What's a Liberal Art and Where Can I Buy One: The Quadrivium

                                            

The Quadrivium

This is part 2/2. Find Part One here.

Last week(ish) we discussed what a liberal art was and introduced the three liberal arts that make up the Trivium. These three arts, grammar, logic, and rhetoric, are the arts that concern the uniquely human powers of reason and speech. Today we'll look at the remaining four arts that make up a group called the Quadrivium. 

The four arts of the quadrivium are Arithmetic, Geometry, Music, and Astronomy. At first glance these may not seem like they're natural companions, well at the very least music seems to be an outlier, but it makes more sense when we understand that the arts of the quadrivium are the arts that combine human reason with number. These are my weakest subjects. I'm a poor student of them for many reasons, not the least of which are my own moral failings. I bring this up because, while I know what they are, they're a bit more difficult for me to explain than the arts of the trivium. 

The Four Arts

Arithmetic

So, let's get this over with: Arithmetic is the study of quantity (number) itself. It includes concepts as basic as even and odd, more and less, counting. It becomes more complex from there. Basically everything we do with quantity stems from the arithmetical art. Addition. subtraction, multiplication, division, fractions, decimals, variables, all of these and their relations are studied in arithmetic. Arithmetic can be as simple as counting and as complex as calculus. Arithmetic is usually considered to be the first and foundational member of the quadrivium as all the others use number as the foundations for their respective arts. 

Geometry

Next is Geometry. Geometry is the art that studies number in space, not the black void of space, but the

concept of space and spatial relations. This is the math with the shapes. Geometry is pretty abstract. It's interested in understanding figures. What are the different shapes? What are their parts? What rules govern their construction? Geometry was pretty set as a discipline for thousands of years. The Greek thinker Euclid literally wrote the book on the subject. There have been more recent developments in the field that I am absolutely unqualified to comment on given my one and only ever 'C' that I won in high school Geometry. However I do know that Descartes, the Renaissance thinker, wrote enough on Geometry that they named a branch after him (Cartesian Geometry aka analytic geometry), and apparently some mathematicians figured out how to break the rules set down by Euclid, so now non-Euclidian geometry is a thing. If anyone's interested in that.

Music

The third of the arts is Music! Wait, you say, these arts have been all about math. There's been numbers and rulers, what can music have to do with that? The way we think about music nowadays, as being very arty and about self-expression doesn't have a lot to do with how music actually works from a compositional standpoint. Music is the study of how number relates to time. Western music has seven notes in eight octaves. Music seeks to take these notes and, in varying combinations, at varying speeds, for varying lengths, create harmony, or sounds that are beautiful.

Astronomy

The last of these is Astronomy. This art is the study of number in both space and time. It looks at how the stars and planetary bodies move. It considers their relationship to one another and to the Earth.

Astronomy gives us our ability to create human notions of time (days, weeks, months, years). In the ancient and medieval world astronomy was studied for its own sake as much as it was studied for its uses in navigation and agriculture. Really until the Renaissance the realm of the stars and planets was considered a place of perfection. The Earth was fallen and broken, but the realm of the heavens was steady, unchanging, and uncorrupted. A lot of time was spent on astronomy rather than on the other sciences precisely because of this belief. Why would you study the mating habits of tree frogs when you could look at a glorious universe unmarred by sin? Of course with advancements in our ability to actually see the cosmos we've come to realize that some of our assumptions about its unchanging nature and indeed ats basic organization were fundamentally flawed. We've also come to realize that there's a great depth and beauty to be found by examining our own world. However, as humans we're still drawn to the stars.

Astronomy has become a difficult art for liberal arts enthusiasts to deal with. Some think that it should now be excluded. Others think that it should be taught with the other sciences like physics, biology, and chemistry. I think that I fall into the latter category, especially when it comes to physics. I think that quantum physics in particular has shown us more and more that the nature of the universe is more complex and potentially less material than we ever imagined. It may be true that the cosmos are not perfect and unchanging, that they cannot be understood in the mind alone, but as we learn more about just how much we don't know about the nature of reality I can't help but wonder if there was more wisdom than we realize in the old ways of thinking. 

