoysal.com.