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A Conversation on Math as a Liberal Art with Topi Heikkerö

April 23, 2018 | By Eve Tolpa

Topi Heikkerö is a tutor at St. John’s College in Santa Fe.

Topi Heikkerö is a tutor at St. John’s College in Santa Fe and author of Ethics in Technology: A Philosophical Study. Heikkerö here discusses math and the liberal arts.

How is math situated within a liberal arts curriculum?

The mathematical arts can be defined as arithmetic, geometry, music, and astronomy. In Plato’s Republic, Socrates called this curriculum the “quadrivium,” and he said it’s likely that this study is really compulsory for us, since it evidently compels the intellect to use the intellect itself on the truth itself. Socrates’ thesis is that the study of four mathematical arts constitutes a necessary intermediate station on the path out of the cave of received opinions and toward philosophical dialectic.

The mathematical arts also give us a model for logical reasoning, although Socrates said that they fall short of producing true philosophical knowledge, because they assume their subject matter without asking the proper questions about it. For example, it’s true that we know that five plus seven equals twelve. It’s twelve in the U.S., it’s twelve in Europe, it’s twelve in Asia. But we can also take a philosophical approach and ask, what is seven? What is five? How do we know they make twelve? People don’t always agree on the answer to those questions, but no one disagrees that five plus seven equals twelve.

How are St. John’s students introduced to math as a liberal art?

St. John's freshmen study a lot of Euclid’s geometry, and we see the necessity of the mathematical proof and reasoning in action. We see that there is a true argument and a false argument—and the true argument is usually quite elegant. Students learn about truth and falsity and elegance. I think it’s more fruitful to explore those issues through math than through logic. People who don’t like math and science tend to try and get a reprieve from studying those subjects in college, but having an experience of being reasonable and having real proof, I think that’s really important.

The self-understanding of modern mathematical-scientific endeavors is in no direct way connected to the idea of liberal education, or learning for its own sake. The modern way is in many ways Baconian: it emphasizes manipulation of nature in service to greater human well-being. When we study math and science in high school, it’s geared as preparation for our professional lives, for being a productive member of society. That’s a very legitimate orientation, because that’s what we at St. John’s also want. If liberal education aims to remain serious, the student of the liberal arts is obliged to encounter modern mathematics and science. Math and science have advanced incredibly in the last 500 years, and the college prepares us to engage in those things—and at the same time to look more deeply, too.

How does the college encourage that deeper perspective?

We want to push students to ask questions that help them reflect. It’s a delicate task. Often our first reaction to math as contemporary people is, “Tell me how it is.” The St. John’s approach requires more care and understanding, but it’s important if you want to become deeply educated people.

One crucial turning point in doing math is to dare to look at ourselves. We are so often focused on the external: How many problems can I solve? Is it right or wrong? At St. John’s we look at ourselves as a person, but also in a community. We want to be compassionate, we want to be civil even when we disagree. We do math and science, but we do that to be human. It’s important to solve problems and also to see myself solving the problem—to ask, what kind of human activity is this?

What often arises is some of the competition and ambition that some of us have learned in our earlier education. If a person can solve a problem or do a proof that is very difficult, that can lead to being slightly tyrannical and ambitious, and it can separate us from each other. But in the St. John’s Program, a student who has a very quick mathematical mind can learn to understand those other students who experience confusion. They might be able to teach them and maybe even learn from them.

We do these things together as human beings, and it’s important to understand ourselves and the other. I think that’s what really matters and what it means to be human.

Heikkerö recently participated in a panel discussion on math in the liberal arts as part of the Dean’s Lecture and Concert Series.