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Lecture, Dr. Joseph Hattrup: The Order of Astronomy and Music to Wisdom

Lecture, Dr. Joseph Hattrup: The Order of Astronomy and Music to Wisdom

Posted: September 3, 2019


The Order of Astronomy and Music to Wisdom[1]


By Dr. Joseph Hattrup
Thomas Aquinas College, California
Opening Lecture
St. Vincent de Paul Lecture and Concert Series
August 30, 2019


It is our custom at Thomas Aquinas College to begin the lecture series each year with a lecture on some aspect of liberal education. Integral to liberal education are what are commonly referred to as the seven “liberal arts.” The last two of these liberal arts are called astronomy and music. Astronomy and music have been considered part of liberal education since the time of ancient Greece, and they have been considered liberal arts since the Middle Ages. But they are both a little bit odd, and, at best, it is not easy to see at first blush what they have to do with philosophy. So, this is the question I want to consider tonight. In particular, I want to consider the order that these two sciences have to wisdom, the goal of all philosophy.


What is wisdom? We could derive a definition from many sources in our academic program, but the two most obvious places are the beginning of Aristotle’s Metaphysics and the beginning of St. Thomas Aquinas’ Summa Theologiae. Each of these texts defines wisdom as the knowledge of the first, highest, and most universal cause of all things, and of the order of all things to this first cause. The curriculum at Thomas Aquinas College is known for its commitment both to wisdom and to the liberal arts. The reason we study the liberal arts is that they have an intrinsic order to wisdom. Wisdom is the purpose of liberal education, for liberal education is the education of the free man. The free man lives for his own sake. That is, he lives for the sake of his own intrinsic perfection and goodness. But the perfection of man as such is wisdom. Therefore, we study the liberal arts for the sake of wisdom.

Not every study has an intrinsic order to wisdom, though there may be an extrinsic order. This is the case with the servile arts, for example. Other studies may have an intrinsic order to wisdom, but less immediately than others. The liberal arts are carefully chosen, therefore. They are those studies that have an intrinsic and immediate order to wisdom.

Tonight, I want to justify this claim with particular respect to astronomy and music. I will therefore do three things. First, in Part I of the lecture, I will briefly define the liberal arts and explain the principle according to which they have been selected. Second, in Part II, I will show the special place that astronomy and music occupy among the liberal arts. I will show that these two, astronomy and music, have a kinship to each other. They go together. Third, in Part III, I will show the intrinsic order that astronomy and music bear to wisdom.

Part I: The liberal arts.

There are seven liberal arts, divided into the so-called trivium and quadrivium. We will dwell for a moment on these names. Of course, they are Latin. “Trivium” means, literally, “the three-fold way”;[2] or, we could say, “the triple road”. Likewise, “quadrivium” means “the four-fold way”. So, we have two ways, or roads, into something. Into what? In his famous quotation, Hugh of St. Victor says that the liberal arts are the ways the lively mind takes into the secrets of philosophy.[3] What is philosophy? It is wisdom, as Aristotle shows at the beginning of his Metaphysics.[4] So even in the names, “trivium” and “quadrivium”, the liberal arts are conceived according to their order to wisdom. Let us begin by manifesting why the order to wisdom demands that the seven liberal arts be divided into these two roads, or pathways: the trivium and the quadrivium.

The arts of the trivium are concerned with human speech. What is speech? In Aristotle’s work On Interpretation, he defines speech as “a vocal sound that is a symbol of something undergone by the soul.”[5] The chief kind of speech is what we might call the argument. An argument is a composition of statements tending necessarily toward some conclusion.[6] There are three ways in which arguments can be considered.[7] First, one can consider an argument simply insofar as it is ordered to some truth. This is the concern of logic. The question the logician asks is, what things must be true of the premises of an argument if the conclusion that follows from them is to be true?[8] Second, one can consider an argument insofar as it is ordered to some action. This is the persuasive speech, and it is the concern of rhetoric. So, we can account for logic and rhetoric among the liberal arts by noting that sometimes arguments are simply ordered to truth, or knowledge, while other times arguments are ordered to bringing about some human action, especially on the political scene.[9] What about grammar? The concern of grammar is the sentence itself.[10] The sentence is the instrument or tool of argument-making, since all arguments are composed of sentences. Grammar deals with the principles according to which sentences are well-formed. Consequently, grammar is related to logic and rhetoric as the art that provides their instruments.[11] This relationship is similar to that in an example that Aristotle employs at the beginning of his Nicomachean Ethics, which the juniors are just now reading, namely, the relationship of bridle-making to horsemanship.[12] The bridle-maker designs the instrument of the rider, or horseman. Similarly, grammar designs the instrument of the logician and of the rhetorician.

So, we have a basic account of the division of the trivium into grammar, rhetoric, and logic. Now, why are these three arts, devoted to human speech, called liberal arts? St. Thomas explains this name in his commentary on Boethius’ De Trinitate. The text is the reply to the third objection in Question 5, Article 1. Let us begin with the following point: Aristotle distinguishes arts from other kinds of knowledge by the order that arts have to making.[13] Every art is a kind of knowledge. But some kinds of knowledge are simply ordered to truth and nothing further. Other kinds of knowledge are ordered beyond the truth to some human action.[14] But arts are ordered to making, or fashioning. Our previous example of bridle-making is an obvious example of an art. Others are carpentry, medicine, and teaching.

