DISQUISITION 1.
Quantity is the object of mathematical science. No one denies this. But no mathematician claims this to be complete or adequate.
Proclus states âčin book 1, chapter 12 of his commentary on Euclidâș,
The Pythagoreans, therefore, thought that the whole mathematical science should receive a fourfold distribution, attributing one of its parts to the how-many, but the other to the how-much; and they assigned to each of these parts a twofold division. For they said, that discrete quantity, or the how-many, either subsists by itself, or must be considered with relation to some other; but that continued quantity, or the how-much, is either stable or in motion. Hence they affirmed, that arithmetic contemplates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immoveable; but spherics [astronomy] contemplates continued quantity as moving from itself. . . .34
And so, according to this, the object of mathematics is supposed to be not only quantity but also number, in either an abstract or a concrete sense, and so forth. Then âčin chapter 13âș Proclus states,35
Again, some think (among whom is Geminus) that the mathematical science is to be divided in a different manner from the preceding. And they consider that one of its parts is conversant with intelligibles only, but the other with sensibles, upon which it borders; denominating as intelligibles whatever inspections the soul rouses into energy by herself, when separating herself from material forms. And of that which is conversant with intelligibles they establish two, by far the first and most principal parts, arithmetic and geometry: but of that which unfolds its office and employment in sensibles, they appoint six parts, mechanics, astrology, optics, geodĂŠsia, canonics, and logistics, or the art of reckoning.36
Then in âčchapter 3 ofâș the same work, Proclus writes that mathematicians must think aboutâ
all considerations respecting proportions, compositions, divisions, conversions, and alternate changes: also the speculation of every kind of reasons . . . together with the common and universal considerations respecting equal and unequal, not as conversant in figures, or numbers, or motions, but so far as each of these possesses a common nature essentially, and affords a more simple knowledge of itself. But beauty and order are also common to all the mathematical disciplines, together with a passage from things more known, to such as are sought for.37
From all these words it is sufficiently evident that the object of mathematics is not only quantity, which is limited to line, surface, and volume. Rather the object is much broader.
DISQUISITION 2.
Certainly, all that can be said in many ways. In one sense the object of mathematics, quantity, is the extension of a whole, of an entity, and not really distinct from the whole. Mathematicians have little concern for this. In another sense, quantity extends per se to be almost a thing itself, pertaining to a class of things.38
And although quantity may be a characteristic of a thing, nevertheless, of those characteristics a thing can have, it is noble to the highest degree. It approaches being a thing that exists of itself. It can be considered and used, like a thing that exists. It falls under no other subject. It is ageless, unchanging, and certain. Nothing stands in opposition to it. It yields to no difficulties of philosophy. It deals in no tricks. It is a certain succinct exhibition of divine power and immense wisdom. It is always finite actually, yet truly infinite potentially. Whether you increase by adding, or you diminish by dividing, you reach a certain numerical value. However, through adding or dividing an infinite multitude of values potentially can be reached. And mathematicians have admired this greatly and have honored this, and the ancients arranged everything on account of this.
DISQUISITION 3.
Quantity can be considered in three ways. Either there can be a whole that consists of discrete parts where each part can stand on its own, or there can be a whole that can be continuously divided into parts, and this in turn either according to a path or succession [like the division of time] or not [like the division of space].39
Mathematicians give over the first way [of discrete parts, of arithmetic] to the theologians and physicists, who merely touch on the parts of things and frequently make suppositionsâlike axiom 8, book 1 of Euclid.40 The mathematicians generally weigh quantity in the other (continuous) way, concerning the measurement of or the ability to measure extension, or shapes, or the growth or division of this or that body, potential, quality, or virtue. All these things barely harmonize with that radical and in a certain way potential quantity,41 strange to the ancient mathematicians, so the teachers of this discipline have very little to do with that.
DISQUISITION 4.
