There remains a kind of philosophical fortress in which the Idealists have taken refuge. At the origin of all proof are Axioms, from which all proofs are derived. Two straight lines cannot enclose a space; two things, equal to a third, are equal to one another; if equals be added to equals, the wholes are equal. These are instructive propositions, for they express, not the meanings of words, but the relations of things. And, moreover, they are fertile propositions; for arithmetic, algebra, and geometry are all the result of their truth. On the other hand, they are not the work of experience, for we need not actually see with our eyes two straight lines in order to know that they cannot enclose a space; it is enough for us to refer to the inner mental conception which we have of them: the evidence of our senses is not needed for this purpose; our belief arises wholly, with its full force, from the simple comparison of our ideas. Moreover, experience follows these two lines only to a limited distance, ten, a hundred, a thousand feet; and the axiom is true for a thousand, a hundred thousand, a million miles, and for an unlimited distance. Thus, beyond the point at which experience ceases, it is no longer experience which establishes the axiom. Finally, the axiom is a necessary truth; that is to say, the contrary is inconceivable. We cannot imagine a space enclosed by two straight lines: as soon as we imagine the space enclosed, the two lines cease to be straight; and as soon as we imagine the two lines to be straight, the space ceases to be enclosed. In the assertion of axioms, the constituent ideas are irresistibly drawn together. In the negation of axioms, the constituent ideas inevitably repel each other. Now this does not happen with truths of experience: they state an accidental relation, not a necessary connection; they lay down that two facts are connected, and not that they must be connected; they show us that bodies are heavy, not that they must be heavy. Thus, axioms are not, and cannot be the results of experience. They are not so, because we can form them mentally without the aid of experience; they cannot be so, because the nature and scope of their truths lie beyond the limits of experience. They have another and a deeper source. They have a wider scope, and they come from elsewhere.
Not so, answers Mill. Here again you reason like a schoolman; you forget the facts concealed behind your conceptions; for examine your first argument. Doubtless you can discover, without making use of your eyes, and by purely mental contemplation, that two straight lines cannot enclose a space; but this contemplation is but a displaced experiment. Imaginary lines here replace real lines: you construct the figure in your mind instead of on paper: your imagination fulfils the office of a diagram on paper: you trust to it as you trust to the diagram, and it is as good as the other; for in regard to figures and lines the imagination exactly reproduces the sensation. What you have seen with your eyes open, you will see again exactly the same a minute afterwards with your eyes closed; and you can study geometrical properties, transferred to the field of mental vision, as accurately as if they existed in the field of actual sight. There are, therefore, experiments of the brain as there are ocular ones; and it is after just such an experiment that you deny to two straight lines, indefinitely prolonged, the property of enclosing a space. You need not, for this purpose, pursue them to infinity: you need only transfer yourself in imagination to the point where they converge, and there you have the impression of a bent line, that is of one which ceases to be straight.[409] Your presence there, in imagination, takes the place of an actual presence; you can affirm by it what you affirmed by your actual presence, and as positively. The first is only the second in a more commodious form, with greater flexibility and scope. It is like using a telescope instead of the naked eye; the revelations of the telescope are propositions of experience; so are those of the imagination. As to the argument which distinguishes axioms from propositions of experience under the pretext that the contraries of the latter are conceivable, while the contraries of axioms are inconceivable, it is nugatory, for this distinction does not exist. Nothing prevents the contraries of certain propositions of experience from being conceivable, and the contraries of others inconceivable. That depends on the constitution of our minds. It may be that in some cases the mind may contradict its experience, and in others not. It is possible that in certain cases our conceptions may differ from our perceptions, and sometimes not. It may be that, in certain cases, external sight is opposed to internal, and in certain others not. Now, we have already seen that in the case of figures, the internal sight exactly reproduces the external. Therefore, in axioms of figures, the mental sight cannot be opposed to the actual; imagination cannot contradict sensation. In other words, the contraries of such axioms will be inconceivable. Thus axioms, although their contraries are inconceivable, are experiments of a certain class, and it is because they are so that their contraries are inconceivable. At every point there results this conclusion, which is the abstract of the system: every instructive or fruitful proposition is derived from experience, and is simply a connecting together of facts.
[Section VII.—Theory of Induction]
Hence it follows that Induction is the only key to nature. This theory is Mill's masterpiece. Only so thorough-going a partisan of experience could have constructed the theory of Induction.
What, then, is Induction?
"Induction is that operation of the mind by which we infer that what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, Induction is the process by which we conclude that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times."[410]
This is the reasoning by which, having observed that Peter, John, and a greater or less number of men have died, we conclude that all men will die. In short, induction connects "mortality" with the quality of "man"; that is to say, connects two general facts ordinarily successive, and asserts that the first is the Cause of the second.
This amounts to saying that the course of nature is uniform. But induction does not set out from this axiom, it leads up to it; we do not find it at the beginning, but at the end of our researches.[411] Fundamentally, experience presupposes nothing beyond itself. No à priori principle comes to authorize or guide her. We observe that this stone has fallen, that this hot coal has burnt us, that this man has died, and we have no other means of induction except the addition and comparison of these little isolated and transient facts. We learn by simple practical experience that the sun gives light, that bodies fall, that water quenches thirst, and we have no other means of extending or criticising these inductions than by other like inductions. Every observation and every induction draws its value from itself, and from similar ones. It is always experience which judges of experience, and induction of induction. The body of our truths has not, then, a soul distinct from it, and vivifying it; it subsists by the harmony of all its parts taken as a whole, and by the vitality of each part taken separately.
"Why is it that, with exactly the same amount of evidence, both negative and positive, we did not reject the assertion that there are black swans, while we should refuse credence to any testimony which asserted that there were men wearing their heads underneath their shoulders? The first assertion was more credible than the latter. But why more credible? So long as neither phenomenon had been actually witnessed, what reason was there for finding the one harder to be believed than the other? Apparently because there is less constancy in the colours of animals than in the general structure of their internal anatomy. But how do we know this? Doubtless from experience. It appears, then, that we need experience to inform us in what degree, and in what cases, or sorts of cases, experience is to be relied on. Experience must be consulted in order to learn from it under what circumstances arguments from it will be valid. We have no ulterior test to which we subject experience in general; but we make experience its own test. Experience testifies, that among the uniformities which it exhibits, or seems to exhibit, some are more to be relied on than others; and uniformity, therefore, may be presumed, from any given number of instances, with a greater degree of assurance, in proportion as the case belongs to a class in which the uniformities have hitherto been found more uniform."[412]