In recent times, liberal education has usually been identified with the liberal arts, but traditionally they are distinguished. “Liberal education” names the whole procedure of the philosophic life, including the study of wisdom itself; “liberal arts,” on the other hand, properly names seven introductory disciplines which, though intrinsically of lesser philosophic interest, are “certain ways by which the lively soul enters into the secrets of philosophy” (Hugh of St. Victor).

Both the trivium and the quadrivium of the liberal arts are intrinsically ordered toward science and philosophy. The trivium provides the universal instruments of all scientific demonstration. The quadrivium, too, moves in the same direction. Pure mathematics is an art, inasmuch as the subject matter is constructed in the imagination, but it is also a science, insofar as the mental constructs are not arbitrary; rather, they are constructed according to the nature of quantity. For example, the equilateral triangle that is constructed in the first proposition of Euclid’s Elements is discovered, not invented. The mathematical sciences are the clearest and most accessible, and thus supply the logician with the readiest paradigm of scientific demonstration. They prepare the student for more difficult sciences and furnish him with rigorous knowledge that may be used to express the order found in non-mathematical objects, as in astronomy and music.

Philosophy is divided broadly into speculative and practical or moral philosophy. Speculative philosophy is interested in nature and its causes. Moral philosophy, on the other hand, is concerned with right reason as it applies to our active life, that is, as it regulates the life of the passions and appetites, which move us. Astronomy and music take up these broadly divided parts of philosophy in an extrinsic way. They recognize an order in the cosmos and in the movements of men’s feelings, and this order is expressed in the light of mathematical principles. It remains for philosophy to account for these things in terms of the intrinsic natural and moral principles.