Part I, Chapter 5: Of Ideas in relation to their Simplicity or Composition, in which the Method of Knowing by Abstraction or Precision is considered
The remark made by the way, in Chap. II., that we are able to consider a mode without making any distinct reflection on the substance of which it is the mode, furnishes us with an opportunity of explaining what is called Mental Abstraction.
The limited extent of our mind renders us incapable of comprehending perfectly things which are a little complex, in any other way than by considering them in their parts, and, as it were, through the phases which they are capable of receiving. This is what may be termed, generally, knowing by means of abstraction.
But since things are differently compounded, and there are some which are composed of parts really distinct, as, for instance, the human body, the different parts of a number; it is in such cases very easy to conceive that our mind can apply itself to consider one part without considering another, since these parts are really distinct; and this is not even called abstraction. It is, however, even in these things so useful to consider the parts separately rather than the whole, that without this, it is scarcely possible to have any distinct knowledge. For example, what means have we of obtaining a knowledge of the human body except by dividing it into all its parts, similar and dissimilar, and giving to each of these different names? All arithmetic is founded on this,—for there is no need of art in order to reckon small numbers, since the mind is able to comprehend them all at once; thus the whole art consists in counting by parts that which we are unable to count as a whole, since it would be impossible, however comprehensive our mind might be, to multiply two numbers of eight or nine figures each, taking them all together at once.
The second knowledge by parts, is when we consider a mode without paying attention to the substance, or two modes which are united together in the same substance, considering them each apart. This is what is done by the geometers, who have taken as the object of their science, body extended in length, breadth, and thickness. For in order to obtain a better knowledge of it, they have first applied themselves to the consideration of it, in relation to one dimension alone, which is length; and they have then given to it the name of line. They have afterwards considered it in respect to the two dimensions of length and breadth, and have called it surface. And, finally, considering all three dimensions, length, breadth, and thickness together, they have called it solid or body.
Hence it may be seen how ridiculous is the argument of certain sceptics, who would call in question the certitude of geometry, because it supposes lines and surfaces which are not in nature; for the geometers do not suppose that there are lines without breadth, or surfaces without depth,—they suppose only that we are able to consider length, without paying attention to breadth; and this is indubitable, as when we measure the distance from one town to another, we measure only the length of the road, without troubling ourselves with its breadth.
Now, the more we are able to distribute things into different modes, the more capable does the mind become of obtaining a thorough knowledge of them; and thus we see, in relation to motion, that as long as the determination towards a certain spot was not distinguished from the motion itself, and from different parts even in the same determination, so long no satisfactory account could be given of reflection and refraction, which is now easily accomplished by that distinction, as may be seen in the second chapter of the Optics of Descartes.
The third way of conceiving things by abstraction is, when a single thing, having different attributes, we think of one without thinking of another, although there may exist between them only a discrimination of reason; and this is brought about as follows: I consider, for example, that I think, and that, consequently, it is myself that is thinking, in the idea which I have of myself thinking, I am able to confine my attention to a thing which thinks, without paying any regard to the fact that it is myself, although within me, myself and he who thinks may be only one and the same thing. And thus the idea which I have conceived of a person who thinks, will be able to represent, not myself alone, but all other persons who think. In the same way, having drawn on paper an equilateral triangle, if I confine myself to the consideration of it in the place where it is, with all the accidents which determine it, I shall have the idea of that triangle alone; but if I detach my mind from the consideration of all these particular circumstances, and consider only that it is a figure bounded by three equal lines, the idea which I form of it will, on the one hand, represent to me more accurately that equality of lines; and, on the other, will be able to represent to me all equilateral triangles. And if, not restricting myself to that equality of lines, but proceeding further, I consider only that it is a figure bounded by three right lines, I shall form an idea which will represent all kinds of triangles. If, again, not confining myself to the number of lines, I simply consider that it is a plane surface, bounded by right lines, the idea which I form will represent all rectilineal figures; and thus, step by step, I am able to ascend to extension itself. Now, in these abstractions, we see that the inferior degree always comprehends the superior, together with some particular determination; as myself comprehends that which thinks, and equilateral triangle comprehends triangle, and triangle, rectilineal figure; but that the superior degree, being less determinate, is able to represent a greater number of things.
Finally, it is clear that, by these abstractions, the ideas of singular things become common, and the common, more common; and thus this gives us the opportunity of passing to what we have to say concerning ideas, considered in relation to their universality or particularity.