Conclusion

The three arts of the Trivium are inherently linguistic arts. The Quadrivium seems to be their opposite in every way. They're incredibly abstract. They certainly have practical uses and applications, but at their core they exist in an ideal world and lose something when they're translated into material reality. The arts of the Trivium, on the other hand, are inherently concrete arts. Human language is a concrete thing. Grammar has to apply to an actual language for it to mean anything. Rhetoric must make use of the words that exist within a specific language and combine and play with them in ways that still make sense. The Trivium lives and dies in human minds and tongues. The arts in the Quadrivium can live in the mind (including the art of astronomy where men decided what the rules of the universe were in abstract and then tried to make what they saw fit their theory)

However, I think that the Trivium and the Quadrivium aren't really so different. I think at bottom they're both linguistic arts. The Trivium deals with human language. The Quadrivium deals with the language of the universe, the language of God, if you will. These arts seek a truth that is inherently more solid and objective than the arts of the Trivium. The Trivium, when used appropriately, tells the truth of what's in a human mind, a human heart, and this is a great and valuable thing. The arts of the Quadrivium tell the truth of what objective reality is. The Quadrivium take us outside of ourselves and show us something objective and solid. 

There's a lot that can be said about how the truths that the quadrivium reveal are more real when they're abstracted and not when we try to use them in our material reality. I'll just leave you with the thought that the realm of the spirit that can be reached by the human mind at the highest levels is a place of great order and perfection, and by peeking behind the curtain of the material world we can bring the lessons we learn there back to do great and wonderful things, even if they pale in comparison. 

Sources

If you're interested in learning more about the seven Liberal Arts and the Quadrivium in particular I recommend these sources.

• The Marriage of Philology and Mercury by Martianus Capella. This is a fun resource because Capella explains what the arts are in an allegory. The character Philology, or the human intellectual life, marries the god Mercury. The seven Liberal Arts visit her on her wedding day to explain their gifts and to offer themselves to her as her handmaidens.
• Institutions of Divine and Secular Learning and On The Soul by Cassiodorus. This one is nice because he provides pretty small descriptions that are easy to understand.
• Etymologies by Isidore of Seville. This one is longer and more complex for those really interested in diving in. 

What's a Liberal Art and Where Can I Buy One? The Trivum

In a case of potentially confusing terminology, classical education is sometimes called a liberal arts education. This is confusing for a couple of reasons. Reason 1. all classical education is technically a liberal arts education, but not all programs that describe themselves as liberal arts programs are classical. This stems from Reason 2. A lot of people don't actually know what the liberal arts are. There are seven liberal arts, but in this post, I will only go over the basic definition of liberal art as a term, and I'll only go over the first three (the linguistic arts of the Trivium). We'll hit the last four next time.

The Liberal Arts

In one sense the answer to the question 'what is a liberal art?' is a relatively simple one. Liberal stems from a Latin word that means 'Free,' and Art stems from a Latin word that means something closer to the English word for 'skill' or 'skilled work.' So, a liberal art is literally a skill needed by a person to function as a free man in society. Remember from a previous post that formal education used to be reserved for society's elite. So a liberal arts education was one designed for men who were not only not slaves, but who were meant to lead their people. Over time the word 'liberal' began to be interpreted in a less literal sense, and now when people talk about the liberal arts they speak of them as the skills that a human being needs to BE truly free. So, the liberal arts over time have shifted from the skills that it was proper for freemen to learn because they were in the class of freemen, and now they are understood as disciplines that create a spiritual and mental freedom in those who study them.

So, that's what the term liberal art means and how it's generally interpreted. There's a little more to it than that, however. According to a tradition that's been handed down since the Middle Ages there are seven liberal arts divided into two categories. There are the three arts of the Trivium (tri - three, vium - way, path, road) and the four arts of the Quadrivium (quad - four, vium - way, path, road). This is kind of my take on them, but the Trivium and the Quadrivium both represent the two different kinds of language that humans use to understand reality.

The Trivium

The three arts of the Trivium are the skills that have to do with human language and our ability to communicate with one another. These arts are Grammar, Logic, and Rhetoric. 

Grammar

Grammar is the art of understanding how language works on a basic level. What are the parts of human language? How do we combine words to make sentences? What are the different kinds of sentences? How does word choice affect meaning? Can we put words in any order we want or is there a point where certain combinations of words destroy coherent meaning? Vocabulary. As anyone who has attempted to learn a new language knows, in order to communicate effectively a person not only needs to have a sufficient number of words from the different parts of speech (nouns, verbs, adjectives, prepositions, etc.), but needs to know how to put those words together in ways that make sense. The sentence "The black dog ate the delicious meal" does not mean the same thing as "The delicious meal ate the black dog." Grammar tells us why. Grammar also tells us why the sentence "Jack gave me a cake." does mean the same thing as "Me Jack a cake gave." Grammar, then, is the ability to understand what language truly is, its ins and outs, the way it works.

