The Great Conversation:
By Matt McKeown
Deceitfully simple on its surface, the idea of quantity reaches into some surprising places in the Great Conversation.
The third section of the Classic Learning Test is titled “Quantitative Reasoning”—a name that has doubtless puzzled many pupils and even teachers. It’s mostly testing math (plus a bit of logic); why not just call it “Mathematics“?
Quantity, or magnitude, is actually a much broader topic than mathematics, and in some ways a more philosophically difficult one. It includes mathematics: numbers are one kind of quantity. However, as Pythagoras learned to his immense grief, not all quantities can be measured in terms of strict numerical relations; irrational numbers are, precisely, magnitudes whose proportions cannot be (in the etymological sense of the word) enumerated. Moreover, while quantity and quality—the answers to the questions “How much?” and “What kind?”—are typically treated as distinct kinds of thing, this isn’t always the case. Certain thresholds of size inspire human beings to treat the objects on either side of those thresholds differently, whether because all or some of those thresholds actually have profound differences underlying them, or simply because human beings are easily impressed by size. This is reflected in everything from place value in mathematics (every time we hit a power of ten as we count up, the way we write numbers changes), to geography (once we get smaller than Australia, we no longer speak of continents but of islands), to chemistry (a cup of wine with a drop of water put in it is still wine, but a cup of wine with nine cups of water put in it is water).
One important aspect of quantities is the idea that they take up space, which is itself a topic of the Great Conversation. Where numbers are discrete (you have either one number or some other number), magnitudes can be continuous; that is, the one consists of a finite number of indivisible parts, whereas the other is in theory infinitely divisible. This distinction underlies the challenges of calculus, and is closely related to Euclid‘s famous definition of a point as “that which has no part.” A point has zero magnitude, because magnitude describes the distance between two points; you haven’t “reached” either kind of quantity, numbers or magnitudes, until you have more than nothing. At least, so it would seem: treating zero as a quantity, rather than just the absence of any quantity, does net us some weird and fascinating results.
This isn’t to say quantities are the same thing as objects, however. Both “take up space”: a two square foot box takes up two cubic feet, which is an amount of space that existed before the box did and will go on existing if the box is destroyed; but the two cubic feet, unlike the box, don’t prevent other things from existing in the same space they do. Indeed, while the space is perfectly happy to just sit there, we can only measure the exact quantity by means of objects—a box, a ruler, what have you.
Speaking of Pythagoras, the traditional study of the seven liberal arts was defined partly in terms of understanding quantities. The quadrivium (the four subjects studied by all educated people after mastering the trivium) consisted in arithmetic, geometry, music, and astronomy. Insofar as mathematics are one branch of the study of quantities, arithmetic and geometry fit into the idea fairly clearly; however, the relation of music and astronomy to the other half of the quadrivium seems a little more esoteric to us today. Proclus, a Neoplatonist philosopher from fifth-century Byzantium, wrote:
The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with number, the other half with magnitude; and each of these they posited as twofold. A number can be considered … by itself, or in relation to another number, and magnitudes as either stationary or in motion. Arithmetic studies numbers as such; music studies the relations between numbers; geometry studies magnitudes at rest; and astronomy studies magnitudes in motion.
The basis for this conception of music was Pythagoras’ discovery that the structure of an instrument showed precise mathematical relations to what notes it produced—for instance, on a lyre, a string that is one-half the length of another string will produce a note one octave higher when struck. (Proclus’ description of astronomy also still works today with surprisingly little modification; we would call it physics rather than astronomy, but obviously astrophysics is one of its branches.) Small wonder, perhaps, that the ancients and medievals assumed that the planets in their serenely regular, proportioned movements through the heavens literally produced music.
Archimedes, Sphere and Cylinder
Euclid, The Elements
Nicomachus of Gerasa, Introduction to Arithmetic
Johannes Kepler, The Harmony of the World
Blaise Pascal, New Experiments With the Vacuum
Matt McKeown is a staff writer for CLT and a proud uncle of seven nephews. He lives in Baltimore.
If you liked this piece, take a look at our other entries in this series on the Great Conversation, from the idea of democracy to varying theories of punishment to beliefs about the soul; you might also be interested in this essay about how to incorporate poetry into our grade school curricula.