Before going further, let us dwell a bit longer on the notion of art. Aristotle distinguishes between kinds of knowledge. One kind of knowledge he calls speculative.[15] When he calls it “speculative”, however, he does not mean what we generally mean by the word today, namely, “uncertain”, or “hypothetical.” Rather, he means that the purpose of such knowledge is simply to see it, simply to look at it, simply to know its truth. This knowledge is a gazing.[16] This feature, in fact, characterizes the noblest knowledge there is. Any knowledge that is desirable for its own sake, simply to possess the truth, is nobler than a knowledge that is desirable only for the sake of some product, separate from itself, which it generates.[17] This is because we consider anything desirable for itself more worthy of desire than something which is desirable only for something else. So, one kind of knowledge, the highest, is speculative knowledge.

The second kind is practical knowledge. Practical knowledge is for the sake of action. Any reasoning that we undertake in order to determine how we should act in some determinate circumstance is ordered to more than the knowledge: it terminates in the action itself. So, we call it practical knowledge,[18] rather than speculative knowledge.

Art is distinct even from practical knowledge in that it is ordered not merely to doing, but to making, and this requires a different sort of habit of mind. Whereas practical knowledge is ordered, properly speaking, to human acts, art is ordered to working with a material.[19] The artist, properly speaking, is the one who takes some material and generates something with a new form out of that material, such as a statue, a house, or a healthy body.[20] One immediate result of this difference is that practical knowledge is ordered directly to human goodness. The question, how ought I to act right now in this given circumstance, is immediately related to virtue and vice, since the act chosen will be either good or evil; but the question, how can I make this sort of something out of this certain material, a statue out of bronze, for example, is not immediately related to virtue and vice.[21] A man could act superbly in virtue of artistic knowledge and still be acting wickedly, as in the case of a painter or a musician whose work is flawlessly executed, but intended for concupiscence, or in the case of a doctor who uses his knowledge to kill, rather than to heal.

It can be seen from this sketch that speculative knowledge and art are, by definition, opposed to each other. Speculative knowledge terminates, in virtue of what it is, in the mind itself, since it is ordered to nothing beyond the truth which it knows. Art terminates in some material outside of the mind, and the perfection of the art is in that thing.[22] This opposition sets up an interesting difficulty we will have to face later on. For we will want to see how it is possible that the members of the trivium and quadrivium are rightly called both speculative sciences and arts at the same time. This point will have everything to do with that quality by which they are called liberal.

Grammar, rhetoric, and logic are all arts. This is because there is, in each of them, some work or product, an opus, as St. Thomas says.[23] In grammar the work is chiefly the sentence. In rhetoric and logic, it is an argument, or syllogism.

So, supposing we grant that grammar, rhetoric, and logic are all arts, why do we call them “liberal arts”, thus implying a direct order to freedom? St. Thomas gives the answer in the De Trinitate commentary. He says that “they involve not only knowledge but also a work that is directly a product of reason itself.”[24] They are called arts because they are ordered to a definite work, or opus. They are called liberal because their works are immediately of reason itself. That is, their works are not corporeal, not of the body, but of the immaterial mind. St. Thomas goes on to distinguish the liberal arts from the servile arts. The servile arts involve corporeal works, like houses and bridles. Of the servile arts, St. Thomas says, “These latter, then, cannot be called liberal arts because such activity belongs to man on the side of his nature in which he is not free, namely, on the side of his body.”[25]

Now, how are the arts of the quadrivium distinct from those of the trivium? Again, the trivium is concerned with human speech. The quadrivium, by contrast, is composed of mathematical sciences. Here is what St. Thomas says about this distinction: “The Philosopher (Aristotle) also says in the Ethics that the young can know mathematics but not physics, because it requires experience. So we are given to understand that after logic we should learn mathematics, which the quadrivium concerns. These, then, are like paths leading the mind to the other philosophical disciplines.”[26] Note first the order implicit in St. Thomas’s statement. It is natural for the young, or the beginners on the road to wisdom, to begin with the sciences concerned with human speech. This is because progress in the other sciences depends on these. Just as grammar fashions the tools employed by logic, so logic fashions the tools employed by anyone engaging in scientific argument.[27] Consequently, before we can adequately progress in mathematics, we must study logic. But I want to draw your attention to a very interesting feature of the account we have just heard St. Thomas give. If we study logic to progress in mathematics, we study mathematics in order to progress in the science of nature, or physics. This is the relationship we will consider next.

If the quadrivium concerns the mathematical sciences we must begin by seeing how these sciences are divided from one another, for there is not just one science here, but four: geometry, arithmetic, astronomy, and music. Then we must consider the order of these sciences to wisdom. Notice here that the very name “mathematics” means “learning.” One might say that mathematics receives its name from its singular order to speculative knowledge. Again, our attention will be given chiefly to astronomy and music, which are, in this connection, of the greatest interest.

Aristotle, in his Categories, shows that there are two genera of quantity, the subject universally treated by mathematics. These are continuous quantity and discrete quantity.[28] There are two quantitative questions that we ask about things, namely, “how much?” and “how many?”[29] When we ask the question, “how much is it?” we are trying to get to know the dimensions of something. How much wheat is there in storage? One bushel. This measure is of length, breadth, and depth. How many potatoes are left? Three potatoes. This measure is of number. Number is a different genus than length, breadth, and depth, and so the science that treats the former necessarily begins from different principles than that which treats the latter. The science of number is arithmetic. The science of dimension is geometry.