Logic

The next art of the Trivium is Logic. Logic takes language and seeks to make it say things that make

sense. Logic is where language tends to look a lot like math. Logic tells us why statements like "A tiger is a cat, but a tiger is not a cat." are nonsense. Grammatically there's nothing wrong with a sentence like that, but it breaks the logical rule of non-contradiction, which says that a statement cannot be true and false at the same time. Logic is a very rigid art. It has a lot of rules and even formulas where variables show the validity and invalidity of certain statements. For example, a classic logical argument is a modus ponens (method of affirming) which says:

p q

p 

 q

 Or in language, If 'P' is true then 'Q' is also true. 'P' is in fact true, therefore 'Q' is also true. P and Q are variables. I told you it was just like math. So, we could substitute actual phrases in for the variables. For example, "If I see a cat then I will eat ice cream. I see a cat! Therefore, I will eat ice cream." Logic is incredibly dry and if it's possible even less interesting to people than grammar is, so it gets skipped a lot. I don't think it should be skipped, but I'll make an apology for logic another time.  For our purposes here, Logic is the art of right thinking, its the art of sense as opposed to nonsense. 

Rhetoric

On to the final member of the Trivium, the sexy one - Rhetoric! Rhetoric gets a bad rap. I mean, honestly, all of the Trivium does. grammar and logic are considered boring, and rhetoric is associated with politicians and therefore has come to be associated with lies and emptiness. That's not really what rhetoric is meant to be. rhetoric is the art of persuasion, it is the art of using language to convince

people to do, or say, or believe something. It seems obvious that rhetoric can easily be abused. However, some of the oldest defenders of rhetoric (i.e. Aristotle) made sure to insert the caveat that REAL rhetoric was concerned with persuading people to do something good or to believe something true. 

Rhetoric is often considered a sort of capstone art because it is meant to combine everything one has mastered from the art of grammar and logic and use them toward the end of persuasion. The best rhetoricians are the ones who know what they want to say, who know how to say it, can say it in a way that makes good sense, and finally can say it in a way that's interesting and moving. It's the last part, interesting and moving, that's unique to rhetoric and that gives rhetoric its unique power.

Conclusion

The Trivium, then, are the human arts. They originate in our humanity, they are unique to us, and they belong to us. There is nothing in nature that can replicate these. If you want to take a theological view they stem from our participation in the image of God. God is a God of language. He gave us reason, which is unique in the natural world (reason being here identified with the ability to think about things beyond our immediate physical needs and desires, the ability to dive into the mysteries of our existence and purpose), and He gave us the ability to take what's inside our heads in all of its complexity and to place it into the heads of other people. Human language is incredible and our ability to use it is an incredible power. The story of Babel illustrates this. Human beings, using language, made themselves so powerful that God, for the sake of their salvation so that they would not orchestrate their own destruction through their hubris, had to confuse and break human language. 

God speaks to us using grammar, logic, and rhetoric. By placing the story of human salvation into a human language He made it possible, through translation, for anyone to understand deep spiritual mysteries. He made His mind and heart known to us. Likewise, in His image, we use language to bear ourselves to one another. Men cannot be truly free unless they can escape from their own minds and connect with other members of their race. We do this with language, with the three arts of the Trivium. There's no denying that language as a human power is incredibly dangerous and easy to abuse, but when language is oriented toward truth, beauty, and goodness it truly shows the best of us. 

Find Part Two here

Sources

If you're interested in reading more on the Liberal Arts or on the Trivium specifically I recommend the following sources (most of these can be found for free online): 

• Institutes of Oratory by Quintillian. If you want to know not only what these arts are in a lot more detail, at least how they were traditionally considered, but also how they were taught you must read Qunitillian. Quintillian is still used as a model in Neo-Classical schools.
• The Marriage of Philology and Mercury by Martianus Capella. This is a fun resource because Capella explains what the arts are in an allegory. The character Philology, or the human intellectual life, marries the god Mercury. The seven Liberal Arts visit her on her wedding day to explain their gifts and to offer themselves to her as her handmaidens.
• Institutions of Divine and Secular Learning and On The Soul by Cassiodorus. This one is nice because he provides pretty small descriptions that are easy to understand.
• Etymologies by Isidore of Seville. This one is longer and more complex for those really interested in diving in. 

