Why Do We Need Mathematics for A Liberal Education?

In their search for the ideal education (paideia), the Ancient Greeks were sometimes split over whether to attribute primacy to mathematical thought or language skills. Pythagoras and Plato defended the primacy of mathematics; the Sophists and Aristotle, in different ways, the importance of language.

Similarly, the Ancient Greeks began to distinguish and systematise fundamental disciplines within each of the two fields.

These disciplines became known as the liberal arts because they equipped the free for their civic duties and the life of the mind.

Medieval scholars, such as John of Salisbury and Hugh of St. Victor, called the language-centred disciplines (grammar, logic, rhetoric) the threefold way (trivium), and the mathematical ones (arithmetic, geometry, music, and astronomy) the fourfold way (quadrivium).

However, the classical liberal arts education was displaced by Renaissance humanism, the rise of modern science, and educational programs of a pragmatist orientation.

Over the last few decades, there has been a movement to retrieve classical liberal arts education. Early proponents of this movement often took inspiration from Dorothy L. Sayers’s advocacy of the trivium in The Lost Tools of Learning. In recent years, however, others have insisted that the quadrivium is equally important for a well-rounded education in the liberal arts.

In this episode, Prof. Peter Ulrickson will discuss five books that can help us learn and appreciate the quadrivium.

Peter Ulrickson, a mathematician, teaches at the Catholic University of America. He is the author of A Brief Quadrivium and Teaching the Quadrivium, books that reveal the enduring significance of the mathematical disciplines of the traditional liberal arts, and make them newly accessible for students and teachers today. Ulrickson also publishes research in various fields of modern mathematics.

1. The Elements (alternative edition, Books I-VI with coloured diagrams)
   by Euclid
2. Almagest
   by Ptolemy
3. Timaeus
   by Plato
4. On order (De ordine) (alternative edition)
   by St. Augustine
5. To Save the Phenomena: An Essay on the Idea of Physical Theory from Plato to Galileo
   by Pierre Duhem

The trivium is clearly important for a Catholic education. It is a requisite of theological reasoning and biblical interpretation. The need for the number-centred quadrivium is not so apparent. Is the quadrivium necessary too?
Yes and no. To sketch out the full seven liberal arts, the mathematical ones are essential. Augustine acknowledges that it is difficult to cover everything. For some, there is going to be more of a focus on language; for others, on the mathematical. As much as I propose the significance of these mathematical disciplines for a liberal education, we have a limited amount of time and attention, There are other goals in education. Ideally, the mathematical disciplines have an important place. What that looks like in practice varies vary according to the person.

What motivated you to write A Brief Quadrivium.
Those who are interested in recovering a classical education, have been confident about the place of the verbal arts and articulated various ways to pursue them. With mathematics, this is more difficult. This difficulty has not just arisen recently.

Take astronomy. We are going to talk about Ptolemy’s Almagest, a very difficult and technical work. However, it has been challenged by heliocentric models of the solar system, not to mention further developments in physics and astronomy. Traditional arithmetic has been challenged, maybe even completed, by the development of algebra and so forth. All these strands go in different directions and develop at different times.

I realized that there was not a nice, single entry-point where you could get, at roughly the same technical level, a presentation of these four disciplines and how they are interrelated. They have their proper boundaries, but they really go together. My book attempts to provide a single accessible entry-point to the four, instead of saying, “Go through all of Euclid's Elements!” That takes a long time and will not teach you some things.

My hope is that someone who studies and works through A Brief Quadrivium will be well prepared to turn to the original sources, which are difficult to get into.

"The practice of these mathematical disciplines is a kind of natural contemplation."

For whom did you write A Brief Quadrivium. For schoolchildren, adults, or both?
It is really for both.

A lot of students could study it profitably around the age of 14. In the United States, that is when they often have a course in geometry.

You can also study it profitably later in life. It is a bit like acquiring a degree of competence, say intermediate level, in a foreign language, such as Latin. Some might do that in high school. That is a great intellectual resource for one’s whole life. However, if you become interested in learning it later on, you can do so profitably. It is accessible but it is not childish. It is fruitful at many ages.

How can Catholics apply the quadrivium in their everyday lives?
The practice of these mathematical disciplines is a kind of natural contemplation. Absolutely speaking, they are relatively low in the order of things. Nevertheless, they have real beauty and clarity. They give the student a chance to contemplate realities that are separated from material things, without putting them to some practical use and experiencing them instead as a kind of end. Of course, we have a further end. But in this life, we also experience intermediate ends. They are signs of our pursuit of that farther end.

