Algebra II. Calculus III. When do the numbers stop?

The great variety of orderly relations throughout the universe is reflected in the multitude of mathematical disciplines, each of which grasps some bit of the whole.

But what should we teach, then? How can we find unity in this apparent diversity?

Under the broad heading of mathematics we can consider two distinctions. One distinction is: order considered in itself vs. order considered in relation to material things. A second distinction is: order in things that are discrete vs. order in things that are extended (spatial).

All four disciplines of the quadrivium tell us about an intelligible order, but each corresponds to one of the four possible combinations of categories.

1. Arithmetic considers the order, in itself, of discrete things.
2. Geometry considers the order, in itself, of extended things.
3. Music considers the order of discrete things realized in sound.
4. Astronomy considers the order of extended things realized in the heavens.

You must have an objection to this conveniently tidy tale. Why stop here? Can't other things be put in those places? If I cannot answer this objection I cannot claim that we have a 'whole' in the quadrivium.

To grasp the whole of quadrivial mathematics, we must think about more than mathematics. Why do we teach and learn? Who is learning?

There is far more mathematical order in the world than in the simple list above. The quadrivium is not a whole because it includes all mathematics. It does not. Instead, the quadrivium is a whole, is complete, when we understand it as one part of liberal education.

The whole to consider, then, is not the universe, but the student. Not the whole cosmos, but the whole person.

Some few students will go on to specific studies and careers in which they learn much more mathematics. For the rest, though, the disciplines of the quadrivium can stand as a culminating mathematical study, one in which the student meets the fundamental structure we find in things and learns to name it clearly.

Among our five senses, two, hearing and sight, are particularly directed to the intelligible. In music and astronomy, these senses are united to arithmetical and geometrical knowledge. We can then begin to apprehend that which is unchanging in the sweetness of a melody or the wonder of a sunset, things which move us with their beauty.

Such a liberal education is directed towards the whole.

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