Now, as you know, arithmetic and geometry are the first two arts of the quadrivium. These are followed by music and astronomy. There are two instructive questions we can ask about music and astronomy. First, why are these added to arithmetic and geometry? In what way are arithmetic and geometry insufficient in liberal studies? Second, why are music and astronomy the only sciences to be added? Surely, there is a whole host of mathematical sciences that could be studied. Why are music and astronomy alone, among all these others, considered liberal arts? Both of these questions, namely, why must music and astronomy be added, and why must these alone be added, tend to this question: in what way are music and astronomy perfective, or completing of the liberal arts?

Before we answer this question, let us consider arithmetic and geometry a little more closely. As with the trivium, we must ask concerning these sciences both why they are considered arts, and then why they are considered liberal arts. As we said above, art is knowledge that terminates in making, in some work, or opus. The opus in grammar was the sentence, in logic, the syllogism. As the freshmen will discover later this year, the opus in the case of rhetoric is what Aristotle will call the enthymeme, a particular variety of syllogism. But what are the opera of arithmetic and geometry?

Here it is important to remind ourselves of a basic distinction. We spoke earlier of the distinction between speculative knowledge and art, and we pointed out that they seem to be opposed to each other insofar as speculative knowledge terminates in a good that is in the intellect alone, while art terminates in a good outside the mind, in some exterior material. However, in the present case this opposition does not exist. It is clear that arithmetic and geometry are both speculative sciences. The reason is that they are pursued simply for the sake of truth. They are not pursued for the sake of any action on our part, and they do not terminate in any product generated in exterior material. However, it is also true that there is a making, or production, that is intrinsic to these two sciences. Geometry, for example, assumes the existence of the things that are first in the various genera it considers. In the genus of plane figure, for example, it assumes the existence of the circle, as in Euclid’s third postulate: “to describe a circle with any given center and radius”. But it demonstrates the existence of the others. For example, through the circle the geometrician demonstrates the existence of the equilateral triangle. That is, he shows how the existence of the triangle is derived from the circle as from a first and immediate principle. However, the geometrician does this precisely by constructing the triangle through the description of circles. Consequently, the demonstration has an artistic aspect.

The difference we find between geometry as an art and the other arts we have been discussing, like carpentry, is that the product, or opus, is not fashioned in any exterior matter. Rather it is fashioned in the human soul itself: either in the intellect, or, if not entirely in the intellect, in the imagination of the reasoner. This note brings us back to the claim made by St. Thomas in the De Trinitate commentary: “We may add that among the other sciences these are called arts because they involve not only knowledge but also a work [or opus] that is directly a product of reason itself.”

We may also add that this artistic aspect is unique to the mathematical sciences. Mathematics is both a speculative science and an art. This is not true of either natural philosophy or of metaphysics, the other two divisions of speculative philosophy.[30] This difference is due to the abstract character of mathematics, which considers its object without reference to the material principles in which that object actually exists.[31] Because reason separates mathematical forms from matter in order to consider them, it is able to treat them as though they were products of reason’s own synthetic activity, and this is why, although nothing considered by mathematics is not received by the mind in some way through sensation, it often appears as though the mind has fashioned these objects independently of any influence from the senses.

So, this amounts to a brief argument that it is appropriate to consider geometry and arithmetic as both speculative sciences and arts, indeed, as liberal arts. Now let us consider music and astronomy. Again, I want to consider two things with respect to these sciences: first, how they complete the quadrivium; second, how they bear an intrinsic order to wisdom. I will consider the first question now in Part II of the lecture and the second afterward in Part III.

Part II: Astronomy and Music as Liberal Arts.

As we have done with the other liberal arts, let us begin by defining and distinguishing music and astronomy. At first sight these seem like very different, even randomly selected, sciences. They don’t look like they have anything to do with each other; indeed, one might even suspect them both of being mere hobbies. Most of us treat them that way. But they have a tremendous kinship. Both can be considered at once as the knowledge of the ratios of periodic movement. In his Republic, Plato says that, whereas geometry is the science of planes and of solids by themselves, astronomy is the science of solids in motion.[32] He then goes on to describe the science of music as the antistrophe of astronomy,[33] for, whereas astronomy is the knowledge of motion as perceived by the eyes, music is the knowledge of motion as perceived by the ears.[34]

But what kind of knowledge are we talking about? In both cases, the knowledge resolves to the appreciation of certain ratios that stand as principles to the motions described. This is most obvious in music. If I take two harp strings made of the same material and identical in gauge, but one is twice as long as the other, and I pluck them both, the longer string generates a tone that is precisely one octave lower than the shorter string does. If the longer string is one and a half times longer, the tones will make a perfect fifth; if it is one and a third times longer, a perfect fourth. These relationships can also be described as the ratios 2:1, 3:2, and 4:3. Western music has always centered on these three intervals, for reasons the juniors will be discussing this year. They are foundational to tonal melody. It is remarkable, therefore, that they are generated by the smallest possible whole number ratios.

Is there a relationship here between music and arithmetic? There certainly is, for we are dealing with whole number ratios. But note this further point: the tones themselves that we hear when we pluck the strings are themselves generated by vibrations. The strings as they vibrate create waves in the air that strike our ears and so make the vibrations of the strings audible. But what are these vibrations? They are periodic movements of the strings, back and forth. When the lengths of the strings have the ratio of 2:1, it also happens that the frequencies of the vibrations of the strings are in the same ratio, the longer string having the lower frequency. And so, it makes sense to describe music as the science of audible periodic motion.

