REPRESENTATIONS 5

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

target.

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

evening.

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