every assertion which is not merely | philosophy is but bady-written poetry verba. forms in effect a couple, that is -Perhaps so.-But what they call rea to say, joins together two facts which, son, or intuition of principles, is only were separate by their nature.

We go even further than you: we think | causes, laws, essences, prin itive prop that there are neither minds nor bodies, erties. They are not new facts added but simply groups of present or possi- to the first, but an essence or extract

ble movements or thoughts. We believe that there are no substances, but only systems of facts. We regard the idea of substance as a psychological illusion. We consider substance, force, and all modern metaphysical existences, as the remains of scholastic entities. We think that there exists nothing but facts and laws, that is, events and the relations between them; and we recognize, with you, that all knowledge consists first of all in connecting or adding fact to fact. But when this is done, a new operation begins, the most fertile of all, which consists in reducing these complex into simple facts. A splendid faculty appears, the source of language, the interpreter of nature, the parent of religions and philosophies, the only genuine distinction, which, according to its degree, separates man from the brute, and great from little men. I mean Abstraction, which is the power of isolating the elements of facts, and of considering them one by one. My eyes follow the outline of a square, and abstraction isolates its two constituent properties, the equality of its sides and angles. My fingers touch the surface of a cylinder, and abstraction isolates its two generative elements, the idea of a rectangle, and of the revolution of this rectangle about one of its sides as an axis. A hundred thousand experiments develop for me, by an infinite number of details, the series of physiological operations which constitute life; and abstraction isolates the law of this series, which is a round of constant loss and continual reparation. Twelve hundred pages teach me Mill's opinion on the various facts of science, and abstraction isolates his fundamental idea, namely, that the only fertile propositions are those which connect ■ fact with another not contained in the first. Everywhere the case is the same. A fact, or a series of facts, can always be resolved into its components. It is this resolution which forms our problem, when we ask what is the nature of an object. It is these components we look for when we wish to penetrate into the inner nature of a being. These we designate under the names of forces,

from them; they are contained in the first, they have no existence apart from the facts themselves. When we dis cover them, we do not pass from one fact to another, but from one to another aspect of the same fact; from the whole to a part, from the compound to the components. We only see the same thing under two forms; first, as a whole then as divided: we only translate the same idea from one language into an other, from the language of the senses into abstract language, just as we express a curve by an equation, or a cube as a function of its side. It signifies little whether this translation be difficult or not; or that we generally need the accumulation or comparison of a vast number of facts to arrive at it, and whether our mind may not often succumb before accomplishing it. However this may be, in this operation, which is evidently fertile, instead of proceeding from one fact to another, we go from the same to the same; instead of adding experiment to experiment, we set aside some portion of the first; instead of advancing, we pause to examine the ground we stand on. There are, thus, fruitful judgments, which, however, are not the results of experience: there are essential propositions, which, however, are not merely verbal: there is, thus, an operation, differing from experience, which acts by cutting down instead of by addition ; which, instead of acquiring, devotes itself to acquired data; and which, going farther than observation, opening a new field to the sciences, defines their nature, determines their progress, completes their resources, and marks out their end.

This is the great omission of your system. Abstraction is left in the background, barely mentioned, con cealed by the other operations of the mind, treated as an appendage of Experience; we have but to re-establish it in the general theory, in order to reform the part cular theories in which it is absent.

To begin with Definitions Mill

a

chemical body by the notion

of equivalent, and a living body by the notion of type. We are striving to transform every group of phenomena into certain laws, forces, or abstract notions. We endeavor to attain in every object the generating elements, as we do attain them in the sphere, the cylinder, the circle, the cone, and in all mathematical loci. We reduce natura bodies to two or three kinds of movement -attraction, vibration, polarizationas we reduce geometrical bodies to twe or three kinds of elements-the point, the movement, the line; and we con sider our science partial or complete, provisional or definite, according as this reduction is approximate or abso lute, imperfect or complete.

IV.