Astronomy has a similar character. The sophomores have probably just finished reading Plato’s Timaeus. In that dialogue, Timaeus is inquiring into the principles of the heavenly bodies, that is, the stars and planets. He gives an account of the origin of the various orbits of the planets. There is a very remarkable feature of this account, that you can’t help but notice. You might not notice that you have noticed it; but it has been nagging at you. This remarkable feature is that Timaeus supposes that the planets and their motions are all carefully conceived parts of a larger whole. Most of us don’t think this way nowadays about the planets. We think of them as separate bodies doing their own things and only involved with one another by chance. That there are eight, nine, or ten planets in the solar system is only a matter of chance, not the decision of a deliberate artist. But this is not Timaeus’ view. There is an account of the number of the planets; but much more importantly, there is an account of the relationships between the speeds of the orbits to one another and of the distances of each orbit from the center of all orbital motion. Notice again the presence of ratio. And, typical of Plato, notice also that not just any ratios are used by Timaeus to account for these orbits; it is once again the smallest possible whole number ratios that are used. That is, the ratios that are used are the musical ones: 2:1, 3:2, and 4:3. (Incidentally, these same ratios dominate the comparison of the five perfect solids at the end of Book XIII of Euclid’s Elements. This is no accident, either for Plato or for Euclid.)

Now, what do these observations show? Just that music and astronomy, as Plato conceives them, are mathematical in character, and that they both involve a mathematical account of the periodic motions of bodies, based on the smallest possible whole number ratios. Now let us pause here to distinctly notice two important points we have just made: first, music and astronomy consist in an application of arithmetic and geometry to the actually existing cosmos as experienced by sight and hearing; second, this application is made explicitly with reference to sensible periodic motion. Let us consider these two points.

First, music and astronomy are mathematical sciences, consisting principally in the knowledge of certain fundamental ratios. I think that in many ways this claim is the point of Plato’s Timaeus. What happens to the human soul when it recognizes that the basic motions of its experience submit themselves to a mathematical order that is very simple and so readily intelligible? This is the basic experience that convinces us that science, that knowledge of the things of our experience, is possible. When we further recognize that this order belongs to the whole cosmos at once, and to its parts through it, rather than to its parts in separation from each other and willy-nilly, we begin to grow in confidence that a scientific account of the whole, what both Plato and Aristotle call “The All”,[35] is also possible. It is precisely this sort of knowledge, that is, the knowledge of the universal whole, that has the character of wisdom. We will develop this thought further in Part III.

But before pursuing this thought, let us notice again the second point just made, namely, that music and astronomy are knowledge precisely of motion. This is a point of enormous importance. Returning for a moment to St. Thomas’ De Trinitate commentary, we recall St. Thomas making the following claim: “The Philosopher (Aristotle) also says in the Ethics that the young can know mathematics but not physics, because it requires experience. So we are given to understand that after logic we should learn mathematics, which the quadrivium concerns. These, then, are like paths leading the mind to the other philosophical disciplines.” Notice the implication of these words. The mathematical sciences have a unique role to play in the education of the young, the beginners on the road to wisdom. Because of their abstract character, they are readily learned without much experience of the world. But this is not true of physics, or the science of nature. Physics requires an extensive experience because it is not dealing with an abstract but with a sensible reality. It is not dealing with forms that are separated from the conditions of their actual existence and separately so considered; rather it is dealing with forms that exist in material conditions precisely insofar as they exist in those conditions. On the other hand, it is precisely physical science that begins to take on the character of wisdom, because it is physical science that first attempts to give an account of the whole cosmic reality of the world. Recall the definition of wisdom with which we began: the knowledge of the first cause of all things, and the knowledge of the order of all things to their first cause. So, we can begin to appreciate the sort of knowledge that music and astronomy are reaching for.

Now, before discussing wisdom as such, we want to see a few more details about music and astronomy that will further assist us. For, really, we have not yet adequately distinguished them as liberal arts. Let us begin with music. There are really three ways in which music can be studied intellectually. Intellectual study of music is distinct from moral training. Your parents regulate the music you listen to because it has a powerful and fundamental effect on your moral character, but this is training, not study. When we study music intellectually, we can see it as a fine art, as a part of politics, or as a liberal art, our topic tonight.

To study music as a fine art is to study it with a view to composition, performance, and so on. The fine arts are distinct from the servile arts in that they are ordered principally to delight, rather than the basic uses of survival. For this reason, the fine arts are closer to the liberal arts than are the servile arts; for example, the fine arts are products of leisure, while the servile arts are unleisurely. Nevertheless, the fine arts are not yet liberal. The reason is that liberal studies always have truth as such as their end, not simply pleasure. Truth is the noble good. Pleasure is noble when it is in accord with right desire, that is, when it is guided by truth.[36] Consequently, because they hover in the neighborhood of the beautiful, the fine arts can act effectively in the service of the truth, and therefore of liberal studies. That is why we have concerts at the College as part of the curriculum. But the fine arts are not themselves liberal studies.