The Quadrivium As Linguistic Arts

My third-grade teacher had to bribe us in order to get us to memorize the multiplication tables. For every table you mastered, you won one part of an ice cream party. The ones got you a bowl, the twos a spoon. If you proved proficient in the threes you won a scoop of cheap vanilla ice cream, the fours got you two, and so on until, once you had demonstrated perfect proficiency of the twelve times, you had amassed a topping laden sundae that was the stuff ten year old's dreams were made of. I coveted that sundae. Unfortunately, my sugar-coated fever dreams were dashed. I believe I earned up to chocolate syrup but was unable to progress satisfactorily past the sixes. If I am being perfectly honest, I still can not. 

As I progressed in my mathematical education, I found myself becoming increasingly bored and frustrated in turns. Long division was difficult, algebra, with its insidious introduction of abstract alphabetical symbols and values on either side of the equal sign, frequently reduced me to tears. I discovered my spatial reasoning issues in geometry, and I can boast that I never passed a single calculus test. 

My father, a nuclear engineer who crafted new equations for his doctoral thesis, would attempt to help me. These tutoring sessions would nearly always end, not in new understanding, but in shouting and tears. The best I could expect from my teachers was enough help to get me to pass classes with a B. It was perhaps inevitable that I would come to the conclusion that I was 'not a math person,' and that 'math was stupid anyway."

Far from being an isolated incident, I find as an educator that many of my students are undergoing similar mathematical experiences. I teach an honors section of eighth-grade students. They're bright children. They are incredibly gifted in nearly every single class, taking to new subjects, information, and skills with relative ease. And yet, mathematically they struggle. One girl came to class with a literal spring in her step. As she flounced through the door she greeted her peers and gaily announced that she and her father had engaged in a shouting match the previous evening as he attempted to help her with her mathematics. The others laughed. Far from being horrified or confused at her struggle, they revealed that they were also part of the culture of mathematical ineptitude. 

Not every student struggles to grasp mathematics, but those who show a natural talent for the subject have become increasingly rare. It's tempting to look at these academic unicorns as opposed to the rest of the unwashed masses and conclude that there are some people who are simply 'mathematically minded.' The best that the rest of us can hope for is to learn our basic sums and rely on calculators to do anything more complicated than single-digit multiplication for us. 

However, if one looks beyond the borders of the U.S. it becomes clear that the issue is not necessarily that there are only a very few men and women who are capable of mastering the mathematical discipline with any sort of competence. Students who live in countries that are not nearly as prosperous as the United States routinely outperform our students in mathematics by an alarming margin. A 2018 PISA study showed that United States (37/78) children test well below children from countries such as China (1/78), Canada (12/78), and even Hungary (34/78) ¹. Perhaps the issue is not that there are 'math people' and everyone else. Perhaps the issue is that American educators have forgotten what math is and other cultures haven't.

I identify myself as part of the American classical education movement. In classical education circles there's a lot of talk about returning to a 'liberal arts' education. What people usually mean when they say this is that children should read old books that comprise a made-up entity called the Western Canon and learn Latin. It is almost entirely centered on a medieval concept called the Trivium. 

Because the pedagogical tools modern classical educators use are unfamiliar to the vast majority of the citizenry, I'll attempt to offer some basic definitions of the relevant terms. When classical educators reference a liberal arts education they are referring to the traditional seven liberal arts classed into two groups - the Trivium and the Quadrivium. The Trivium refers to the three arts Grammar (the art of symbol creation and combination), Logic (the art of reason), and Rhetoric (the art of expressing oneself persuasively). The Quadrivium refers to the four arts of Arithmetic (the art of number theory), Music (the art of number theory applied to time), Geometry (the art of spatial theory), and Astronomy (the art of spatial theory practically applied). These are not definitions that are set in stone. There is a messy and dizzying debate within the classical community about the precise nature of each of these arts, their importance, how they should be understood and taught, if they should be taught, etc. 

The point I'm attempting to make, however, is that classical schools tend to focus on the Trivium - Grammar, Logic, and Rhetoric, placing a larger emphasis on the subjects that seem to apply to these easily - namely Literature, History, Art, and Latin. The Quadrivial arts tend to be underemphasized with mathematics and the sciences treated like embarrassing bastard children. Many classical schools simply seem not to know what to do with them. 

