Elements of the philosophy of the human mind. By Dugald Stewart. Rev. and abridged, with critical and explanatory notes, for the use of colleges and schools. By Francis Bowen ...

REASON1ING AND DEDUCTIVE EVIDENCE. 427
sllhould be followed; and, indeed, if it were copied as a mlodel
in any other branch of human knoNwledge, it would lead to errors
fully as dangerous as any which result from the imperfections
of language. The real fact is, that it has been copied much
more than it ought to have been, or than would have been
attempted, if the peculiarities of mathematical evidence had
been attentively considered.
That in mathematics there is no such thing as an ambiguous
word, and that it is to the proper use of definitions we are indebted for this advantage, must unquestionably be granted.
But this is an advantage easily secured, in consequence of the
very limited vocabulary of mathematicians, and the distinctness
of the ideas about which their reasonings are employed. The
difference, besides, in this respect, between mathematics and the
other sciences, however great, is yet only a difference in degree;
and is by no means sufficient to account for the essential distinction, which every person must perceive between the irresistible cogency of a mathematical demonstration and that of any
other process of reasoning.
Proper use of definzitions. -From the foregoing considerations it appears, that in mathematics, definitions answer two
purposes; first, to prevent ambiguities of language; and, secondly, to serve as the principles of outr reasoning. It appears
further, that it is to the latter of these circumstances, (I mean
to the employmnent of hypotheses instead of facts, as the data on
which we proceed,) that the peculiar force of demonstrative evidence is to be ascribed.* It is, however, only in the former
* [Mr. Stewart shows with sufficient clearness, that the definitions are
the true premises of mathematical reasoning, and that it is only upon the
supposition or hypothesis of these definitions being granted, that the
reasoning holds good. But he does not show very clearly whiy the employment of definitions and hypotheses should give to a mathematical
demonstration the irresistible cogency which distinguishes it from every
other species of reasoning. In another work, the editor of this volume
has endeavored to supply this defect by the following considerations.
Demonstrative reasoning is employed exclusively about the relations of
ideas, or abstract ideas, and its conclusions are always abstract; the induc