1. The One
a) Formation of the theory. The metaphysics of the one
has a twofold source. The first is Parmenides, with his keen
sense for the unity of being. Being is, and it is one, so much
so that it admits of no change and no diversity: thus the
Parmenidean idea of it. The other source is the Pythagorean
philosophy, specifically its speculations on the role of number
in the constitution of material realities. Most important
were its views on the role of the numerical unit, principle of
number. Plato draws on both traditions, which also provide
the setting for Aristotle's theory of transcendental one.
In elaborating his theory Aristotle was above all intent on
securing the distinction between numerical and transcendental
unity, a distinction which, though more or less
inherited, he nevertheless saw the necessity of establishing
more firmly. In addition, he was at pains to show that
transcendental one is reducible to being, of which, however,
he clearly saw it was but a property in the sense spoken of
above, and not the essence (as the Pythagoreans had it).
Thus, where the Platonico-Pythagorean school clung to the
idea that number is the essence of things, and plied their
minds (and imaginations) in support of it, Aristotle's more
perceptive view found being prior to one, a priority he
labored successfully to bring home and which, in effect, left
the whole number-essence idea shorn of foundation.
St. Thomas, in a quite deliberate manner, also begins by
exposing the confusion of numerical and transcendental
unity. Addressing himself to the point he says, among other
things:
Some philosophers failed to distinguish between unity which is convertible with being, and unity which is the principle of number, and thought that in neither sense does unity add anything to substance, and that in either sense it denotes the substance of a thing. From this it followed that number which is composed of units is the substance of all things: this was the opinion of Pythagoras and Plato.
Others who failed to distinguish between unity that is convertible with being and unity that is the principle of number held the contrary opinion, namely that in either sense unity adds a certain accidental being to substance, and that in consequence all number is an accident pertaining to the genus of quantity. This was the opinion of Avicenna, and apparently all the teachers of old followed him; for they did not understand by one and many anything else but something pertaining to discrete quantity. . . . The above opinions, then, were based on the supposition that the one which is convertible with being is the same with that which is the principle of number, and that there is no plurality but number that is a species of quantity. Now this is clearly false.12
The confusion and the consequent error of which St. Thomas here speaks, stem from the failure to grasp the true nature of metaphysical unity, which consists in the absence of division, as well as from not having seen that there is a division which transcends the order of quantity; to this division, then, is opposed a unity which likewise transcends that order.
b) Transcendental unity. For Aristotle and St. Thomas transcendental unity denotes the nondivision of being, in other words, the negation of its division. The steps by which the mind comes to the notion of this unity are outlined by St. Thomas as follows:
First understood is being, and then nonbeing, and then division, and then the kind of unity which is the privation of division, and then plurality, whose concept includes the notion of division, just as the concept of unity includes the notion of nondivision. 14
As this passage indicates, the concept of "one" designates none other than being, conceived however as undivided, ens indivisum. And how is the mind led to conceive of being as undivided? Through its activity of negation, which in this case is twofold: negation of being (nonbeing) and negation of division (one). Thus, what one adds to being is but a thing of reason, moreover a purely negative thing, a privation; yet not a privation in the strict sense, which refers to the absence of a property or quality that the subject ought to have - as the absence of sight in the case of blindness. The absence of division in being is obviously not of this kind; indeed, it were an absurdity to say that being ought to have the division which its unity prevents it from having. If one, then, adds nothing real to being, it follows that they are really identical, hence also logically convertible. This does not mean, however, that the two concepts (of being and of one) are the same, but that the reality to which they point is the same.
c) Modes of unity. Needless to say, the concept of one must be, as is that of being, analogous. Since, moreover, its analogy necessarily parallels that of being, there will be as many modes of unity as there are of being. St. Thomas, even as Aristotle before him, made a noticeable effort to reduce the complexity of them to some order." His first distinction is that of essential and accidental unity - unum per se, unum per accidens. Though five modes of per se unity are advanced, primarily this unity refers to that of substantial or essential being, as opposed to accidental - a man is one in substance, though many in accident." Accidental unity results from the union of accidents, or of accident with substance: musician-mathematician-man constitutes an accidental unity.
Essential unity, moreover, may be real or logical. If real, it follows the distinction of the predicaments, where we note in particular the following: unity of substance (identity), unity of quantity (equality), unity of quality (similarity or likeness).
d) Measure: property of the unit." Transcendentally speaking, unity is always defined by the absence of division; this is its essential or most formal nature. The same is true of the numerical unit, a mode of unity pertaining to the predicament of quantity; it, too, is most properly defined by its undividedness. However, with respect to number, which proceeds from it, the numerical unit has this special property, that it is the measure of number, measure being taken for what makes a thing known. How, for example, do I know how much a certain number is, say the number ten? Only by comparing it (measuring it) against the unit. That is why I say it numbers ten, meaning ten units. Thus, the number to is not understood without reference to the unit that measures it. Nevertheless, the most proper constitutive of the numerical unit is not that it measures number but that, like everything one, it is undivided. The absence of division, that, as we have said, is the first essential of whatever unity.
Furthermore, the property of measure, though primarily vested in the numerical unit, is found proportionally in other modes of the unit as well. This is readily seen in anything that involves continuous quantity: length, motion, time. For each of these there is a measure thanks to which my knowledge of them takes shape - so many yards, so many seconds, and I know at once how much is meant. But the notion of measure is not limited to the quantitative order; analogically it can be applied to other predicaments. And not to be overlooked is its relevance in the order of knowledge. In a sense, science (and knowledge generally) does measure reality; but more fundamentally reality is the measure, for in its capacity as object it measures the faculties of knowledge. All in all, the notion of measure has many important applications; while derived from the relation of number to the unit, its service to the mind goes far beyond that.
c) The many: opposition to the one.18 Just as unity follows on the notion of nondivision, so plurality or multiplicity follows on the notion of division. The multiple is simply divided being. Between one and many (or divided being) there is, as mentioned earlier, an opposition of privation. The "one" implies privation or absence of division, the "many" privation of nondivision. Hence, to every mode of unity corresponds a mode of multiplicity. There is, however, an important distinction to be kept in mind concerning multiplicity itself; namely between numerical multiplicity, which is number and therefore limited to the predicament of quantity, and transcendental multiplicity, which pertains to every mode of being as divided. Nor, in its more general sense, should transcendental multitude (multiplicity) be identified solely with multitude as it applies to separate substances (angels), which constitute but one kind, albeit pre-eminent, of transcendental multitude.. Finally, we take note of a possible difficulty. One, we have said, is defined as absence or privation (in the large sense) of division, hence of multiplicity. It would seem, then, that the many (the multiple) is prior to one; if so, one could hardly be the measure or, in any way, the principle of the many. There is no doubt (in reply) that negation of division is the very essence of unity; however, the division which is negated does not yet imply the formal or express recognition of multitude as such. This recognition can come only after the mind has adverted to the unity of each member of the group. Consequently, the true sequence in the formation of our first notions is as described by St. Thomas, to wit:
First apprehended by our intellect is being, and then division, and next unity, which is the privation of division, and finally multitude, which is a composite of units. For even though things which are divided are many, they do not have the formal note of a many until the fact of being one is attributed to each of the particular things concerned.19