So that in these we
cannot miss of a certain and undoubted reality.
6. Hence the reality of mathematical knowledge. I doubt not but it
will be easily granted, that the knowledge we have of mathematical
truths is not only certain, but real knowledge; and not the bare empty
vision of vain, insignificant chimeras of the brain: and yet, if we
will consider, we shall find that it is only of our own ideas. The
mathematician considers the truth and properties belonging to a
rectangle or circle only as they are in idea in his own mind. For it
is possible he never found either of them existing mathematically,
i.e. precisely true, in his life. But yet the knowledge he has of
any truths or properties belonging to a circle, or any other
mathematical figure, are nevertheless true and certain, even of real
things existing: because real things are no further concerned, nor
intended to be meant by any such propositions, than as things really
agree to those archetypes in his mind. Is it true of the idea of a
triangle, that its three angles are equal to two right ones? It is
true also of a triangle, wherever it really exists. Whatever other
figure exists, that it is not exactly answerable to that idea of a
triangle in his mind, is not at all concerned in that proposition. And
therefore he is certain all his knowledge concerning such ideas is
real knowledge: because, intending things no further than they agree
with those his ideas, he is sure what he knows concerning those
figures, when they have barely an ideal existence in his mind, will
hold true of them also when they have a real existence in matter:
his consideration being barely of those figures, which are the same
wherever or however they exist.
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