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.