teaches that there is no definition of define a things, and that when you define sphere as the solid generated by the revolution of a semicircle about its diameter, you only define a name. Doubtless you tell me by this the meaning of a name, but you also teach me a good deal more. You state that all the properties of every sphere are derived from this generating formula; vou reduce an infinitely complex system of facts to two elements; you ransform sensible into abstract data; you express the essence of the sphere, that is to say, the inner and primordial cause of all its properties. Such is the nature of every true definition; it is not content with explaining a name, it is not a mere description; it does not simply indicate a distinctive property; it does not limit itself to that ticketing of an object which will cause The same alteration is required in it to be distinguished from all others. the Theory of Proof. According to There are, besides its definition, sev- Mill, we do not prove that Prince Al eral other ways of causing the object bert will die by premising that all men to be recognized; there are other are mortal, for that would be asserting properties belonging to it exclusively: the same thing twice over; but from we might describe a sphere by saying the facts that John, Peter, and others. that, of all bodies having an equal sur- in short, all men of whom we have face, it occupies the most space; or in ever heard, have died. I reply that many other ways. But such descrip- the real source of our inference lies tions are not definitions; they lay neither in the mortality of John, Peter, down a characteristic and derived and company, nor in the mortality of property, not a generating and primi- all men, but elsewhere. We prove a tive one; they do not reduce the thing fact, says Aristotle,* by showing its to its factors, and reconstruct it before cause. We shall therefore prove the our eyes; they do not show its inner mortality of Prince Albert by showing nature and its irreducible elements. the cause which produces his death. A definition is a proposition which And why will he die? Because the hu marks in an object that quality from man body, being an unstable chemical which its others are derived, but which compound, must in time be resolved ; is not derived from others. Such a in other words, because mortality is proposition is not verbal, for it teaches added to the quality of man. Here the quality of a thing. It is not the is the cause and the proof. It is this affirmation of an ordinary quality, for abstract law which, present in nature, it reveals to us the quality which is will cause the death of the prince, prince, and source of the rest. It is an assertion which, being present to my mind, shows of an extraordinary kind, the most fer- me that he will die. It is this abstract ile and valuable of all, which sums up proposition which is demonstrative; a whole science, and in which it is the it is neither the particular nor the gen aim of every science to be summed up. eral propositions. In fact, the abstract There is a definition in every science, proposition proves the others. If John, and one for each object. We do not Peter, and others are dead, it is because in every case possess it, but we search mortality is added to the quality of man. for it everywhere. We have arrived If all men are dead, or will die, it is at defining the planetary motion by the still because mortality is added to the tangential force and attraction which * See the Posterior Analytics, which are much compose it; we can aiready partially,perior to the Prior-δί αἰτίων καὶ προτέρων.

the

quality of man. Here, again, the part tangle which generated t. It will not played by Abstraction has been over- do to say that a straight line is the looked. Mill has confounded it with shortest from one point to another, Experience: he has not distinguished for that is a derived property; but i the proof from the materials of the may say that it is the line described by

proof, the abstract law from the finite or indefinite number of its applications. The applications contain the law, and the proof, but are themselves neither law nor proof. The examples of Peter, John, and others, contain the cause, but they are not the cause. It is not sufficient to add up the cases, we must extract from them he law. It is not enough to experimentalize, we must abstract. This is the great scientific operation. Syllogism does not proceed from the particular to the particular, as Mill says, nor from the general to the particular, as the ordinary logicians teach, but from the abstract to the concrete; that is to say, from cause to effect. It is on this ground that it forms part of science, the links of which it makes and marks out; it connects principles with effects; it brings together definitions and phenomena. It diffuses through the whole range of science that Abstraction which definition has carried to its summit.

V.

com

second

a point, tending to approach towards another point point, and towards that point only: which amounts to saying that two points suffice to determine a straight line; in other words, that two straight lines, having two points in mon, coincide in their entire length; from which we see that if two straight lines approach to enclose a space, they would form but one straight line, and enclose nothing at all. Here is a method of arriving at a knowledge of the axiom, and it is clear that it differs much from the first. In the first we verify; in the second we deduce it. In the first we find by experience that it is true; in the second we prove it to be true. In the first we admit the truth; in the second we explain it. In the first we merely remark that the contrary of the axiom is inconceivable; in the second we discover in addition that the contrary of the axiom is contradic tory Having given the definition of the straight line, we find that the axiom that two straight lines cannot enclose a space is comprised in it, and may be derived from it, as a consequent from a principle. In fact, it is nothing more than an identical proposition, which means that the subject contains its attribute; it does not connect two separate terms, irreducible one to the other; it unites two terms, of which the second is a part of the first. It is a simple analysis, and so are all axioms. We have only to decompose them, in order to see that they do not proceed from one object to a different one, but are concerned with one object only. We have but to resolve the notions of equality, cause, substance, time, and space into their abstracts, in order to demonstrate the axioms of equality, substance, cause, time, and space. There is but one axiom, that of identity. The others are only its applications or its conse

Abstraction explains also axioms. According to Mill, if we know that when equal magnitudes are added to equal magnitudes the wholes are equal, or that two straight lines cannot enclose a space, it is by external ocular experiment, or by an internal experiment by the aid of imagination. Doubtless we may thus arrive at the conclusion that two straight lines cannot enclose a space, but we might recognize it also in another manner. We might reprevent a straight line in imagination, and ve may also form a conceptior of it by reason. We may either study its form or its definition. We can observe it in itself, or in its generating elements. { can represent to myself a line ready drawn, but I can also resolve it into its e'ements. I can go back to its for- quences. When this is admitted, we

matior, and discover the abstract elements which produce it, as I have watched the formation of the cylinder and discover the revolution of the rec

at once see that the range of our mind is altered. We are no longer merely capable of relative and limited knowledge, but also of absolute and infinite

knowledge; we posses in axioms facts | produced by all varieties of texture, all which not only accompany one another, diversities of substance all inequalities but one of which includes the other. If, of temperature, all complications of