Some draw consequences well from a few principles, and this is rightness of thinking.
Others draw well the consequences of things involving many principles. For example, the former understand well the effects of water, which involve few principles; but these consequences are so subtle that only acute right thinking can get to them. And, in spite of that, these people might not be great geometers, because geometry comprises a great number of principles, and a mind may be of such a nature as to be able to penetrate easily to the bottom of a few principles without in the least being able to penetrate those things involving many principles.
There are therefore two kinds of minds: the one, penetrating rapidly and deeply the consequences of principles, is the intuitive mind [esprit de finesse]; the other, grasping a great number of principles without confusing them, is the geometric mind [esprit de géométrie]. The first has strength and rightness of mind, the second breadth of mind. Now, one can exist without the other; the mind can be strong and narrow, and also broad and weak.
Difference between the geometric and the intuitive mind
In the one, principles are obvious, but removed from ordinary use, so that we find it difficult to turn our head in that direction, for lack of habit. But if it was turned that way ever so little, we would see the principles fully and would need to have a wholly defective mind to reason wrongly about such principles so obvious that it is almost impossible for them to notice.
But with the intuitive mind, principles are in common use and before everybody's eyes. You have only to look, and no effort is necessary; it is only a question of good sight. But it must be good, because the principles are so subtle and numerous that it is almost impossible but that some escape notice. Now the omission of one principle leads to error. Thus you must have very clear sight to see all the principles, as well as an accurate mind to avoid reasoning falsely from known principles.
All geometers would then be intuitive if they had clear sight, for they do not reason wrongly from principles known to them. And intuitive minds would be geometric if they could bend their thinking to the principles of geometry to which they are unaccustomed.
The reason, then, that some intuitive minds are not geometric is that they simply cannot turn their attention to the principles of geometry. But the reason that geometers are not intuitive is that they do not see what is before them, and that, accustomed to the exact and plain principles of geometry, and not reasoning until they have clearly seen and handled their principles, they are lost in matters of intuition, where the principles do not allow such handling. The principles are scarcely seen; they are felt rather than seen; there is endless difficulty in making them felt by those who do not themselves apprehend them. These principles are so delicate and numerous that a delicate and clear sense is needed to apprehend them, and to judge rightly and correctly according to this feeling, without most often being able to demonstrate them in order as in geometry; because the principles are not known to us in this way and because it would be an endless task to undertake it. We must see the matter at once, at a glance, and not by a process of reasoning, at least up to a point. And thus it is rare for geometers to be intuitive and for intuitive people to be geometers, because geometers want to treat matters of intuition geometrically and make themselves ridiculous, wanting to begin by definitions and then by principles, which is not the way to proceed in this kind of reasoning. Not that the mind does not do this, but it does so tacitly, naturally, and without art, for its expression surpasses all men, and only a few can apprehend it.
Intuitive minds, on the contrary, being thus accustomed to judge at a single glance, when presented with propositions of which they understand nothing and the paths to which consist of such sterile definitions and principles, which they are not accustomed to see in such detail, are so surprised that they are repelled and disheartened.
But false minds are never either intuitive or geometrical.
Geometers who are only geometers, then, have rightness of mind, provided all things are explained to them by means of definitions and principles; otherwise they are wrong and insufferable, for they are only right when the principles are quite clear.
And intuitive minds that are only intuitive cannot have the patience to reach the first principles of speculative and imaginative things, which they have never seen in the world and which are altogether out of the common.
Saturday, May 3, 2014
esprit de géométrie & esprit de finesse
Pascal, Blaise. Pensées. Translated by Roger Ariew. Hackett Publishing Company, Inc., 2004: