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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 contradictory. 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 consequences. When this is admitted, we 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 possess in axioms facts which not only accompany one another, but one of which includes the other. If, as Mill says, they merely accompanied one another, we should be obliged to conclude with him, that perhaps this might not always be the case. We should not see the inner necessity for their connection, and should only admit it as far as our experience went; we should say that, the two facts being isolated in their nature, circumstances might arise in which they would be separate; we should affirm the truth of axioms only in reference to our world and mind. If, on the contrary, the two facts are such that the first contains the second, we should establish on this very ground the necessity of their connection; wheresoever the first may be found, it will carry the second with it, since the second is a part of it, and cannot be separated from it. Nothing can exist between them and di

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vide them, for they are but one thing under different aspects. Their connection is therefore absolute and universal; and we possess truths which admit neither doubt nor limitation, nor condition, nor restriction. Abstraction restores to axioms their value, whilst it shows their origin; and we restore to science her dispossessed dominion, by restoring to the mind the faculty of which it had been deprived.

Section VI.-The Methods of Induction

Induction remains to be considered: which seems to be the triumph of pure experience, while it is in reality, the triumph of abstraction. When I discover, by induction, that cold produces dew, or that the passage from the liquid to the solid state produces crystallization, I establish a connection between two abstract facts. Neither cold, nor dew, nor the passage from the liquid to the solid state, nor crystallization, exist in themselves. They are parts of phenomena, extracts from complex cases, simple elements included in compound aggregates. I withdraw and isolate them; I isolate dew in general from all local, temporary, special dews which I observe; I isolate cold in general from all special, various distinct colds, which may be produced by all varieties of texture, all diversities of substance, all inequalities of temperature, all complications of circumstances. I join an abstract antecedent to an abstract consequent, and I connect them, as Mill himself shows, by subtractions, suppressions, eliminations; I expel from the two groups, containing them, all the proximate circumstances; I discover the couple under the surroundings which obscure it; I detach, by a series of comparisons and experiments, all the subsidiary accidental circumstances which have clung to it, and thus I end by laying it bare. I seem to be considering twenty different cases, and in reality I only consider one; I appear to proceed by addition, and in fact I am performing subtraction. All the methods of Induction, therefore, are methods of Abstraction, and all the work of Induction is the connection of abstract facts.

Section VII.-Experience and Abstraction

We see now the two great moving powers of science, and the two great manifestations of nature. There are two operations, experience and abstraction; there are two kingdoms, that of complex facts, and that of simple elements. The first is the effect, the second the cause. The first is contained in the second, and is deduced from it, as a consequent from its principle. The two are equivalent: they are one and the same thing considered under two aspects. This magnificent moving universe, this tumultuous chaos of mutually dependent events, this incessant life, infinitely varied and multiplied, may be all reduced to a few elements and their relations. Our whole efforts result in passing from one to the other, from the complex to the simple, from facts to laws, from experiences to formulæ. And the reason of this is evident; for this fact, which I perceive by the senses or the consciousness, is but a fragment, arbitrarily severed by my senses or my consciousness, from the infinite and continuous woof of existence. If they were differently constituted, they would intercept other fragments; it is the chance of their structure which determines what is actually perceived. They are like open compasses, which might be more or less extended; and the area. of the circle which they describe is not natural, but artificial. It is so in two ways, both externally and internally. For, when I consider an event, I isolate it artificially from its natural surroundings, and I compose it artificially of elements which do not form a natural group. When I see a falling stone, I separate the fall from the anterior circumstances which are really connected with it; and I put together the fall, the form, the structure, the color, the sound, and twenty other circumstances which are really not connected with it. A fact, then, is an arbitrary aggregate, and at the same time an arbitrary severing;1 that is to say, a factitious group, which separates things connected, and connects things that are separate. Thus, so long as we only regard nature by observation, we do not see it as it is: we have only a provisional and illusory idea of it. Nature is, in reality, a tapestry, of which

1 An eminent student of Physical Science said to me "A fact is a superposition of laws."

we only see the reverse; this is why we try to turn it. We strive to discover laws; that is, the natural groups which are really distinct from their surroundings, and composed of elements really connected. We discover couples; that is to say, real compounds and real connections. We pass from the accidental to the necessary, from the relative to the absolute, from the appearance to the reality; and having found these first couples, we practice upon them the same operation as we did upon facts, for, though in a less degree, they are of the same nature. Though more abstract, they are still complex. They may be decomposed and explained. There is some ulterior reason for their existence. There is some cause or other which constructs and unites them. In their case, as well as for facts, we can search for generating elements into which they may be resolved, and from which they may be deduced. And this operation may be continued until we have arrived at elements wholly simple; that is to say, such that their decomposition would involve a contradiction. Whether we can find them or not, they exist; the axiom of causation would be falsified if they were absent. There are, then, indecomposable elements, from which are derived more general laws; and from these, again, more special laws; and from these the facts which we observe; just as in geometry there are two or three primitive notions, from which are deduced the properties of lines, and from these the properties of surfaces, solids, and the numberless forms which nature can produce, or the mind imagine. We can now comprehend the value and meaning of that axiom of causation which governs all things, and which Mill has mutilated. There is an inner constraining force which gives rise to every event, which unites every compound, which engenders every actual fact. This signifies, on the one hand, that there is a reason for everything; that every fact has its law; that every compound can be reduced to simple elements; that every product implies factors; that every quality and every being must be reducible from some superior and anterior term. And it signifies, on the other hand, that the product is equivalent to the factors, that both are but the same thing under different aspects; that the cause does not differ in nature from the effect; that the generating powers are but elementary properties; that the active force, by which

we represent Nature to our minds, is but the logical necessity which mutually transforms the compound and the simple, the fact and the law. Thus we determine beforehand the limits of every science; and we possess the potent formula, which, establishing the invincible connection and the spontaneous production of existences, places in Nature the moving spring of Nature, whilst it drives home and fixes in the heart of every living thing the iron fangs of necessity.

Section VIII.-Idea and Limits of Metaphysics

Can we arrive at a knowledge of these primary elements? For my part, I think we can; and the reason is, that, being abstractions, they are not beyond the region of facts, but are comprised in them, so that we have only to extract them from the facts. Besides, being the most abstract, that is, the most general of all things, there are no facts which do not comprise them, and from which we cannot extract them. However limited our experience may be, we can arrive at these primary notions; and it is from this observation that the modern German metaphysicians have started, in attempting their vast constructions. They understood that there are simple notions, that is to say, indecomposable abstract facts, that the combinations of these engender all others, and that the laws for their mutual union or contrarieties, are the primary laws of the universe. They tried to attain to these ideas, and to evolve, by pure reason, the world as observation shows it to us. They have partly failed; and their gigantic edifice, factitious and fragile, hangs in ruins, reminding one of those temporary scaffoldings which only serve to mark out the plan of a future building. The reason is, that with a high notion of our powers, they had no exact view of their limits. For we are outflanked on all sides by the infinity of time and space; we find ourselves thrown in the midst of this monstrous universe like a shell on the beach, or an ant at the foot of a steep slope. Here Mill is right. Chance is at the end of all our knowledge, as on the threshold of all our postulates: we vainly try to rise, and that by conjecture, to an initial state; but this state depends on the preceding one, which depends on another, and so on; and thus we are forced to accept it

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