— 565 —
Hence the Reality of mathematical Knowledge.
      §6. I doubt not but it will be easily granted, that the Knowledge we
may have of Mathematical Truths, is not only certain, but real Know-
ledge; 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 ’tis 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 farther
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, where-ever it really exists.
Whatever other Figure exists, that is not exactly answerable to that
Idea of a Triangle in his Mind, is not at all concerned in that Proposi-
tion. And therefore he is certain all his Knowledge concerning such
Ideas, is real Knowledge: because intending Things no farther 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, where-ever, or however they exist.
Locke Hum IV, 4, §6, p. 565