the doctrine is based.

9. MATHEMATICS.--Let us now turn to take a glance at the mathematical
sciences. Of course, these have to do with things sooner or later, for
our mathematical reasonings would be absolutely useless to us if they
could not be applied to the world of things; but in mathematical
reasonings we abstract from things for the time being, confident that
we can come back to them when we want to do so, and can make use of the
results obtained in our operations.

Now, every civilized man who is not mentally deficient can perform the
fundamental operations of arithmetic. He can add and subtract,
multiply and divide. In other words, he can use _numbers_. The man
who has become an accomplished mathematician can use numbers much
better; but if we are capable of following intelligently the intricate
series of operations that he carries out on the paper before us, and
can see the significance of the system of signs which he uses as an
aid, we shall realize that he is only doing in more complicated ways
what we have been accustomed to do almost from our childhood.

If we are interested, not so much in performing the operations, as in
inquiring into what really takes place in a mind when several units are
grasped together and made into a new unit,--for example, when twelve
units are thought as one dozen,--the mathematician has a right to say:
I leave all that to the psychologist or to the metaphysician; every one
knows in a general way what is meant by a unit, and knows that units
can be added and subtracted, grouped and separated; I only undertake to
show how one may avoid error in doing these things.

It is with geometry as it is with arithmetic. No man is wholly