Chapter LXXXIV: Section VI: Theory of Axioms

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.