Classical schools may produce better test scores in mathematics than other public educational models, I am not sure. If they do, however, I would not be convinced that this was not a happy accident - that by teaching certain subjects well, the mental acumen needed to have some success in the quadrivial arts bleeds over. 

I have recently experienced a kind of revelation that has made me reevaluate my juvenile attitude toward mathematics. I have long suspected that the mathematical arts are not a waste of time, and may, perhaps, even have some value inherent in themselves. I have also come to believe that the liberal arts, all of the liberal arts, are necessary in order to create men and women who have free minds. I have struggled, however, to understand how math fits into the creation of a free and happy man. The arts of the Trivium seem obvious. Understanding how to speak well and think well can obviously help a person to avoid being taken in by jargon and propaganda. Written and verbal language can also be used to persuade one of the truth, to debate and explore the subjects that define our civilization and to some extent out humanity. They can be used to communicate deep and meaningful truths about the nature of reality and our experience of it. My understanding of the Quadrivium, on the other hand, has been hampered, not only by my deep ignorance of the subjects themselves, but by my inability to understand what they truly are.

I cannot, therefore, credit myself with this insight, if indeed it is any sort of insight at all. If there is any truth in it, it did not come from me. The Biblical picture of reality, given in the creation account, then Genesis 1:3 is the most profound statement in the whole of human history as regards our understanding of the nature of reality: "And God said, Let there be light, and there was light." Creation was accomplished through Language. 

Now, no one should suppose that the language God used in the act of creation was anything like the verbal language we use to speak with one another, to express our thoughts, ideas, and feelings. How did God communicate Being? I do not think that we will ever truly know what exactly constitutes divine language. However, God in His graciousness allows us to understand something of what it must have been like through mathematics.  I do not think that this is a new idea. I am well aware that as far back as Pythagoras number was associated with the divine. Men like Kepler and Newton knew that God spoke about His universe through the language of mathematics.

What I want to offer is this: The classical movement, in its attempts to reunify knowledge, has struggled to articulate a philosophy of education that truly unites learning into a cohesive unit. Their issue is that, for many, they still see the arts of the Trivium and Quadrivium as functionally different rather than facets of one Master Art. 

The Trivium and Quadrivium are both one half of the way human beings are able to understand and interact with Truth. Reality can only be understood by human beings through language, because creation was, at bottom, a linguistic act. It is our view of what language is that is far too narrow. The Trivium and Quadrivium cannot be divided into the linguistic arts and the mathematical arts. Written expression, verbal expression, and mathematical expression are all aspects of one divine Language. Our troubles understanding this stem from our finite natures. We are not God. We are contrained by the fall, by sin, by the inherently limited nature of our own being. Even still, we are inherently linguistic creatures. We can only understand reality through language.

All of the Liberal Arts represent ways in which human beings can master language. The Trivium contains the arts that give humans mastery over moral and relational reason and by extension human speech. Although it seems limited now, it is possible that human speech once had godlike power. In the wake of Babel and the intervening linguistic corruptions of thousands of years, we will never know for certain. However, there is still great power in human speech for good. God used human speech when he sent his prophets to declare His kingdom and to call men to repentance. He demonstrated this power in His own person through the incarnation of Christ. There is also, however, power in the spoken language for great evil. The devil used speech to tempt Eve to sin. History is littered with tyrants and demagogues who seduced whole countries to great evil using spoken language. The arts of the Trivium are truly human arts. They belong to us. They are limited in the ways that we are limited and they are corruptible in the ways that we are corruptible.

The language of the Quadrivium, however, is of a different kind. This mathematical language seems less corrupted and more universal than the linguistic arts of the Trivium. Grammar, Logic, and Rhetoric give us mastery of the language of man. Arithmetic, Geometry, Music, and Astronomy allow us to speak the language of the universe. 

I do not know to what extent this language, in the hands of man, is corrupted and corruptible, but it seems to me mathematical language is not inherent to man. Instead, man must learn to discern and understand mathematical language as if it were a language one had known as a child but had forgotten through years of disuse. 

If, then, this is the case, let the student master the spoken, human word, but do not allow him to neglect the language of the stars. Mathematics is not a mere series of numbers and equations and theorems but is a linguistic expression of the mind of God- a reverberating echo from the beginning of time. The Word crying "Let there be Light."

¹https://factsmaps.com/pisa-2018-worldwide-ranking-average-score-of-mathematics-science-reading/