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.
- The Elements (alternative edition, Books I-VI with coloured diagrams)
by Euclid - Almagest
by Ptolemy - Timaeus
by Plato - On order (De ordine) (alternative edition)
by St. Augustine - 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.
1.
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.
2.
For your second book we turn to the Almagest (Syntaxis mathematica) of Claudius Ptolemy (c. 100-c. 170). How would you justify the inclusion of the Almagest on this list to those who might deem it an eccentric recommendation? After all, modern science has disproved many of the fundamental theses and conclusions of Ptolemy’s geocentric astronomy, with its cosmology of heavenly spheres.
Rather than seeing opposing camps, the one saying, “X!”, the other, “Not X!”, there is a successive refinement of approximations. A more modern example would be Newtonian mechanics as a limiting case of relativistic mechanics for things that move slowly. I would encourage an attitude like the one we display toward both Newton and Einstein, without seeing them as combatants in a war.
There are things that we can see when we look up into the sky. Ptolemy gives a nice explanation of them. He makes it clear that he needs to set some principles but is open to being corrected. He gives us good mathematical models for these phenomena. He also gives us a model for how to proceed, relating respectfully what his predecessors have done while trying to advance beyond them and recognising that his own methods might need some correction.
I propose him not simply as an example of method. Something very nice that he does is apply mathematics to ordinary observation.
Say you are 40º north. Where is the sun going to rise on the horizon in the middle of summer? That is a relatively simple phenomenon. It is familiar to and people have known about it for a long time. But it is nice to be able to give an account of it. The sun is not exactly east. Where is it then? Ptolemy can tell you.
You recommend it, then, for its value for explaining everyday immediate experience?
Yes. In the more elaborate, more modern kinds of physics and astronomy, you are dealing with things that people have never seen. You just take it all on. It is what you read in the book. However, it is good to unite the mathematical account with things that have really come through your own senses, not just someone else’s.
" In Euclid, we have both geometry and arithmetic. In Ptolemy, we have astronomy. Music is present in this Platonic dialogue."
3.
The nineteenth-century Anglican cleric, Benjamin Jowett, whose translations of Plato are still among the most widely-read, claimed that, “Of all the writings of Plato, the Timaeus is the most obscure and repulsive to the modern reader, and has nevertheless had the greatest influence over the ancient and medieval world.” Jowett was referring to the work’s cosmology and the influence it exerted on medieval theologians, thanks to Calcidius, who translated it into Latin. Why have you chosen the Timaeus and which of the four number-centred liberal arts does it cover?
There are a few reasons.
One is, as you mentioned, its significance for medieval cosmology. For a while, it was really the only work of Plato that was known.
Another reason is that it has a spirit of playfulness. Timaeus, the primary speaker for much of the work, says that he shall give a likely story. This is a reminder to temper our desire to give a mathematical account of the world's order. He reminds us playfully that there is a variety of ways in which we can account for it.
The disciplines that are especially present are music and geometry.
As you mentioned, in Euclid, we have both geometry and arithmetic. In Ptolemy, we have astronomy. Music is present in this Platonic dialogue in that, at one point, Timeus talks about the numerical relation between tones and how they correspond to octaves, fifths, fourths, and so forth. He does not just allude to this but treats it explicitly.
Geometry is also treated somewhat explicitly: in connection with the five Platonic solids. These are five symmetrical shapes that are made out of a single regular polygon. Euclid's Elements culminates in classifying the five Platonic solids. Part of what is going on in Timaeus is the correlating of the solids to the elements.
There is water, air, fire, and earth in the heavens. The most familiar Platonic solid is the cube. A square is a regular shape. Six squares make a cube. For the Timaeaus, this corresponds to the earth, which, like a cube, is very stable. The next solid is thetetrahedron. It is somewhat like a pyramid. Its base is a triangle. It comes from four vertices and four points. It has four faces, each of which is an equilateral triangle. There is also the octahedron. You get it by taking two pyramids and stacking them against one another, base to base. The icosahedron, which has twenty equilateral triangles as faces, and the dodecahedron, which has twelve pentagonal phases, are more elaborate still. The Timaeus suggests that this regularity in the mathematical order corresponds to the regularity that exists in material things.
Here is a marvelous aspect of Plato's artful writing, though. These things do not line up perfectly. There are these five solids, but there are four elements. There is something going on in the heavens that is less clear. There is this tension between saying, “Yes, there is a mathematical pattern!” and “Though these mathematical patterns are so orderly and compelling, they are only approximative.”
"For Augustine, however, pondering things that separate from the material, as we do in mathematics, is a stage whereby our study ceases to be just an undirected relaxation and really leads us toward our end."
4.
For your fourth book we pass from a philosophical dialogue of Plato to one of Augustine’s. De ordine was one of four dialogues he held the autumn before his baptism. As Augustine specifies in his Retractions, this dialogue aims to show how all good and evil falls within the order of divine providence but, due to the inherent difficulty of the issue, focuses instead on the order of study by which one can advance from the corporeal to the incorporeal. Have you included this work for Augustine’s advocacy of the liberal arts or his cosmology?
For his advocacy of the liberal arts. At an important point in the dialogue, he treats the order of learning explicitly for the first time. He gives his interlocutors a division of the arts. He argues that, once one has studied the verbal arts, the soul begins to yearn for divine things. One needs to retrace the same steps as the mathematical arts. He then discusses the four arts of the quadrivium.
Importantly, this great thinker, a rhetorician by training, whose predominant focus was on the verbal arts, acknowledged that, once we have accomplished these early stages of learning, we want to move on to the highest things, yet need take the time to study the mathematical arts. In answering your first question, I mentioned how he qualifies this. A little later, Augustine acknowledges that it is hard to study all the liberal arts. Maybe you study only the verbal arts. Maybe you study only the mathematical ones. For Augustine, however, pondering things that are separate from the material, as we do in mathematics, is a stage whereby our study ceases to be just an undirected relaxation and really leads us toward our end.
It involves an abstraction from the corporeal that disposes us to contemplate the incorporeal God?
Yes.
5.
Finally, there is Pierre Duhem’s To Save the Phenomena. Duhem was a major theoretical physicist, but also a major scholar of the history of science, a pioneer of the study of medieval science, and a devout Catholic. Fr Stanley Jaki describes this work as a summary of Duhem’s five-volume Systême du monde. It traces the debate from Plato to Galileo over whether mathematics or empirical evidence is the standard of truth in astronomy. How does reading this book help one master or appreciate the quadrivium?
When we were talking about Ptolemy, you asked whether he had been superseded and it was eccentric to put him in here. Eccentric is an apt term here. At heart, Duhem is discussing how multiple accounts of the same apparent motion are possible. The epicyclic account is one. Take something relatively simple such as the apparent motion of the sun. In an epicycle, one circle turns on another. There is also an eccentric account, where there is just one circle, which has been displaced out of the center.
Duhem wrestles with the question of whether these mathematical tools, that we use to account for the things that we see, are actually real or just useful fictions. It isn't so much that something has changed and now, with contemporary science, we are doing something that is totally foreign. Rather, Duhem traces the question and shows how it is rooted in in the science of two millennia ago. He, and especially Stanley Jaki in the preface, help you see how we need to chart a central course, balancing the need for correspondence with sense experience with the utility of mathematical accounts.