In metaphysics one abstracts from particulars to generals [in pursuit of the supreme feat of unifying all classes of reality together in relation to each other, and thus perfecting knowledge].42 In physics the abstraction is from particular relations to general relations in material thingsâbetween the material that comprises things and the structures that govern that material, and within the structures themselves (considering structure clearly). In mathematics the abstraction is separated from the material things and considers the quantitative relations found in things. The metaphysicist considers being as a thing itself; the physicist considers relations in material things; the mathematician, in considering immaterial relationships, clearly walks the middle path between the other two, being closer to the material than the metaphysicist, yet further than the physicist. Thus Proclus âčin chapter 10 of book 1 of his work on Euclidâș writes,
After the same manner, the mathematical science is indeed the second from the first of all sciences, and, with reference to it, imperfect: but it is, nevertheless, a science, not as being free from supposition, but as knowing the peculiar reasons resident in the soul, and as bringing the causes of conclusions, and containing the reason of such things as are subject to its knowledge. And thus much for the opinion of Plato respecting mathematics.43
A little above this he writes,
We must not say, therefore, that Plato expels mathematical knowledge from the number of the sciences, but that he asserts it to be the second from that one science, which possesses the supreme feat of all: nor must we affirm, that he accuses it as ignorant of its own principles, but that receiving these from the master science dialectic, and possessing them without any demonstration, it demonstrates from these its consequent propositions.44
Therefore metaphysics has being itself as its object; mathematics has as its object what is possible; physics, what actually exists.
DISQUISITION 5.
Mathematics demonstrates her conclusions scientifically, through axioms, definitions, postulates, and suppositions. From this it is clear that mathematics should truly be called a science. If anyone would demand that a science treat physical matters, he must also exclude metaphysics from the ranks of the sciences, and even more so logicâand yet these are not excluded from the sciences.
DISQUISITION 6.
That stuff of mathematics that deals with establishing the higher and more universal principles, such as what treats the mathematical elements, especially emulates metaphysical abstraction. Euclid, in his books on the elements âč(see Proclus book 2, chapter 2)âș, thoroughly describes lines, angles, shapes, sizes, and their limits, divisions, ratios, mathematical relations, equalities, applications, inequalities of greater and less, and the nature of numbers. He is not concerned with a triangle made from a physical element, such as earth or air or fire or aether, but with the triangle in general, in the abstract sense, however that may beânot some sort of utopian triangle, as some people incorrectly think, but the triangle as a quantity that has existed since the beginning of time.
DISQUISITION 7.
Other mathematical sciences are less universal and deal (indirectly, not directly) with particular objects and material things; with measuring their size, shape, number, and so forth. Thus the accountant applies numbers to money; the musician applies them to sound; the architect applies them to symmetry, the configurations of buildings, and so on; and the astronomer applies the ratios of circles and spheres to celestial objects. So the astronomerâs principle objective, then, is quantity, as it pertains to the attributes of a celestial body or to learning about a celestial body.
DISQUISITION 8.
It is commonly supposed that any sort of speculation concerning things in the heavens is astronomy. However, speculation that seeks to divine or judge the influence of heavenly bodies on earthly events, and to gain insight into future events based on the positions of the stars and planets, is astrology.
Accepted astronomy is more deliberate. It is that which studies absolute and inherent qualities of the heavensânumber, shape, position, motion, time of occurrence, time of duration, qualities of light such as color or brilliance, and so forth. It is the most noble and most ancient science, as old as humankind itself, almost an offshoot of the human soul âč(see Proclus book 1)âș,45 and always held in the greatest esteem by all. It is cultivated by kings,46 the foundation of chronology, the basis of sundials and of the computation of the liturgical calendar. It records and preserves celestial phenomena. It is the one friend with whom the heavens share their secrets. Elegant geometry and subtle arithmetic give it wings.
Its paths become known to those who faithfully and carefully, through long and repeated experience, come to know its phenomena. Fine craftsmanship sustains their hands and strengthens their arms. Keen o...