Archimedes encountered solutions to individual problems in the calculus, but the idea that the integral and the differential were a new order of number that could be manipulated like any other number lay outside the boundaries of the Hellenic imagination. The concept of higher-order number separates the mathematics of classical antiquity from modern mathematics beginning with the calculus. Bonaventure in the thirteenth century and embraced by Nicholas of Cusa in the fifteenth, Augustine’s “numbers of judgment” point to the mathematical revolution of Newton and Leibniz in the seventeenth century. Augustine’s assertion is arresting in all three of its parts: first, that neither our sense perception nor even our memory explains how we hear music second, that the faculty by which we judge the numbers (rhythms or harmonies) of music is also a kind of number and third, that this higher-order number comes from God.Ĭhampioned by St. But what sort of number is it? In the sixth book of his De Musica, Augustine asserted the existence of a higher order of number that in some way stands above the senses, the numeri iudiciales or “numbers of judgment” which “come from God” and enable the mind to judge what it perceives and remembers, as well as what it expects. Music employs number both in its harmonic foundation and its metrical presentation in time. Augustine’s fifth-century treatise on music, and a red thread links it to Leibniz’ invention of the calculus in the seventeenth. The deeper affinity between mathematics and music, though, is less consoling and more challenging: The modern concept of a higher-order number begins with St. The eternal relation of math and music has been a perennial question since Plato, from Boethius and Cassiodorus in late antiquity, through Dante’s celestial harmony in Paradiso and Shakespeare’s discussion in The Merchant of Venice.

It is consoling to think that the emotions that music arouses in us have something to do with the makeup of the universe.

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