Euclid’s method of demonstration has brought forth from its own womb its
most striking parody and caricature in the famous controversy on the
theory of parallels, and the attempts, which are repeated every year, to
prove the eleventh axiom. This axiom asserts, and indeed supports its
assertion by the indirect evidence of a third intersecting line, that two
lines inclining towards each other (for that is just the meaning of “less
than two right angles”) if produced far enough must meet—a truth which is
supposed to be too complicated to pass as self-evident, and therefore
requires a demonstration. Such a demonstration, however, cannot be
produced, just because there is nothing that is not immediate. This
scruple of conscience reminds me of Schiller’s question of law:—
“For years I have used my nose for smelling. Have I, then, actually a
right to it that can be proved?” Indeed it seems to me that the logical
method is hereby reduced to absurdity. Yet it is just through the
controversies about this, together with the vain attempts to prove what is
_directly_ certain as merely _indirectly_ certain, that the
self-sufficingness and clearness of intuitive evidence appears in contrast
with the uselessness and difficulty of logical proof—a contrast which is
no less instructive than amusing. The direct certainty is not allowed to
be valid here, because it is no mere logical certainty following from the
conceptions, thus resting only upon the relation of the predicate to the
subject, according to the principle of contradiction. That axiom, however,
is a synthetical proposition _a priori_, and as such has the guarantee of
pure, not empirical, perception, which is just as immediate and certain as
the principle of contradiction itself, from which all demonstrations first
derive their certainty. Ultimately this holds good of every geometrical
theorem, and it is quite arbitrary where we draw the line between what is
directly certain and what has first to be demonstrated. It surprises me
that the eighth axiom is not rather attacked. “Figures which coincide with
each other are equal to each other.” For “coinciding with each other” is
either a mere tautology or something purely empirical which does not
belong to pure perception but to external sensuous experience. It
presupposes that the figures may be moved; but only matter is movable in
space. Therefore this appeal to coincidence leaves pure space—the one
element of geometry—in order to pass over to what is material and
empirical.
The reputed motto of the Platonic lecture-room, “Αγεωμετρητος μηδεις
εισιτω,” of which mathematicians are so proud, was no doubt inspired by
the fact that Plato regarded the geometrical figures as intermediate
existences between the eternal Ideas and particular things, as Aristotle
frequently mentions in his “Metaphysics” (especially i. c. 6, p. 887, 998,
_et Scholia_, p. 827, ed. Berol.) Moreover, the opposition between those
self-existent eternal forms, or Ideas, and the transitory individual
things, was most easily made comprehensible in geometrical figures, and
thereby laid the foundation of the doctrine of Ideas, which is the central
point of the philosophy of Plato, and indeed his only serious and decided
theoretical dogma. In expounding it, therefore, he started from geometry.
In the same sense we are told that he regarded geometry as a preliminary
exercise through which the mind of the pupil accustomed itself to deal
with incorporeal objects, having hitherto in practical life had only to do
with corporeal things (_Schol. in Aristot._, p. 12, 15). This, then, is
the sense in which Plato recommended geometry to the philosopher; and
therefore one is not justified in extending it further. I rather
recommend, as an investigation of the influence of mathematics upon our
mental powers, and their value for scientific culture in general, a very
thorough and learned discussion, in the form of a review of a book by
Whewell in the _Edinburgh Review_ of January 1836. Its author, who
afterwards published it with some other discussions, with his name, is Sir
W. Hamilton, Professor of Logic and Metaphysics in Scotland. This work has
also found a German translator, and has appeared by itself under the
title, “_Ueber den Werth und Unwerth der Mathematik_” _aus dem Englishen_,
1836. The conclusion the author arrives at is that the value of
mathematics is only indirect, and lies in the application to ends which
are only attainable through them; but in themselves mathematics leave the
mind where they find it, and are by no means conducive to its general
culture and development, nay, even a decided hindrance. This conclusion is
not only proved by thorough dianoiological investigation of the
mathematical activity of the mind, but is also confirmed by a very learned
accumulation of examples and authorities. The only direct use which is
left to mathematics is that it can accustom restless and unsteady minds to
fix their attention. Even Descartes, who was yet himself famous as a
mathematician, held the same opinion with regard to mathematics. In the
“_Vie de Descartes par Baillet_,” 1693, it is said, Liv. ii. c. 6, p. 54:
“_Sa propre expérience l’avait convaincu du peu d’utilité des
mathématiques, surtout lorsqu’on ne les cultive que pour elles mêmes....
Il ne voyait rien de moins solide, que de s’occuper de nombres tout
simples et de figures imaginaires_,” &c.