The Eternal Essence of Number

The eternal essence of number is the most providential principle of the universe, etc.

The great Syrianus noted down how the Pythagoreans philosophized about numbers, which I wrote in my Theoretic Arithmetic.

The Pythagoreans turned from the vulgar paths and delivered their philosophy in secret to those who were worthy to receive it. They expressed it through mathematical names.

Forms and numbers are things which are the first separated from impartible union.

• The natures above the forms are also above separation.[103]

They obscurely signified the all-perfect multitude of forms through the duad.

But they said that the first formal principles as the monad and duad are not being numbers.

also by the first triad and tetrad, as being the first numbers, the one being odd, and the other even, from which by addition the decad is generated; for the sum of 1, 2, 3, and 4, is ten.

But after numbers, in secondary and multifarious lives, introducing geometrical prior to physical magnitudes.

These also they referred to numbers, as to formal causes and the principles of these;

The monad was the indivisible point and the first cause.

The duad was the first line or interval.

The triad had more intervals and lines.

The tetrad was solid.

Aristotle says that they called the first length as the duad. It is not simply length, but the first length, in order that by this they might signify cause.

In a similar way, they called the first width as the triad. The first depth was the tetrad.

They also referred to formal principles all psychical knowledge.

And intellectual knowledge indeed, as being contracted according to impartible union, they referred to the monad;

But scientific knowledge, as being evolved, and as proceeding from cause to the thing caused, yet through the inerratic, and always through the same things, they referred to the duad.

The triad was opinion, because its power is not always directed to the same thing, but at one time inclines to the true, and at another to the false.

Tetrad was the sense, because it has an apprehension of bodies.

In the duad, there is one interval from one monad to the other. But in the triad, there are two intervals from any one monad to the rest.

In the tetrad there are three.

They referred, therefore, to principles every thing knowable, viz. beings, and the gnostic powers of these.

But they divided beings not according to breadth, but according to depth; into intelligibles, objects of science, objects of opinion, and sensibles.

In a similar manner, also, they divided knowledge into intellect, science, opinion, and sense.

Plato, in the Timaeus, calls the extremity of the intelligible triad, or animal itself, is assumed from the division of the objects of knowledge, manifesting the intelligible order, in which forms themselves, viz. the first forms and the principles of these, are contained, viz. the idea of the one itself, of the first length, which is the duad itself, and also the ideas of the first breadth and the first depth; (for in common the term first is adapted to all of them), viz. to the triad itself, and the tetrad itself.

Again, the Pythagoreans and Plato did not denominate idea from one thing, and ideal number from another.

They assert that all things are similar to number. Thus, number, and especially every ideal number, was denominated on account of its paradigmatic peculiarity.

If any one, however, wishes to apprehend this from the appellation itself, it is easy to infer that idea was so called, from rendering as it were its participants similar to itself, and imparting to them form, order, beauty, and unity; and this in consequence of always preserving the same form, expanding its own power to the infinity of particulars, and investing with the same species its eternal participants.

Number also, since it imparts proportion and elegant arrangement to all things, was allotted this appellation. For the ancients, says Syrianus,[104] call to adapt or compose αρσαι arsai, whence is derived αριθμος arithmos number. Hence αναρσιον anarsion among the Greeks signifies incomposite.

Hence too, those Grecian sayings, you will adapt the balance, they placed number together with them, and also number and friendship. From all which number was called by the Greeks arithmos, as that which measures and orderly arranges all things, and unites them in amicable league.

Farther still, some of the Pythagoreans discoursed about inseparable numbers alone, i. e. numbers which are inseparable from mundane natures, but others about such as have a subsistence separate from the universe, in which as paradigms they saw those numbers are contained, which are perfected by nature.

But others, making a distinction between the two, unfolded their doctrine in a more clear and perfect manner.

If it be requisite, however, to speak concerning the difference of these monads, and their privation of difference, we must say that the monads which subsist in quantity, are by no means to be extended to essential numbers; but when we call essential numbers monads, we must assert that all of them mutually differ from each other by difference itself, and that they possess a privation of difference from sameness.

Those which are in the same order, are contained through mutual comparison, in sameness rather than in difference, but that those which are in different orders are conversant with much diversity, through the dominion of difference.

The Pythagoreans asserted that nature produces sensibles by numbers. But then these numbers were not mathematical but physical.

They spoke symbolically, it is not improbable that they demonstrated every property of sensibles by mathematical names.

However, says Syrianus, to ascribe to them a knowledge of sensible numbers alone, is not only ridiculous, but highly impious. For they received indeed, from the theology of Orpheus, the principles of intelligible and 311 intellectual numbers, they assigned them an abundant progression, and extended their dominion as far as to sensibles themselves.”

Again, their conceptions about mathematical and physical number, were as follow:

According to Aristotle’s doctrine, one thing corresponds to matter, and another to form, in any number, as for instance the pentad, its five monads, and in short its quantity, and the number which is the subject of participation, are derived from the duad itself.

