CHAPTER VI. THE ‘PENSÉES.’

“We may have three principal objects in the study of truth—one to discover it when we seek it, another to demonstrate it when we possess it, and a third and last to discriminate it from the false when we examine it. . . .  Geometry excels in all three, and especially in the art of discovering unknown truths, which it calls *analysis*. . .  There is a method which excels geometry, but is impossible to man, *for whatever transcends geometry transcends us* [in natural science, as he explains elsewhere].  This is the method of defining everything and proving everything. . .  A fine method, but impossible; since it is evident that the first page 166terms that we wish to define, suppose precedent terms necessary for their explanation—and that the first propositions that we wish to prove, suppose others which precede them; and so it is clear we can never arrive at absolutely first principles.  In pushing our researches to the utmost, we necessarily reach primitive words that admit of no further definition, and principles so obvious, that they require no proof.  Man can never, therefore, from natural incompetency, possess an absolutely complete science. . . .  But geometry, while inferior in its aims, is absolutely certain within its limits.  It neither defines everything, nor attempts to prove everything, and must, so far, yield its pretension to be an absolute science; but it sets out from things universally admitted as clear and constant, and is therefore perfectly true, because in consonance with nature.  Its function is not to define things universally clear and understood, but to define all others; and not to attempt to prove things intuitively known to men, but to attempt to prove all others.  Against this, the true order of knowledge, those alike err who attempt to define and to prove everything, and those who neglect definition and demonstration where things are not self-evident.  This is what geometry teaches perfectly.  It attempts no definition of such things as *space*, *time*, *motion*, *number*, *equality*, and the like, because these terms designate so naturally the things which they signify, that any attempt at making them more clear ends in making them more obscure.  For there is nothing more futile than the talk of those who would define primitive words. ³ . . . . . . . . “In geometry the principles are palpable, but removed from common use. . . .  In the sphere of natural wit or acuteness, the principles are in common use and before all eyes—it is only a question of having a good view of them; for they are so subtle and numerous, that some are almost sure to escape observation. . . .  All geometers would be men of acuteness if they had sufficient insight, for they page 167never reason falsely on the principles recognised by them.  All fine or acute spirits would be geometers if they could fix their thoughts on the unwonted principles of geometry.  The reason why some finer spirits are not geometers is, that they cannot turn their attention at all to the principles of geometry; but geometers fail in finer perception, because they do not see all that is before them, and being accustomed to the plain and palpable principles of geometry, and never reasoning until they have well ascertained and handled their principles, they lose themselves in matters of intellectual subtlety, where the principles are not so easily laid hold of.  Such things are seen with difficulty; they are felt rather than seen.  They are so delicate and multitudinous that it requires a very delicate and neat sense to appreciate them. . . .  So it is as rare for geometers to be men of subtle wit as it is for the latter to be geometers, because geometers like to treat these nicer matters geometrically, and so make themselves ridiculous; they like to commence with definition, and then go on to principles—a mode which does not at all suit this sort of reasoning.  It is not that the mind does not take this method, but it does so silently, naturally, and without conscious art.  The perception of the process belongs only to a few minds, and those of the highest order. . . .  Geometers, who are only geometers, are sure to be right, provided the subject come within their scope, and is capable of explanation by definition and principles.  Otherwise they go wrong altogether, for they only judge rightly upon principles clearly set forth and established.  On the other hand, subtle men, who are only subtle, lack patience, in matters of speculation and imagination, to reach first principles which they have never known in the world, and which are entirely beyond their beat. . . . “There are different kinds of sound sense.  Some succeed in one order of things, and not in another, in which they are simply extravagant. . . .  Some minds draw consequences well from a few principles, others are more at home in drawing conclusions from a great variety of principles.  For example, some understand well the phenomena of water, with page 168reference to which the principles are few, but the results extremely delicate, so that only very great accuracy of mind can trace them.  Such men would probably not be great geometers, because geometry involves a multitude of principles, and because the mind which may penetrate thoroughly a few principles to their depth may not be at all able to penetrate things which combine a multitude of principles. . . .  There are two sorts of mind: the one fathoms rapidly and deeply the consequences of principles—this is the observant and accurate mind; the other embraces a great multitude of principles, without confounding them—and this is the mathematical mind.  The one is marked by energy and accuracy, the other by amplitude.  But the one may exist without the other.  The mind may be powerful and narrow, or it may be ample and weak.” ⁴