To study music as a part of politics is to come to know it just insofar as it tends to result in certain characters and certain actions. Every political community is ruled by a regime, or constitution, and every regime seeks to make its citizens of the sort that will perfect the regime.[37] Again, music of different kinds tends to generate different characters. The reason for this is that moral character consists in a certain formation of the passions of the soul. But music powerfully imitates the passions of the soul, through an intrinsic likeness.[38] Therefore, musical training is a component of the education of citizens employed by any political regime, and the regime must know music to the degree that it sees what sort of music will tend to generate that character that is most consistent with the regime.[39]

Now, what is music as a liberal study? First, a note about the name. The name “music” refers principally to poetic song, the Iliad, for example. This is the gift inspired by the Muses. When Socrates turns to music as a liberal study in Book 7 of the Republic, he calls it “harmonics”. The reason for this is that its principle subject is what the Greeks called “harmonia,”[40] which is the name of the various modes, or musical scales, of which there are seven, that are the matrices of any tonal melody. We are most familiar with two of them, the so-called Ionian and Aeolian modes, which very nearly approximate to what we would call the major and minor keys. And here is, perhaps, the chief thing the Greeks were thinking about when they approached music as a liberal art: the principle work, or opus, of the science of harmonics is the musical scale, and the scale is built out of the very numerical ratios we just discussed, namely, 2:1, 3:2, and 4:3. Timaeus describes the manner in which the scale is so built.[41] By considering the musical scale through the numerical principles that generate it in the first place, we can see the principles of music precisely as a nature, and, therefore, with reference to nothing more than the truth about the cosmos. This is music as a liberal study. So, this is a point about music that will help to move us forward in the quest for wisdom.

Now, how is astronomy a liberal art?  First, let us notice a difference. Unlike the case of music, there is no fine art of astronomy, so far as I know, nor is there a political art of astronomy. The reason is that we can produce the musical scale with our voices and on our instruments, but we cannot produce the celestial motions in this way. However, there is a manner in which we can produce them, and this production is what gives astronomy the character of a liberal art. As with all mathematical sciences, astronomy involves constructions. Just as the geometrician constructs triangles and squares in order to know them, so the astronomer constructs geometrical models of motions. There is no single thing that Ptolemy, for example, does in the Almagest that is more important than the construction of the eccentric and epicyclical circles that form the backbone of the hypotheses he uses to explain what we see in the night sky. These constructions give astronomy the character of an art, and it is clear that this art is liberal, because its goal is nothing more than truth.

This point, by the way, is worth emphasizing here, though it is a little incidental to the lecture: namely, the brilliance of Ptolemy’s use of circles in the Almagest, especially when you get to Books 9 and 12. You will often hear people say that we study Ptolemy at the College, “in spite of the fact that he is wrong.” Well, this makes it sound like we are only reading Ptolemy for his historical value. There is, indeed, much historical value to the reading of Ptolemy; nevertheless, nothing could be further from the truth than this opinion. You will find, if you attend closely to the matter, that Ptolemy’s original demonstrations concerning the organization of circles guide the work of astronomers into the twentieth century. Even authors like Kepler and Newton are still working out the consequences of Ptolemy’s principles. This is a sign of a genius of the first order, one who has discovered fundamental truths about the nature of the cosmos. This comment is just given by way of encouragement. There are things that Ptolemy sees about the orbits of the planets with such sharpness of insight that you will find both Kepler and Newton returning to them, even in the 17th century.

So, these reflections show how we can think of music and astronomy as liberal arts. But how are they perfective of the quadrivium? First, notice that the fact that Ptolemy uses circles to explain the motions of stars and planets also necessitates that the motions explained be periodic, that is, returning back to their beginning. So, what music and astronomy really amount to, as Plato says in the Republic, is mathematics applied to periodic motion. I would like to put forward a thesis, which I will immediately defend in Part III of the lecture, namely, that the purpose of the quadrivium is to introduce the beginner in philosophy, that is, you all, to motion in a universal manner, and, therefore, to the world of natural philosophy, or physics. Neither music nor astronomy is natural philosophy, at least, not when they are treated like liberal arts; but they do provide us with a powerful introduction to natural philosophy, and natural philosophy is the first knowledge that can truly be called wisdom.

Part III: Music, Astronomy, and Wisdom.

We have been working up to the claim that astronomy and music have a unique ability to prepare the mind for the consideration of physical or natural science. So, let us begin this part of the lecture by distinguishing physics from the other sciences. Aristotle describes physics as the science of natural being, that is, of beings whose intrinsic first principle is nature. Because nature, as the sophomores will argue later this semester, is the intrinsic principle of motion and rest within natural beings, physics is also the science of mobile things precisely insofar as they are mobile. Now, in his Metaphysics, Aristotle makes the following remark: “If there is no substance other than those which are formed by nature, natural science will be the first science; but if there is an immovable substance [and therefore not a natural or physical substance], the science of this must be prior and must be first philosophy, and universal in this way, because it is first.”[42] In this and in other texts Aristotle implies that natural philosophy, or the speculative knowledge of nature and natural being, has the character of wisdom, albeit in a position secondary to theology. The reason for this is the universality of the study of nature. When you understand nature in principle, you understand the whole world at once through its most universal causes, and so you are on the doorstep of theology. This is the kind of knowledge proposed in Plato’s Timaeus. But this knowledge is not mathematical in nature, at least, not through itself. It uses mathematics to make its knowledge more accurate, but it is not itself mathematics.

Again, the difference is due to the abstract character of mathematics. In Book 2 of the Physics, Aristotle distinguishes mathematics from physics along these lines: “The mathematician, then, is also concerned with these things, [i.e., the shapes of the moon, sun, and earth] but not as each is a limit of natural body, nor does he consider their accidents insofar as they occur in such beings. Whence, he also separates [his objects]. For they are separable from motion in thought. And this makes no difference, nor, separating, do they become false.”[43] Mathematics considers features like bulk and figure, or shape. Shape belongs to physical bodies, all of which are susceptible of motion. But shape can be considered without any attention to motion. Physics, or natural science, considers bodies precisely insofar as they are subject to motion, because nature is precisely the intrinsic source of motion in bodies, as I have just indicated. But mathematics abstracts from motion in its consideration of shape and extension.

