LINEAR ASSOCIATIVE ALGEBRA, Benjamin Peirce...
"Mathematics is the science which draws necessary conclusions."
This definition of mathematics is wider than that which is ordinarily given,
and by which its range is limited to quantitative research. The ordinary
definition, like those of other sciences, is objective ; whereas this is subjective.
Recent investigations, of which quaternions is the most noteworthy instance,
make it manifest that the old definition is too restricted. The sphere of mathe
matics is here extended, in accordance with the derivation of its name, to all
demonstrative research, so as to include all knowledge strictly capable of dog
matic teaching. Mathematics is not the discoverer of laws, for it is not
induction ; neither is it the framer of theories, for it is not hypothesis ; but it is
the judge over both, and it is the arbiter to which each must refer its claims ;
and neither law can rule nor theory explain without the sanction of mathematics.
It deduces from a law all its consequences, and develops them into the suitable
form for comparison with observation, and thereby measures the strength of the
argument from observation in favor of a proposed law or of a proposed form of
application of a law.
Mathematics, under this definition, belongs to every enquiry, moral as well
as physical. Even the rules of logic, by which it is rigidly bound, could not be
deduced without its aid. The laws of argument admit of simple statement, but
they must be curiously transposed before they can be applied to the living speech
and verified by,observation. In its pure and simple form the syllogism cannot
be directly compared with all experience, or it would not have required an
Aristotle to discover it. It must be transmuted into all the possible shapes in
which reasoning loves to clothe itself. The transmutation is the mathematical
process in the establishment of the law. Of some sciences, it is so large a
portion that they have been quite abandoned to the mathematician, which
may not have been altogether to the advantage of philosophy. Such is the
case with geometry and analytic mechanics. But in many other sciences, as in
all those of mental philosophy and most of the branches of natural history, the
deductions are so immediate and of such simple construction, that it is of no
practical use to separate the mathematical portion and subject it to isolated
discussion.