Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

VIII
INTRODUCTION
fore, that the soul is the generator of mathematical forms,
and the source of the productive principles with which the
mathematical sciences are replete.
If, however, she contains these as paradigms, she gives them
subsistence essentially, and the generations are nothing else
than the projections of the forms which have a prior existence
in herself. And thus we shall speak conformably to the doctrine
of Plato, and discover the true essence of mathematical
objects. But if the soul, though she neither possesses nor caus
ally contains these productive principles, yet weaves together
so great an immaterial order of things, and generates such an
admirable theory, how will she be able to judge whether her
productions are stable, or things which the winds may dissipate,
and images rather than realities? What standard can she
apply as the measure of their truth? And how, since she does
not possess the essence of them, can she generate such a variety
of principles productive of truth? For from such an hypothesis
we shall make their subsistence fortuitous, and not referred
to any scientific bound.
In the second place, therefore, if by a downward process,
and from sensibles we collect mathematical principles, must
we not necessarily say, that those demonstrations are more excellent,
which derive their composition from sensibles, and not
those which are always framed from more universal, and more
simple forms. For we say, in order to the investigation of the
thing sought, that the causes should be appropriate and allied
to the demonstrations. If, therefore, partial natures are the
causes of universals, and sensibles of the objects of the reasoning
power, why does the boundary of demonstration alwavs
refer to that which is more universal, instead of that which is
partial? And how can we evince that the essence of the objects
of the reasoning power is more allied to demonstrations