Aristotle goes on in the same text to compare optics, astronomy, and harmonics, two of which are our liberal arts, to physics, or natural science. Concerning them he says, “The more natural of the mathematical sciences, like optics and harmonics and astronomy, can also make this clear. For these are in a way the converse of geometry. For geometry looks into natural lines, but not as natural, but optics looks into mathematical lines, but not as mathematical, but as natural.”[44] St. Thomas in commenting on this text describes astronomy and harmonics as “middle sciences.” The character of a middle science is that it is concerned with a physical subject matter, the path of a planet, or the vibration of a string, for example, but it demonstrates truths about this physical subject matter through abstract, mathematical middle terms.[45] Indeed, this is just how Ptolemy characterizes astronomy at the beginning of the Almagest.[46]

Now this distinction is of great importance for our purposes. What St. Thomas and Aristotle are saying is that there are sciences that stand in between the strictly mathematical sciences, geometry and arithmetic, and natural science, or natural philosophy. These “middle sciences” are sciences such as optics,[47] harmonics, and astronomy. The middle sciences are about natural things, and so they begin to penetrate to the truth about the natural world. But they use mathematical concepts to do so: concepts that are derived from arithmetic and geometry. Notice, by the way, an interesting fact about St. Thomas’s characterization of optics, harmonics, and astronomy in this text from the Physics: he says that optics takes its middle terms from geometry; harmonics takes its middle terms from arithmetic; and astronomy takes its middle terms from both arithmetic and geometry.[48] The sophomores and juniors will very soon have some experience of this division.

Now why should the fact that harmonics and astronomy take their middle terms from arithmetic and geometry be so interesting? Let’s go back to a text from St. Thomas’s De Trinitate commentary: “The Philosopher (Aristotle) also says in the Ethics that the young can know mathematics but not physics, because it requires experience. So we are given to understand that after logic we should learn mathematics, which the quadrivium concerns. These, then, are like paths leading the mind to the other philosophical disciplines.” We can begin to see now what St. Thomas has in mind, I think. Harmonics and astronomy are not present among the liberal arts simply as some extra math. Rather, harmonics and astronomy are those mathematical sciences that begin to penetrate the natural world of sense experience. Consequently, they prepare us for discourse about the natural world: the first grade, if you will, of wisdom.

But there is another crucial point to make about these two arts, and that is that they are both concerned with the natural world as a whole. You don’t really appreciate what natural philosophy is until you understand that it is concerned with the whole material cosmos as a whole. What the natural scientist hopes to accomplish is to see all material being through its first and original causes. The sophomores will see this while studying Aristotle’s Physics this semester. This is an ambitious and radical undertaking that depends on generation after generation of effort. The reason why harmonics and astronomy are placed among the liberal arts of the quadrivium is that they, too, seek to know the world as a whole, not simply one of its parts, as other mathematical sciences might do.

There is a final point we must add to this consideration. As we just noted a moment ago, Aristotle thinks of natural science as a knowledge that might have been accounted wisdom, had it been the case that bodies were the only things that exist. But this is not true. There is a spiritual reality above the corporeal reality available to our senses. Not only is there a spiritual reality above the corporeal one, but it is infinitely more vast and beautiful and intelligible than the corporeal one is. This is the reality that the theologian aspires to know, in the manner in which such realities can be known by those of us still in the pilgrim way. Now, natural science has two fundamental works to perform. The first is to demonstrate the truth about material being through its own proper and intrinsic causes. The second is to demonstrate the existence and certain aspects of the first and universal cause of material being; that is, God most holy, the knowledge of whom is most perfectly called “Wisdom”.[49] Different sorts of demonstration must be marshalled to accomplish these two works.[50] The freshmen will define these sorts of demonstration next semester in the philosophy tutorial, and the juniors will recall them this semester as they approach St. Thomas’s five ways of demonstrating God’s existence in the theology tutorial. So, natural science is not complete until it has laid the ground for theology by revealing God as the source of natural being. This is natural science’s noblest work. You can divide up Aristotle’s book called the Physics according to these two demonstrations,[51] as the seniors will discover this semester.

Now, harmonics and astronomy also introduce us to this theological aspect of natural science. The reason is that they introduce us to discourse about motion that is already pure and elevated. The motions that astronomy considers are not messy and chaotic, but obviously orderly and therefore intelligible. But not only that, they are also the motions that are the true and universal causes of all other motions of our experience. Aristotle and Plato both clearly think of the motions of the stars as true and original causes of the generation of plants and animals on the earth. You will see this in Plato’s Timaeus[52] and in Aristotle’s Physics[53] and Metaphysics.[54] In fact, the seniors will end the year with this consideration in Book 12 of the Metaphysics. So, we can reasonably say that harmonics and astronomy are good introductions to the study of natural science both because they consider motion universally, as natural science does, but also because they consider it as something leading the mind swiftly to the anticipation of divine things, which is natural science’s ultimate goal.