If I have understood you correctly, you are saying that the primary aim of the quadrivium is to help us contemplate the order present in the universe and, ultimately, the divine source of that order. Its primary aim is not, as occurs in much modern science, to work out how nature works so as to develop better technology. It is primarily a contemplative rather than a practical endeavour.
Yes! A couple of years back, Elon Musk tweeted, “no such thing as a ‘rocket scientist’, only rocket engineers.” Of course, he is not opposed to building rockets, but that is a simple way of framing the distinction.

Some might fight it surprising that music is one of the four liberal arts of the quadrivium. We associate the appreciation of music with listening to works, say, by Bach, Mozart, Beethoven, or, if you want, The Beatles and Jazz. We associate it with playing an instrument or singing, not with mathematics. What do you mean by "music" when you include it within the quadrivium?
Understanding the way that certain relations between numbers make pleasant relations of pitch intelligible.

The simplest example is an octave. You have a relation of two to one. This is seen in, say, a guitar string, whose frets allow you to stop the string at various lengths. You are not going to play an octave up on the same string. You would go to a different string. But if you go carefully up the neck, you shall find the distance to the note that is an octave higher is half as long as the distance to the original note. From there, you can proceed to other pleasing combinations or sequences of sound. These can be understood through ratios: the relations of whole numbers.

We also see number in the rhythm of music. Augustine emphasized this. He thought a lot about poetry and, so, about long and short sounds. Latin poetry, unlike ours, emphasises long and short vowels, not just accent. Both rhythm and pitch come together in the way that we teach little children to count. We say, “Ooone, Twooo, Threee….” We say the numbers in this dramatic or performative way for children. Very naturally, we make it rhythmic, with even times, and a bit of pitch variation.

So, there is number in the ordering of pitches and times. Those are the ways in which music is mathematical.

What is the focus of your own teaching and research?
Recently, my teaching and research recently has focused on the quadrivium and on mathematics as a liberal art.

My doctoral research was on the mathematics inspired by quantum field theory. This is a fairly abstract area of mathematics and uses a lot of technical tools.

These are two very different areas: the mathematical aspects of quantum field theory and the general foundation for liberal education. To me, they are complementary. Throughout history, mathematics has been connected with many philosophical disputes. One of the clearest ways to appreciate ancient mathematics is to study mathematics, as practiced by professional mathematicians today. You see the continuity between the two and appreciate the value of the older traditions. They are complementary.

Some may find your booklist surprising, if not eccentric. The first four books that you have chosen are likely to figure in a great books program but not in the curriculum of a science major. They may be of interest to the intellectual historian or to the philosopher, but surely much of the science is outdated, or, even when the mathematics is still sound, as in Euclid’s Elements, we would do better to read the state-of-the-art literature?
The two questions are “Who is reading?” and “When?” Certainly, some people need to be reading the state-of-the-art literature. However, in thinking about the teaching of mathematics, I like to use the metaphor of physical training. There is the workout that an Olympic decathlete will be doing right now. It is an incredible feat. Most of us could not dream of ever doing it. There is another kind of workout, such as taking a fifteen-minute walk during lunchtime or doing some stretches and push-ups in the evening. That is not particularly good training for the decathlete, but it is proportional to our needs.

The works that I propose are meant to be proportional to the learner who is a beginner and does not aim at attaining specific technical competence.

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The first book you have chosen is Euclid’s great treatise on geometry, the Elements. Have you recommended it as a classic treatise on geometry or for its exposition of arithemetic too (Books 7-10).
In some way, all four disciplines are present in the books that I recommend. However, you are right. The Elements is principally about geometry but has some significant arithmetic in those intermediate books. If you can get through the arithmetical books, that is great.

The nice thing about the geometry of the Elements, even if you just look at Book One, is its method. Euclid states clearly the first principles that he will use and the proposition that is to be proved. Then, he goes through the proof. His mathematics is focused on proof rather than computation. That comes through in an especially clear way in plane geometry. This is also the method for speaking of whole numbers, integral quantities, and proving things about them.

We get caught up in computation with numbers: to do our taxes or figure out how much a home improvement project is going to cost. Numbers are tied up with these things in the ordinary practice of life. By beginning with the geometrical, we imbibe the spirit of mathematical proof.

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