truth is known without verification by experience. Having
proved the proposition about the isosceles triangle, we do
not go about measuring the angles of triangular objects to
make sure there is no exception. We know it without any
experience at all. And if we {215} were sufficiently clever,
we might even evolve mathematical knowledge out of the
resources of our own minds, without its being told us by
any teacher. That Caesar was stabbed by Brutus is a fact
which no amount of cleverness could ever reveal to me. This
information I can only get by being told it. But that the
base angles of an isosceles triangle are equal I could discover
by merely thinking about it. The proposition about Brutus
is not a necessary proposition. It might be otherwise. And
therefore I must be told whether it is true or not. But the
proposition about the isosceles triangle is necessary, and
therefore I can see that it must be true without being told.
Now Plato did not clearly make this distinction between
necessary and non-necessary knowledge. But what he did
perceive was that mathematical knowledge can be known
without either experience or instruction. Kant afterwards
gave a less fantastic explanation of these facts. But Plato
concluded that such knowledge must be already present in
the mind at birth. It must be recollected from a previous
existence. It might be answered that, though this kind of
knowledge is not gained from the experience of the senses,
it may be gained from teaching. It may be imparted by an-
other mind. We have to teach children mathematics, which
we should not have to do if it were already in their minds.
But Plato’s answer is that when the teacher explains a ge-
ometrical theorem to the child, directly the child under-
stands what is meant, he assents. He sees it for himself.
But if the teacher explains that Lisbon is on the Tagus,
the child cannot see that this is true for himself. He must
either believe the word {216} of the teacher, or he must
go and see. In this case, therefore, the knowledge is really
imparted from one mind to another. The teacher transfers
to the child knowledge which the child does not possess.
But the mathematical theorem is already present in the
child’s mind, and the process of teaching merely consists in
making him see what he already potentially knows. He has
only to look into his own mind to find it. This is what we
mean by saying that the child sees it for himself.In the “Meno” Plato attempts to give an experimental proof
of the doctrine of recollection. Socrates is represented as
talking to a slave-boy, who admittedly has no education in
mathematics, and barely knows what a square is. By dint
of skilful questioning Socrates elicits from the boy’s mind
a theorem about the properties of the square. The point
of the argument is that Socrates tells him nothing at all.
He imparts no information. He only asks questions. The
boy’s knowledge of the theorem, therefore, is not due to the
teaching of Socrates, nor is it due to experience. It can only
be recollection. But if knowledge is recollection, it may be
asked, why is it that we do not remember at once? Why is
the tedious process of education in mathematics necessary?
Because the soul, descending from the world of Ideas into
the body, has its knowledge dulled and almost blotted out
by its immersion in the sensuous. It has forgotten, or it
has only the dimmest and faintest recollection. It has to be
reminded, and it takes a great effort to bring the half-lost