I want to end with a point that I take to be a crucial consideration for liberal studies, especially at a school like Thomas Aquinas College. I have argued tonight that the reason harmonics and astronomy have been placed among the liberal arts is that they uniquely prepare the mind for natural science, where natural science is a true wisdom. It is a true wisdom, first because it is utterly universal and primal knowledge about the material world, and second because it prepares the mind through demonstration for theology, which is the noblest knowledge of the highest things. But what does it mean to really have knowledge like harmonics and astronomy? These sciences, as we have described them, are not mere knacks or hobbies. They are not a merely private interest. The knowledge of them is the beginning of philosophy. It therefore implies conversion of the soul, and, consequently, it is not a common, but a very rare possession, as with anything that requires conversion. This is another point that Plato makes in the Republic, indeed, immediately before he introduces the liberal arts. You will also find it at the heart of St. Augustine’s Confessions. Most men live as slaves to the senses and their pleasures, as practical, if not philosophical, materialists. As Aristotle says at the beginning of the Metaphysics, the human race is in many ways in bondage, due to the influence of the body over the mind.[55] A result of the way of life that is based on corporeal pleasure is that a man so enslaved does not even see that the kind of knowledge we have been discussing tonight is possible. Even scientists usually don’t see it. We’re talking about a knowledge that prepares us for the awakening to spiritual reality. So, what is the study of harmonics and astronomy really about? It is about a spiritual turning away from commitment to the senses and the sensible as the most important things in our lives toward the intellect and the intelligible, which is the true common good of the human race, and therefore that which will make us free and happy.


[1] In this lecture, I employ the following translations: Ross’s Metaphysics and Nicomachean Ethics; Coughlin’s Physics; Barnes’s Posterior Analytics; Lord’s Politics; and Maurer’s De Trinitate. The edition of C. S. Lewis’s Discarded Image is the 1967 Cambridge paperback edition.

[2] Cf. C. S. Lewis, The Discarded Image, c. 7, p. 186.

[3] Eo quod his quasi quibusdam viis vivax animus ad secreta philosophiae introeat. Hugh of St. Victor, Didascalion 3.3; quoted by St. Thomas Aquinas, De Trinitate 5. 1. ad 3.

[4] Aristotle, Metaphysics 1.1-2, esp. 982b18-19.

[5] Esti men oun ta en te phone ton en te psyche pathematon symbola. Aristotle, De Interpretatione 1.1 (16a3-4).

[6] Syllogismos de esti logos en ho tethenton tinon heteron ti ton keimenon ex anangkes symbainei to tauta einai: “A syllogism is speech in which one thing having been posited something different from the thing laid down follows of necessity from its being.” Aristotle, Prior Analytics 1.1 (24b18-20).

[7] Cf. J. F. Nieto, “Grammar as a Liberal Art”, #’s 6-9.

[8] Cf. Aristotle, Posterior Analytics 1. 4-6. See, for example, 1. 6 (74b5-13): “Now, if demonstrative understanding depends on necessary principles (for what one understands cannot be otherwise), and what belongs to the objects in themselves is necessary (for in the one case it belongs in what they are; and in the other they belong in what they are to what is predicated of them, one of which opposites necessarily belongs), it is evident that demonstrative deduction will depend on things of this sort; for everything belongs either in this way or accidentally, and what is accidental is not necessary.”

[9] Cf., for example, Aristotle, Rhetoric 1.1 (1354b23-8).

[10] Cf. J. F. Nieto, “Grammar as a Liberal Art”, # 7. Note, however, that this claim is true only in principle; grammar is, in fact, concerned with the full length and breadth of the art of composition. Cf. C. S. Lewis, The Discarded Image, c. 7, pp. 186-8; also, Nieto #5. Both authors point out that that prosody, for example, is a grammatical concern.

[11] “The grammarian, however, does not consider speech precisely as it attains any of these ends. Rather, like many ministerial arts, it considers the making of the instrument as such. A lower art commissioned by a higher art to make its instrument knows the order of this instrument to that end, although it does not know the proper causes of that end.” Nieto, #5.

[12] Aristotle, Nicomachean Ethics, 1.1-2.

[13] Ibid. 6.4 (1140a1-5)

[14] See Nicomachean Ethics 6.1 (1139a4-15) and 6.2 (1139a21-32).

[15] “It is right also that philosophy should be called knowledge of the truth. For the end of theoretical knowledge is truth, while that of practical knowledge is action (for even if they consider how things are, practical men do not study the eternal, but what is relative and in the present).” Aristotle, Metaphysics 2.1 (993b19-22).

[16] Pantes anthropoi tou eidenai orengontai phusei: “All men by nature desire to know.” Aristotle, Metaphysics 1.1 (980a22).

[17] “But where such arts fall under a single capacity… in all of these the ends of the master arts are to be preferred to all the subordinate ends; for it is for the sake of the former that the latter are pursued.” Aristotle, Nicomachean Ethics, 1.1 (1094a10-16).

[18] From the Greek “praxis”, comparable to the Latin “actio”. Cf. St. Thomas Aquinas, De Trinitate, Q.5, A.1, c.

[19] “If, however, art imitates nature, and it is of the same science to know the species and the material up to a point, as it is for the doctor to know health and bile and phlegm, in which health is, and, so too, it is for the house-builder to know the species of house and the material, that it is bricks and timber, and so in other arts, so too, it would be for the student of nature to know both natures.” Aristotle, Physics 2.2 (194a22-7). (Coughlin translation)

[20] “Of things that come to be, some come to be by nature, some by art, some spontaneously. Now everything that comes to be comes to be by the agency of something and from something and comes to be something… all things produced either by nature or by art have matter; for each of them is capable both of being and of not being, and this capacity is the matter in each.” Aristotle, Metaphysics 7.7 (1032a13-22).

                There is a further qualification: in generation through art, the principle of generation is in the artist, not in the thing coming to be, whereas in natural generation, the principle of generation is intrinsic to the thing coming to be: “in general, both that from which they are produced is nature, and the type according to which they are produced is nature (1032a22-3),” but, “from art proceed the things of which the form is in the soul of the artist (1032b1).”

