THE REMAINING LAWS OF NATURE.
There are, we have seen, five facts, one of which every proposition must
assert, viz. Existence, Order in Place, Order in Time, Causation, and
Resemblance. Causation is not fundamentally different from Coexistence
and Sequence, which are the two modes of Order in Time. They have been
already discussed. Of the rest, Existence, if of things in themselves,
is a topic for Metaphysics, Logic regarding the existence of _phenomena_
only; and as this, when it is not perceived directly, is proved by
proving that the unknown phenomenon is connected by _succession or
coexistence_ with some known phenomenon, the fact of Existence is not
amenable to any _peculiar_ inductive principles. There remain
Resemblance and Order in Place.
As for Resemblance, Locke indeed, and, in a more unqualified way, his
school, asserted that all reasoning is simply a comparison of two ideas
by means of a third, and that knowledge is only the perception of the
agreement or disagreement, that is, the resemblance or dissimilarity, of
two ideas: they did not perceive, besides erring in supposing ideas, and
not the phenomena themselves, to be the subjects of reasoning, that it
is only sometimes (as, particularly, in the sciences of Quantity and
Extension) that the agreement or disagreement of two things is the one
thing to be established. Reasonings, however, about _Resemblances_,
whenever the two things cannot be directly compared by the virtually
simultaneous application of our faculties to each, do agree with Locke's
account of reasoning; being, in fact, simply such a comparison of two
things through the medium of a third. There are laws or formulae for
guiding the comparison; but the only ones which do not come under the
principles of Induction already discussed, are the mathematical axioms
of Equality, Inequality, and Proportionality, and the theorems based on
them. For these, which are true of all phenomena, or, at least, without
distinction of origin, have no connection with laws of Causation,
whereas all other theorems asserting resemblance have, being true only
of special phenomena originating in a certain way, and the resemblances
between which phenomena must be derived from, or be identical with, the
laws of their causes.
In respect to Order in Place, as well as in respect to Resemblance, some
mathematical truths are the only general propositions which, as being
independent of Causation, require separate consideration. Such are
certain geometrical laws, through which, from the position of certain
points, lines, or spaces, we infer the position of others, without any
reference to their physical causes, or to their special nature, except
as regards position or magnitude. There is no other peculiarity as
respects Order in Place. For, the Order in Place of effects is of course
a mere consequence of the laws of their causes; and, as for primaeval
causes, in _their_ Order in Place, called their _collocation_, no
uniformities are traceable.
Hence, only the methods of Mathematics remain to be investigated; and
they are partly discussed in the Second Book. The directly inductive
truths of Mathematics are few: being, first, certain propositions about
existence, tacitly involved in the so-called definitions; and secondly,
the axioms, to which latter, though resting only on induction, _per
simplicem enumerationem_, there could never have been even any apparent
exceptions. Thus, every arithmetical calculation rests (and this is what
makes Arithmetic the type of a deductive science) on the evidence of the
axiom: The sums of equals are equals (which is coextensive with nature
itself)--combined with the definitions of the numbers, which are
severally made up of the explanation of the name, which connotes the way
in which the particular agglomeration is composed, and of the assertion
of a fact, viz. the physical property so connoted.
The propositions of Arithmetic affirm the modes of formation of given
numbers, and are true of all things under the condition of being divided
in a particular way. Algebraical propositions, on the other hand, affirm
the equivalence of different modes of formation of numbers generally,
and are true of all things under condition of being divided in _any_
way.
Though the laws of Extension are not, like those of Number, remote from
visual and tactual imagination, Geometry has not commonly been
recognised as a strictly physical science. The reason is, first, the
possibility of collecting its facts as effectually from the ideas as
from the objects; and secondly, the illusion that its ideal data are not
mere hypotheses, like those in now deductive physical sciences, but a
peculiar class of realities, and that therefore its conclusions are
_exceptionally_ demonstrative. Really, all geometrical theorems are laws
of external nature. They might have been got by generalising from actual
comparison and measurement; only, that it was found practicable to
deduce them from a few obviously true general laws, viz. The sums of
equals are equals; things equal to the same thing are equal to one
another (which two belong to the Science of Number also); and, thirdly
(what is no merely verbal definition, though it has been so called):
Lines, surfaces, solid spaces, which can be so applied to one another as
to coincide, are equal. The rest of the premisses of Geometry consist of
the so-called definitions, which assert, together with one or more
properties, the real existence of objects corresponding to the names to
be defined. The reason why the premisses are so few, and why Geometry is
thus almost entirely deductive, is, that all questions of position and
figure, that is, of quality, may be resolved into questions of quantity
or magnitude, and so Geometry may be reduced to the one problem of the
measurement of magnitudes; that is, to the finding the equalities
between them.
Mathematical principles can be applied to other sciences. All causes
operate according to mathematical laws; an effect being ever dependent
on the quantity or a function of the agent, and generally on its
position too. Mathematical principles cannot, indeed, as M. Comte has
well explained, be usefully applied to physical questions, whenever the
causes are either too inaccessible for their numerical laws to be
ascertained, or are too complex for _us_ to compute the effect, or are
ever fluctuating. And, in proportion as physical questions cease to be
abstract and hypothetical, mathematical solutions of them become
imperfect. But the great value of mathematical training is, that we
learn to use its _method_ (which is the most perfect type of the
Deductive Method), that is, we learn to employ the laws of simpler
phenomena to explain and predict those of the more complex.