In his article, "True Education Liberates" (January 2008), Dr. Lasseter’s explanation of the quadrivium was murky. For example, Dr. Lasseter leads the reader to believe that the art of harmonics teaches aesthetic balance. Aesthetic balance is good indeed, but the topic isn’t part of harmonics, as historically understood.
The arts of the quadrivium (arithmetic, geometry, astronomy, and harmonics) are best explained in terms of the dimensions with which they deal. Dimensions are parameters required to adequately describe a thing for a particular purpose. For example, if our purpose is to lift a block, four dimensions adequately describe it: height, width, length, and mass. Now, the art of arithmetic teaches numeracy apart from any dimension. In other words, two and two are four regardless of whether we are discussing meters, liters, seconds, or volts. The art of geometry introduces the most tangible dimension of length and its derivatives, including area and volume. In geometry, one learns how to apply arithmetic to the physical world in a limited way: Units of length, area, and volume are understood to follow the rules of arithmetic, and some concepts of arithmetic, such as the Pythagorean Theorem, are taught most effectively within the context of geometry. Also, because the physical context is limited, geometry is an ideal forum for teaching rigorous deductive reasoning. However, because the art of geometry is limited to the base dimension of length, things are not taught as responding to cause in a predictable way. Such teaching is encountered in the art of astronomy, wherein one applies the rules of arithmetic to cause–effect relationships involving the dimensions of force, length, mass, and time. As with geometry, some concepts of arithmetic, such as differentiation, are taught most effectively within the context of astronomy. Finally, the art of harmonics specially emphasizes the dimension of frequency, which is the inverse of time. This is because many physical phenomena, such as sound, are best described in terms of frequency. (A common unit of frequency is cycles per second or Hertz.) Again, some concepts of arithmetic, such as the imaginary number, are taught most effectively within the context of harmonics. At most universities, the arts of the quadrivium are covered adequately at the undergraduate level within the following courses: three semesters of calculus, one semester of differential equations, one semester of linear algebra, and three semesters of physics.
I also question Dr. Lasseter’s portrayal of the quadrivium as progressively more and more abstract. I find it rather sacramental that force, matter, and time all manifest the fundamental truths of arithmetic. Moreover, within astronomy, we learn that force causes mass to accelerate, but what causes force? And what causes that which causes force? In this way, the orderly behavior of matter inevitably leads one to the idea of first cause. Indeed, God’s invisible qualities are clearly seen and understood from what has been made, so that men are without excuse (Rom. 1:20).
Rollin Lasseter replies: What an encouraging response—indeed the universe’s rational order is sacramental, an "outward and visible sign of an inward and spiritual grace." We believe in a rational God, a God who limits himself to his own order and plan. All Catholic education should take this thought to heart and direct students toward that final goal, our first and our final "cause."
But the writer may be misreading two points of my article: Most importantly, she misses the point of the sequence of the liberal arts (though of course that sequence has been debated since Boethius), that the arts are incremental and increasingly complex, from concretes to abstracts, percepts to concepts, and lead the soul and mind to the consideration of philosophical and metaphysical truth. I hope that was the point of what I was saying about educational order.
To miss that the quadrivium is incrementally abstract and thus complex is to miss the "art" of the "liberal" thing. One is freed, liberated, by the seven arts—from literalism, concretism, and sensory percepts. The art, not the craft, of number is the gradual discovery of the complexity of reality, and naming of its movements and workings. Thus the first scientists, not yet convinced of the reductionism of the later Enlightenment, could say with Galileo, "the Universe is writ in number."
The letter writer also understands number in the Enlightenment fashion prevalent for the last 300 years, the approach and understanding that has crippled both education in numeracy, and perhaps also science, for those long centuries. She refers to the arts of the quadrivium as best explained "in terms of the dimensions with which they deal." And of harmonics as specially emphasizing "the dimension of frequency." Is her word "dimensions," or the concept of "frequency," not another synonym for measure, and thus counting? That is what the arts of numeracy are designed to overcome, number as mere counting.
Counting depends on number, but number exists outside of counting. What number is, is still a matter of debate, but that it is, and is not just count, is what medieval arithmetic assumed and what many branches of modern mathematics explore today.
There’s some fuzziness here between two ideas: one that numbers have existence (constructed or Platonic) independently of the idea of counting, and second that there exist systems of number for which counting is not even possible. It’s this second idea that topology explores in its study of "continuous functions."
Actually, topology is just one of several disciplines within modern mathematics that explores the structure of "numbers," in the absence of countability: Others include number theory, real analysis, logic, set theory, lattice theory, and category theory. Abstract algebra also drops by to visit on occasion.
The idea of number sets too large to count (which implies that some infinitely large sets are larger than others) is perhaps the most forceful demonstration of the independence of numbers from the concept of counting, but it’s a relatively new one. It was introduced to mathematics by Cantor’s work in the late 19th century, and was unknown to the medievals, whose study would mainly have been limited to some number theory and geometry.
That harmonia has been discussed as the relations of sounds, is the classic example of proportional relationships, the true subject of the art of music, or harmonia. Historical opinion has focused on sounds as exemplary of proportion and relationship, as medieval thought focused on the stars as exemplary of motion and change. Harmony is the final stage in the preparation of the mind and soul to accept philosophy and then theology. Only the grammar of these two disciplines, philosophy and theology, can be learned before that moment of awakening that is harmonia, or "aesthetic balance."
Thus, discussions of force that might explain what force may be, or why it is at all, are the prerogative of philosophy and its metaphysics, not of physics, the scientific discipline. And teaching the facts of observable force is part of the grammar of the discipline. Discussions of the nature of force, without that preparation in harmonia, is necessarily difficult, if not harmful.
Modern subjects do cover a great deal of what the liberal arts contain, but they do not treat of the inner order and sequence of learning that the seven indicate as the way of the soul. That may be why the Enlightenment dismissed them in favor of subjects and individual disciplines—the soul being not a subject of "rational" exploration.
Every discipline has its seven, even its numeracy. That education would improve if this were recognized and curricula structured accordingly is my contention. And the vision of sacramentality of the Creation would then be the final goal of education and the salvation of the soul in God’s world.
Thus our educational programs should take into account the natural sequence of the human brain as an organ of the soul, soul-journeying, and approach all disciplines as spiritual exercise as well as incremental accumulation of data.
Thanks to Ms. Hay for insisting that I clarify my stance on the quadrivium.