But its form, i. e. the pentad itself, is from the monad; for every form is a monad, and unites its subject quantity.

The pentad itself is a monad. It proceeds from the principal monad and forms its subject quantity, which is itself formless, and connects it to its own form.

For there are two principles of mathematical numbers in our souls= the monad, which comprehends in itself all the forms of numbers, and corresponds to the monad in intellectual natures; and the duad, which is a certain generative principle of infinite power, and which on this account, as being the image of the never-failing and intelligible duad, is called indefinite.

While this proceeds to all things, it is not deserted in its course by the monad, but that which proceeds from the monad continually distinguishes and forms boundless quantity, gives a specific distinction to all its orderly progressions, and incessantly adorns them with forms.

In mundane natures, there is neither anything formless, nor any vacuum among the species of things. Likewise in mathematical number, neither is any quantity left innumerable; for thus the forming power of the monad would be vanquished by the indefinite duad, nor does any medium intervene between the consequent numbers, and the well-disposed energy of the monad.

The pentad does not consist of:

• substance and accident, as a white man
• genus and difference, as man of animal and biped
• five monads mutually touching each other, like a bundle of wood
• things mingled, like a drink made from wine and honey
• things sustaining position, as stones by their position complete the house
• things numerable, for these are nothing else than particulars

But it does not follow that numbers themselves, because they consist of indivisible monads, have nothing else besides monads, (for the multitude of points in continued quantity is an indivisible multitude, yet it is not on this account that there is a completion of something else from the points themselves).

But this takes place because there is something in them which corresponds to matter, and something which corresponds to form. Lastly, when we unite the triad with the tetrad, we say that we make seven.

The assertion, however, is not true. This is because monads conjoined with monads, produce indeed the subject of the number 7, but nothing more.

Who then imparts the heptadic form to these monads? Who is it also that gives the form of a bed to a certain number of pieces of wood?

Shall we not say that the soul of the carpenter, from the art which he possesses, fashions the wood, so as to receive the form of a bed, and that the numerative soul, from possessing in herself a monad which has the relation of a principle, gives form and subsistence to all numbers?

But in this only consists the difference, that the carpenter’s art is not naturally inherent in us, and requires manual operation, because it is conversant with sensible matter; but the numerative art is naturally present with us, and is therefore possessed by all men, and has an intellectual matter which it instantaneously invests with form. And this is that which deceives the multitude, who think that the heptad is nothing besides seven monads.

For the imagination of the vulgar, unless it first sees a thing unadorned, afterwards the supervening energy of the adorner, and lastly, above all the thing itself, perfect and formed, cannot be persuaded that it has two natures, one formless, the other formal, and still further, that which beyond these imparts form; but asserts, that the subject is one, and without generation.

Hence, perhaps, the ancient theologists and Plato ascribed temporal generations to things without generation, and to things which are perpetually adorned, and regularly disposed, privation of order and ornament, the erroneous and the boundless, that they might lead men to the knowledge of a formal and effective cause.

It is, therefore, by no means wonderful, that though seven sensible monads are never without the heptad, these should be distinguished by science, and that the former should have the relation of a subject, and be analogous to matter, but the latter should correspond to species and form.

“Again, as when water is changed into air, the water does not become air, or the subject of air, but that which was the subject of water becomes the subject of air, so when one number unites itself with another, as for instance the triad with the duad, the species or forms of the two numbers are not mingled, except in their immaterial reasons (or productive principles), in which at the same time that they are separate, they are not impeded from being united, but the quantities of the two numbers which are placed together, become the subject of the pentad.

The triad, therefore, is one, and also the tetrad, even in mathematical numbers= for though in the ennead or number nine, you may conceive a first, second, and third triad, yet you see one thing thrice assumed; and in short, in the ennead there is nothing but the form of the ennead in the quantity of nine monads. But if you mentally separate its subject, (for form is impartible) you will immediately invest it with forms corresponding 315 to its division; for our soul cannot endure to see that which is formless, unadorned, especially as she possesses the power of investing it with ornament.

“Since also separate numbers possess a demiurgic or fabricative power, which mathematical numbers imitate, the sensible world likewise contains images of those numbers by which it is adorned; so that all things are in all, but in an appropriate manner in each.

The sensible world, therefore, subsists from immaterial and energetic reasons, and from more ancient causes.

But those who do not admit that nature herself is full of productive powers, lest they should be obliged to double things themselves, these wonder how from things void of magnitude and gravity, magnitude and gravity are composed; though they are never composed from things of this kind which are void of gravity and magnitude, as from parts.

But magnitude is generated from essentially impartible elements; since form and matter are the elements of bodies; and still much more is it generated from those truer causes which are considered in demiurgic reasons and forms. Is it not therefore necessary that all dimensions, and all moving masses, must from these receive their generation? For either bodies are unbegotten, like incorporeal natures; or of things with interval, things without interval are the causes; of partibles impartibles; and of sensibles and contraries, things insensible and void of contact= and we must assent to those who assert that things possessing magnitude are thus generated from impartibles.