slogan on a T-shirt can represent an idea, and a mental image
can represent a beloved person. In mathematics, the situation is
different. All the mathematical entities we encounter or invent are
considered to be on the same plane and have the same degree and
type of reality or ideality: They are all mathematical entities.
What are representations used for? They explain one thing by
means of another. The object we want to understand is the “thing”:
the thing-in-itself, the source. The object that we know quite a bit
about already, to which we compare the source via a representation,
we call the standard object. It is the site of appearance, the
Our conventions might not correspond to your expectations. The
target, the object at the head of the arrow, is the piece of the picture
that we understand better. We will derive information about the
source by using properties of both the arrow and the target.
An Example: Countin g
We look at the simplest possible example, one that goes back to
prehistory: counting. Suppose we have a sack of potatoes or a ﬂock
of sheep. We want to know how many potatoes or sheep we have.
This is a much more sophisticated question than knowing
whether they are the same in number as another sack of potatoes
or another ﬂock of sheep. We start with the less sophisticated
question. Suppose we want to know whether the ﬂock of sheep being
herded home this evening is the same size as the herd we let out to
the pasture in the morning. In the morning, we put a small pebble
in our pouch for each sheep as it went out of the fold. Now we take
a pebble out of the pouch as each sheep returns to the fold.
We were careful to make sure the pouch was e mpty in the
morning before we began, and careful not to put anything in or
take anything out during the day. So if the pouch becomes empty
exactly as the last sheep comes in, we are happy. A mathematician
says that we have demonstrated the existence of a one-to-one
correspondence from the sheep in the morning to the sheep in the