By Annie Frost (SF28) | November 19, 2025

Lia McGrath is not a sailor and has little to no real-life experience with knot-tying, but the 2025 Santa Fe graduate is currently doing some light research in knot theory on top of her post-baccalaureate studies at Smith College.

What exactly is knot theory? McGrath explains: “If you wanted it in math-speak, it’s about embeddings of the circle in three-dimensional space, or of an n minus 2-dimensional manifold in n space.” The main question in knot theory, however, can be boiled down to, “How can you tell if two knots are the same thing?” she asks. “Because they could be tangled up differently, but how do you know if you can get from one knot to another?” 

McGrath’s research and coursework at Smith is the latest stage in her long-term interest in math, which began during high school and deepened throughout her four years at St. John’s. She completed the Program on a high note, co-winning the school’s annual Richard D. Weigle Prize for best senior essay for her paper entitled, “On the Limits of Mathematical Construction and Proof: An Investigation of Axioms, Models, and Forcing Through Godel and Cohen’s Proof of the Independence of the Generalized Continuum Hypothesis.” Antonio Fox (SF25), the prize’s other recipient, wrote on Friedrich Nietzsche’s The Gay Science.

McGrath's curiosity was first sparked by a high school pre-calculus class, where she was introduced to infinite sequences and series—a series being the sum of the terms of an infinite sequence of numbers. It really took off, however, during the summer after her junior year of high school. Looking for a way to fill her free time, she naturally enrolled in a linear algebra class at a local university in her hometown of Tucson, Arizona.

It was during this course that McGrath found herself getting lost in the experience of seeing the math in her mind. “It was definitely a visualization thing,” she says. “I realized you can visualize vector spaces and stuff like this, and also, with taking derivatives, that you can visualize functions and their slopes.”

In retrospect, she adds, it’s not surprising that this type of visualization is what initially drew her in, since math itself is a discipline that first began with geometry. So, of course, does math at St. John’s College.

McGrath found herself drawn to proof composition during her freshman year when, while studying Euclid’s Elements in her math tutorial, she went to the Meem Library to feed a desire to learn real analysis and came across the proof of the Heine-Borel theorem, which, according to her, states that “compact sets in Euclidean space are those which are both closed and bounded.” Three years later, McGrath's senior essay included a proof of her own in an addendum entitled “A Proof of the Closure of Constructible Places with Respect to Set Union.”

For those who don’t spend their free time pursuing real analysis, McGrath describes the starting point of inquiry for her essay in simple terms: “I think that the question really was that, when doing math, we feel like we can know math by visualization, by just actually picturing it in our mind, and we can know that a proof is true. Or that something works in a certain way because we can develop a visual intuition for it.”

At some stage in the historical development of mathematics, we lose the very visualization ability that drew her into the field, and are forced to rely on something else. “Maybe it was when people wanted to reason about geometry in more than three dimensions, or stuff in analysis, and then also with these set theoretical paradoxes,” McGrath says. “We need the symbols of math to work on their own; it can’t just be a way to describe something that we’re thinking. We need to be able to think with the symbols.”

McGrath was fascinated by the next step in this process, in which, for set theory—which needs to be axiomatized—the symbols are further divorced from our thinking. Otherwise, she says, we wind up with paradoxes like Russell’s paradox: “The system of symbols as set theory gets kind of separated from set theory as I imagine things with things in them … And once that separation is made, what is math? Can I get back from the symbols to a more visual, intuitive understanding, or am I stuck over here? That’s kind of the question.”

While working on her paper and reflecting on this narrative through mathematics, McGrath noticed something similar happening with the quantum physics unit of senior lab, specifically regarding axiomatization with symbols.

“You read Huygens or Newton or whatever, and they say light is a wave, or light is a particle,” she says. “You think of it in this clear way in your head. The same thing happens where our models—‘model’ in the more general sense, not the math sense—don’t really cohere with the phenomena of how the world works any longer. It starts to kind of pull apart in that same sort of way.”

Despite having moved far beyond freshman year geometry, McGrath employed an example in her senior essay familiar to all Johnnies: “We are still acting completely inside of the Euclidean mode, which says, ‘A point is that which has no part,’ and in doing so grounds its concepts in a shared experience of language.” This is the opening definition in Euclid’s Elements, and a freshman student's deceptively simple introduction to the St. John’s math program.

McGrath didn’t originally intend to invoke the start of the math program with this example, but found it to be fitting in the end. “I do feel like this, to me, completes the narrative of the math program, this slow pulling apart that I’ve referenced a little bit. And it did fit the purpose, because it illustrates how definitions kind of changed their function: that the definitions were just getting at the thing in your mind, versus something which just gets at something in the formal symbols, where I don’t actually know it in the same sort of way.”

As someone interested primarily in mathematics, McGrath opted to attend St. John’s because she thought that studying math alone would be too limiting for her: “I remember looking at the senior seminar list and thinking that I wanted to read Marx and Hegel.”

Having completed said wish—and after spending four years immersed in the history and evolution of the field of mathematics spanning more than 2,000 years—McGrath is ready to transition onto more modern material. She is planning on attending a PhD program next year after supplementing her knowledge through Smith’s one-year post-bac program.

When forced to choose between dinner with the 20th-century mathematicians Kurt Gödel and Paul Cohen, McGrath chose Gödel: “He didn’t betray much personality in his writings I’ve read, but I get the sense that he found mathematics and logic something more than simply an exercise for the mind,” she concludes. “I’d like to maybe ask him about his ontological proof of God.”