FUNDAMENTAL EXPLANATION OF THE ABSTRACT IDEA OF THE INFINITE.
49. Supposing that our idea of the infinite is not intuitive but abstract, let us see how its true nature may be explained.
We have the ideas of being and of its opposite, not-being; these ideas considered in themselves are general, indeterminate, and may be applied to every thing which is subjected to our experience.
We may affirm and deny something of every limited being: we may affirm what it is: we may deny what it is not: the limit is only conceived as such when something is denied of it.
50. The activity of our being is unceasing, but it is limited by the absence or the resistance of objects; the external world is an assemblage of beings presenting a great variety of limitations.
Therefore both internal and external experience give us the idea of the finite, that is, of a being which involves some not-being. The brute has sensible perception, but no understanding: itissensitive, and herein it has being; itis notintelligent, and herein it is limited. Man is sensitive and intelligent; the limit of the brute is not the limit of man. Among intelligent beings some understand more than others; therefore the limit of all is not the same.
51. Since we find a limit in both internal and external experience, it is evident that we can form the general idea of limit, that is, of a negation applied to an object.
52. The same experience teaches that what is the limit of some things is not the limit of others, and that the limit applied to one object must be denied of another. When we compare different beings together, we frequently find ourselvesdenying certainlimits. As our understanding has the faculty of generalizing, it is evident that we may conceive in general the negation ofcertainlimits, and form an indeterminate conception, including the two ideas ofnegationandlimit.
53. I do not see what objection can be made either to the possibility or to the existence of this conception; but as this fact is necessary for the explanation of the idea of infinity,I shall make some further observations for the purpose of confirming it.
We have an idea of negation in general; this is a primitive fact of our mind: without it no negative judgments would be possible, nor could we even know the principle of contradiction. It is impossible for any thing to beand not beat the same time; when we saynot bewe express a negation, we therefore have the conception of negation. This conception is general, because it involves no determination; we speak of not-being without applying it to any particular object, nor even to any determinate species or genus. Therefore the conception of negation is general and absolutely undetermined.
54. We have the idea of limit; for, as we have seen, it is a negation applied to a being. We have also the idea of the negation of limit; for just as we conceive the limit as applied or applicable, we may and do conceive it as not applied or not applicable. At every moment we deny certain limits; this idea generalized becomes the negation in general of limit in general.
55. After these remarks we may establish what is contained in the idea of the infinite. This idea is a general conception involving the conception of being in general, and the negation of limit in general. The union of these two conceptions constitutes the abstract idea of the infinite.
56. The general conception of the negation of limit gives us an idea of infinity in the abstract, but not any infinite thing. Without the intuitive cognition of an infinite object, and with only a very imperfect idea of it, we may speak of infinity without falling into contradiction, and determine the cases in which it may be applied to a being or to an order of beings, whether real or possible. Man has many ideas of this vague kind, which nevertheless answer his necessities. We shall make this palpable by examples.
57. Suppose we take an uneducated person and point out to him a number of learned men, telling him that one of them knows more than all the rest. The uneducated person has no idea of what the man knows who knows the most, nor the man who knows the least; he has no idea of the degrees of science, nor of what science itself is; but he possesses the general ideas of degree, of more and less, and also of knowledge, and this enables him to speak, without contradiction or confusion, of the greater science of the one and the less science of the others, and even to solve with certainty the questions concerning the science of those individuals, in so far as these questions are contained in the general idea that the science of one is greater than that of all the others.
A servant in an establishment where the most beautiful products of art are collected, may speak of them all without contradiction or confusion, although he may be incapable of knowing their merit, and entirely ignorant of the circumstances which constitute the beauty of the objects. It is sufficient for him to have the idea of perfection or beauty in general, and to arrange by certain arbitrary signs the degrees of perfection or beauty of the objects, in order to be able to point them out to visitors, and talk of the greater skill of one artist, the poorer success of another; the greater effect and value of the works of the former, and the inferiority of those of the second, and to make other remarks of a similar nature, which at first might make us suppose him a consummate artist, or, at the least, an amateur of a great intellect and exquisite taste.
58. It would be easy to show by other examples, how fruitful some general ideas are, and how they may undergo innumerable combinations, without presenting any thing determinate to the intellect. This is precisely what happens with the idea of the infinite: in vain we ask whatthere is within us which corresponds to it: the conception of being in general and of the negation of limit present nothing fixed, except certain abstract conditions to which we continually reduce the objects which come under our intuition, or are presented to us with certain characteristic properties which permit us to form a less vague idea of the negation of limit.