[21] “Again, the case of the arts and that of the virtues are not similar; for the products of the arts have their goodness in themselves, so that it is enough that they should have a certain character, but if the acts that are in accordance with the virtues have themselves a certain character it does not follow that they are done justly or temperately.” Aristotle, Nicomachean Ethics, 2.4 (1105a26-30).

[22] Ibid.

[23] Vel ideo hae inter ceteras scientias artes dicuntur, quia non solum habent cognitionem, sed opus aliquod, quod est immediate ipsius rationis. St. Thomas Aquinas, De Trinitate, Q.5, A.1, ad 3.

[24] Ibid.

[25] Compare this statement to that of Aristotle in Metaphysics 1.2 (982b29-32): “Hence the possession of it [wisdom] might be justly regarded as beyond human power; for in many ways human nature is in bondage, so that according to Simonides ‘God alone can have this privilege’, and it is unfitting that man should not be content to seek the knowledge that is suited to him.”

[26] St. Thomas Aquinas, De Trinitate, Q.5, A.1, ad 3.

[27] “Now reason is able to direct not only the acts of inferior faculties, but also its own acts. For the capacity to reflect upon itself is proper to the intellectual power; the intellect understands itself and, similarly, reason can reason about itself. Now, if by reasoning about the acts of the hand, we discovered the art of building, and this art enables us to build easily and in an orderly way, then, for the same reason, we need an art to direct the acts of reason, so that in these acts also we may proceed in an orderly way, easily, and without error. This art is logic, the science of reason.” St. Thomas Aquinas, Commentary on Aristotle’s Posterior Analytics, proemium.

[28] Aristotle, Categories, c.6 (4b20).

[29] Cf. Aristotle, Metaphysics 5.13 (1020a8-14).

[30] “Other sciences (such as divine and natural science) either do not involve a work produced but only knowledge, and so we cannot call them arts, because, as the Metaphysics says, art is “productive reason”; or they involve some bodily activity, as in the case of medicine, alchemy, and other sciences of this kind.” St. Thomas Aquinas, De Trinitate Q.5, A.1, ad 3.

[31] Cf. Aristotle, Physics 2.2 (193b31-5) and Metaphysics 6.1 (1026a7-10). See also St. Thomas Aquinas, De Trinitate 5.1.c: “On the other hand, there are some things that, although dependent upon matter for their being, do not depend upon it for their being understood, because sensible matter is not included in their definitions. This is the case with lines and numbers – the kind of objects with which mathematics deals.”

[32] Plato, Republic, Book 7 (528d).

[33] Ibid., 530d.

[34] Cf. Plato, Republic, Book 7 (530d) and Timaeus (47a-e).

[35] Variously referred to either as to holon or as to pan, as in Metaphysics 12.10, 1075a11 and 1076a1, respectively.

[36] Cf. Aristotle, Nicomachean Ethics 6.2 (1139a21-5).

[37] “Similarly, although citizens are dissimilar, preservation of the community is their task, and the regime is this community; hence the virtue of the citizen must necessarily be with a view to the regime.” Aristotle, Politics 3.4 (1276b28-30).

[38] “For in rhythms and tunes there are likenesses particularly close to the genuine natures of anger and gentleness, and further of courage and moderation and of all the things opposite to these and of the other things pertaining to character. This is clear from the facts: we are altered in soul when we listen to such things.” Aristotle, Politics 8.5 (1340a19-23). See also Marcus Berquist, Learning and Discipleship, p. 239.

[39] “It is evident from these things, then, that music can render the character of the soul of a certain quality. If it is capable of doing this, clearly it must be employed and the young must be educated in it.” Ibid., 1340b10-14.

[40] The word harmonia originally signifies a joint, and then a covenant. It is then extended to signify a musical scale, in which several notes are “hinged” together into melodies around one key note. Note how easily this term might be transferred, therefore, from a musical scale to a system of planets, or even to the soul itself, as Simmias does in the Phaedo.

[41] Plato, Timaeus, 35b-36b.

[42] Aristotle, Metaphysics 6.1 (1026a27-30).

[43] Aristotle, Physics 2.2 (193b31-5).

[44] Ibid., 194a7-12.

[45] “Those sciences are called intermediate sciences which take principles abstracted by the purely mathematical sciences and apply them to sensible matter. For example, perspective applies to the visual line those things which are demonstrated by geometry about the abstracted line; and harmony, that is music, applies to sound those things which arithmetic considers about the proportions of numbers; and astronomy applies the consideration of geometry and arithmetic to the heavens and its parts.” St. Thomas Aquinas, Commentary on Aristotle’s Physics, Book 2, Lectio 3 #164.

[46] Ptolemy, Almagest, 1.1.

[47] Geometrical Optics, at least as Euclid understood it, seems to have principally to do with questions such as, “How are the relative sizes, positions, and shapes of bodies affected by the distance between the eye and them?” Perhaps one reason why it is not included among the liberal arts is that there is no occasion in answering these questions to consider periodic motion as such.

[48] St. Thomas Aquinas, Commentary on Aristotle’s Physics, Book 2, Lectio 3 #164.

[49] St. Thomas Aquinas, Summa Theologiae, I.1.7.c.

[50] Ibid., I.2.2.c.

[51] Books 3-6, and 7-8, respectively.

[52] Plato, Timaeus, 41a-d.

[53] Aristotle, Physics, 4.14 (223b21-3) and 8.7 (260a20-25).

[54] Aristotle, Metaphysics 12.6 (1072a9-17).

[55] Aristotle, Metaphysics 1.2 (982b25-30).

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