Chapter 10

Sixth special law: the term of total, complex thought can never be indefinite

1372. Finite is that than which something greater can be thought.

1373. Infinite , absolutely speaking, is that than which nothing greater can be thought.

1374. Theinfinite must not be confused with theindefinite .(209) That which is indefinite can always receive a further increase; it has no fixed measure because it is considered susceptible of a continuously increasing series. Hence that which is indefinite does not express an ens but an abstract idea, for example, the generic idea of number . This idea corresponds to all numbers which, no matter how large, can always be increased by a unit. Clearly, an ens can never be indefinite , because what is abstract (formed by abstraction strictly understood) is not an ens but, as we saw, an object of abstractive reflection, something seen by the spirit. This reflection limits our attention to a quality of an ens and always presupposes in the mind information about the ens from which the abstraction is made and in which what has been abstracted is seen.

1375. I have been accused of accepting a universal, indeterminate and abstract being (ideal being) as the first object of intuition, but I have explained my opinion in many places. For example, in A New Essay I said the following about universality and indetermination:

Again it is said that I attribute abstraction to the idea of being. But in A New Essay I explained as follows:

But I think it a waste of time to go on recalling what is already in print. I would simply refer those who honour me with their objections to many other places in my works. I am convinced that the public and I would best be served if critics were more attentive in their severe judgment. Let me add a few new considerations, or considerations presented in a new way.

1376. Is it possible for us to think of the abstract idea of colour without knowing some particular colour, or to think abstractly of sound without ever having known a particular sound, and so on for all other sensible things? I do not think so. Note, I do not induce this from experience alone, but from the intimate nature of colour, sound, taste, etc. In fact, abstract colour, abstract sound or abstract taste simply mean that which is common to particular sensations of colour, sound, taste, etc. Now, that which is common, purely common, to several things and not proper to each, cannot be thought without reference in some way to the particular things in which it resides.

Is this also true of being in general? - At first sight, it seems so, because being in general is extremely common to everything that is particular and real. But closer examination shows that this is not the case. Being in general is not purely common in such a way that it excludes what is proper . Rather, it includes what is proper in a common way. Ideal being is that which is realised not only in the substance of things but in accidents as well; not only in what is generic and abstract-specific to things, but what is full-specific, that is, proper to them. Consequently ideal being extends to the whole of an ens and to all that is in it (although not in the same way); it is not purely a common element of the ens to the exclusion of what is proper to it. Ideal being has an entirely different nature from abstractions. These express only what is generic or abstract-specific to an ens, and exclude differences. Abstractions cannot exist entirely on their own, contemplated by thought, without their seeking some support in perceptions or in the full species which perceptions leave in the spirit; abstractions alone are not ideas of entia. On the contrary, the idea of an ens has eminently and essentially the characteristic by which it reveals an ens together with all that it must have in itself to be an ens, although a part of this 'all' which the ens must have is only virtually contained in it.

This first difference gives rise to a second difference which shows the supreme diversity between abstractions strictly speaking andideal being in general . Abstractions express something about an ens, something which does not have and cannot have its own act of existence. In fact no abstraction, considered in itself, could supply an artist with the model for a statue or with the sketch for a painting. The act of being is outside what is abstract, or certainly made impossible by it. No one can ever conceive any act of being proper to an abstract colour or sound, or even to an abstract substance (excluding the accidents). On the contrary, the idea of being is precisely that which manifests every act of being; its object lacks nothing for its intuition by thought. As I have noted, thought neither determines anything special in the idea of being, nor excludes anything; on the contrary, it presupposes and demands something special, and expects to find it whenever opportune.

The characteristics of ideal being and of abstractions are not therefore the same, but rather total opposites. Ideal being, precisely as the idea of an ens and conceivable by itself, has everything to constitute an ens. Abstractions on the other hand lack and even exclude many things necessary for the ens to which they relate; as such, they cannot be an object of complex but only of partial thought, of abstracting reflection.

1377. This enables us see what is true and what is false in the teaching of Stewart and other nominalists. They maintain that abstractions are only names which the mind uses to pass freely from one particular idea to another. As an example they point to the use which algebraists make of the letters of the alphabet for their calculations. But these philosophers are mistaken:

1.They do not see that an idea, although revealing an ens with all its conditions and accidental qualities, is universal , and can be called particular only when considered in the perception joined to it. This condition is extrinsic to the idea and relative to the spirit that joins it to the perception. Although an idea is naturally universal , it is not naturally abstract , because it can manifest everything that can occur in an ens. We can think of ideal being, that is, a non-abstract ideal being , without having to think of a subsistent being. We do not need signs to think of a non-abstract being, that is, a full idea which must be either given to our spirit by nature or extracted by the spirit from perception. Names are by no means necessary for this.(212)

1378.2.Abstract ideas cannot be thought by the mind if full-ideas , to which abstract ideas are related, are totally lacking to the mind. However, it is sufficient for full-ideas to be present in the mind without the spirit's giving them any attention. In fact, as we saw, abstraction is simply concentration and limitation of the mind's attention on some quality present in the full-idea, to the exclusion of everything else in it. The psychological fact that ideas, or part of them, can exist in the mind without the mind's attention is certain and of supreme importance. Such ideas are continuously intuited but without our advertence, or without advertence to one rather than another. We can therefore pass freely and with varying ease from an abstract idea to awareness of the full ideas to which it relates(213) - this is the truth seen or glimpsed by our nominalists. However, when they said that an abstraction was nothing in the mind, but purely a sign outside the mind, they made a mistaken induction from the fact that an abstraction can be thought in the mind without the full-idea (which they confused with a particular idea). This mistake was also the result of the following.

