Descartes - Rules For The Direction of The Mind
Philosophy from Descartes

RULE XI

If, after gaining intuitive knowledge of several simple propositions, we are to draw some further inference from them, it is useful for us to run through them in a continuous and uninterrupted movement of thought, to reflect on their interrelations and to form, sofar as we can, distinct conceptions of several at once. For this adds much to the certainty of our knowledge, and it greatly increases the scope of our mind.

It is in place here to give a clearer exposition of what I said before about intuition (Rules III and VII). In the one place I contrasted intuition with deduction; in the other, merely with enumeration. (I defined enumeration as an inference made from many separate data put together; the simple deduction of one thing from another is made, I said, by intuition.) This procedure was necessary because intuition must satisfy two conditions: first, our understanding of a proposition must be clear and distinct; secondly, it must be one simultaneous whole without succession. Now if we are thinking of the act of deduction, as in Rule III, it has not the appearance of being a simultaneous whole; rather, it involves a movement of the mind in which we infer one thing from another. Here, then, we were justified in distinguishing it from intuition. If on the other hand we attend to deduction as something already accomplished, as in the notes on Rule VII, then the term does not stand any longer for such a movement, but for the result of the movement. In that sense, then, I assume that a deduction is something intuitively seen, when it is simple and clear, but not when it is complex and involved; for that, I used the term ' enumeration' or 'induction '. For the latter sort of deduction cannot be grasped all at once; its certainty depends in a way on memory, which must retain judgments about the various points enumerated in order that we may put them all together and get some single conclusion.

All these distinctions had to be made in order to bring out the meaning of the present Rule. Rule IX dealt only with intuition, and Rule X only with enumeration; then comes this Rule, explaining how these two activities cooperate-operate and supplement one another-seem, in fact, to merge into a single activity, in which there is a movement of thought such that attentive intuition of each point is simultaneous with transition to the next.
I mention two advantages of this: the greater certainty in our knowledge of the conclusion we have in view, and the greater aptitude of our mind for making further discoveries. As I said, when conclusions are too complex to be held in a single act of intuition, their certainty depends on memory; and since memory is perishable and weak, it must be revived and strengthened by this continuous and repeated movement of thought. For example, suppose I have learnt, in a number of successive mental acts, the relations

between magnitudes 1 and 2, magnitudes 2 and 3, magnitudes 3 and 4, and, finally, magnitudes 4 and 5; this does not make me see the relation between magnitudes 1 and 5, nor can I deduce it from the ones I already know, unless I remember them all; accordingly, I must run over them in thought again and again, until I pass from the first to the last so quickly that I have hardly any parts to the care of memory, but seem to have a simultaneous intuition of the whole.

In this way, as no-one can fail to see, the slowness of the mind is remedied, and its capacity enlarged. But it must further be noticed, as the chief advantage of this Rule, that by reflection upon the interdependence of simple propositions we acquire the practice of rapidly discerning their degrees of derivativeness and the steps of their reduction to what is underived. For example, if I run through a series of magnitudes in continued proportion, I shall reflect on all the following points: it is by concepts of the same level that I discern the ratio of term 1 to term 2, of term 2 to term 3, of term 3 to term 4, and so on, and there are no degrees of difficulty in conceiving these ratios; but it is more difficult for me to conceive the way that term 2 depends on terms 1 and 3 together, and still more difficult to conceive how the same term 2 depends on terms 1 and 4, and so on. This shows me the reason why, given merely terms 1 and 2, 1 can easily find terms 3, 4, etc.; for this is done by means of particular and distinct concepts. But given merely terms 1 and 3, I cannot so easily find their (geometric) mean; this can be done only by means of a concept involving two together of the concepts just mentioned. Given only terms 1 and 4, it is still more difficult to get an intuition of the two mean (proportionals), since this involves three simultaneous concepts. Consequently it might seem to be even more difficult to find three mean (proportionals) given terms 1 and 5; but, for a further reason, this is not the case. Although we have here four concepts joined together, they can be separated, because 4 is divisible by another number; so I can begin by trying to find term 3 from terms 1 and 5, and then go on to find term 2 from terms 1 and 3 <and then term 4 from terms 3 and 5>. He who is accustomed to reflect on such matters recognizes at once, when he examines each new problem, the source of the difficulty and the simplest method <of solution> ; and this helps very much towards knowledge of the truth.