The great error lies in supposing that even the
truths of what is called pure algebra, are abstract or general truths.
And this error is so egregious that I am confounded at the
universality with which it has been received. Mathematical axioms
are not axioms of general truth. What is true of relation --of form
and quantity --is often grossly false in regard to morals, for
example. In this latter science it is very usually untrue that the
aggregated parts are equal to the whole. In chemistry also the axiom
falls. In the consideration of motive it falls; for two motives,
each of a given value, have not, necessarily, a value when united,
equal to the sum of their values apart. There are numerous other
mathematical truths which are only truths within the limits of
relation. But the mathematician argues, from his finite truths,
through habit, as if they were of an absolutely general
applicability --as the world indeed imagines them to be. Bryant, in
his very learned 'Mythology,' mentions an analogous source of error,
when he says that 'although the Pagan fables are not believed, yet
we forget ourselves continually, and make inferences from them as
existing realities.' With the algebraists, however, who are Pagans
themselves, the 'Pagan fables' are believed, and the inferences are
made, not so much through lapse of memory, as through an unaccountable
addling of the brains.
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