1379.3.To ask 'What is necessary for an idea to be thought or thinkable?' is one question; to ask 'What is necessary for a person to form an idea or be able to form and think it?' is another. An abstraction is thinkable when the mind has the full-idea to which the abstraction relates and from which it is taken. For the spirit to move to this act, it must be propelled by an object, term or motive, because its activity is always aroused by the term. But an abstraction, purely as such, does not exist and therefore cannot draw the spirit to itself. On the other hand, if the abstraction is joined to a sensible sign, it can stimulate and activate the mind's attention towards itself. I have already used this fact to prove the usefulness of language, or better, of signs for forming abstractions. This usefulness consists solely in presenting to the spirit a stimulus and term which moves it to concentrate and fix its attention in the way I have explained at greater length in A New Essay , and which I will again discuss further on.(214) Nominalists erred because this fact too escaped their observation. They moved from the usefulness of language to the formation of abstractions, and concluded that abstractions were nothing in themselves and therefore could not be formed or thought without the signs of language.

1380.4.Finally the example they used to confirm their teaching (the use made by the algebraists of the letters of the alphabet), far from proving their teaching, proves its contrary. Indeed the meaning given to the letters of the alphabet by the algebraist is one thing; the truth he wishes to establish with their use is another. Algebraic symbols certainly indicate abstract quantities (a discrete quantity, even when determinate, remains an abstraction). However, the algebraist does not use them simply to indicate these quantities, but to discover their relationships.

What in fact is his intention when he writes a + b , and d - c ? In the first case he wants to express the relationship of addition between any two quantities (abstract and indeterminate) indicated by the two letters a and b . In the second case he wants to express the relationship of subtraction between any two quantities indicated by the two letters d and c . When he makes an equation between the two functions of a + b and d -c {(a + b ) = (d -c )}, he finds that a = d - (c + b ), that is, that the value of a equals the value of d less the sum of c and b . When performing this task, his mind directed its attention to the relationship of equality between the two functions, and the result of his attention was to join them with the sign of equality. He then directed his attention to the result, which was a discovery of the value of a relative to the other three letters. If the algebraist made his calculation by directing his attention to the relationship, and after noting the relationship joined the letters with various signs, it is clear that before he put the signs on paper, his mind thought of the relationship and the results, and that he wrote down the signs expressing the relationships only after he had thought of the latter. In other words, he had thought of the relationships without their signs; these followed as a result of the relationships already thought by his mind. But relationships are themselves abstractions, far superior to the simple quantities involved.

The use of algebraic signs therefore clearly demonstrates that abstractions are thinkable per se without the need of signs. This use would in fact be impossible if a mind did not think of abstractions without signs. Their use implies that the mind already possesses abstractions at a very high level, but does not in any way explain how the mind formed them, and much less does it answer the question 'What is necessary for abstractions to be thinkable?' Signs simply help the mind to keep its attention on a series of relationships, which because of its length and multiplicity would easily be lost.

Notes

(209) Ancient languages did not make this distinction, which is so necessary in philosophy. The Latin word infinitum means both that than which we cannot think anything greater, and that than which we can always think something greater without determining the amount. The Greek (*) means both what is limited and what is not indefinite or indeterminate, although nothing greater than it can be thought. Hence Parmenides applies this epithet to ens , although he calls it (*) and acknowledges nothing outside it. By calling it (*), he clearly intended to exclude indetermination. Melissus, on the other hand, a philosopher of the same school, calls it (*), which also has a double meaning: it expresses both that which has limits, but indeterminate limits (the indefinite), and that which naturally has no limits because nothing limits or circumscribes it in such a way that some addition is possible. This explains why these two philosophers are in apparent contradiction. Aristotle himself, possibly following his usual practice of understanding preceding philosophers literally, takes Melissus' infinite in the sense of indefinite , and therefore commends Parmenides for having called the all, that is, ens, finite in his condemnation of Melissus (Physic ., 3, 1).

(210) NE, vol. 2, fn. 139.

(211) NE, vol. 3, 1455.

(212) NE, vol. 2, 514-427.

(213) Note that when we perceive with our different sensory organs a corporeal ens and acquire the full idea of it, we do not need to clothe the corporeal ens with all its sensible qualities but simply with those found in each perception of this ens. A perception is always limited to a single sensation. The fact that we understand how an ens given us by several perceptions of the same sensory organ or by several organs, is identical, results from the association of sensations and perceptions through the identity of space, and through reasoning. Cf. NE, vol. 2, 941-960.

(214) Loc. cit .