122 Functions, limits, derivatives
called a black box and is sometimes called a transformer T. When an element x
of the domain is selected and fed into the black box or transformer, the x is called
an input and the black box or transformer is supposed to produce an output
which is an element y of a set R which is called a range. Thus to each x in D
there corresponds exactly one y in R which is called the transform of x and is
denoted by T(x) so that y = T(x). Thus we have a transformer T which trans-
forms each x in D into a transform T(x) in R. So far we have used the words
"transformer" and "transform," but we have not used the word "transformation."
Our domain and transformer and range determine and are determined by the
set S of ordered pairs (x,y) for which x is an element of D and y is the element of
R for which y = T(x), and we call this set S a transformation Ts. The domain
(set of inputs) and range (set of outputs) of the transformer Tare also the domain
and range of the transformation Ts. We now have adequate terminology and
notation. The transformer T is the active "operator" that converts each ele-
ment x of D (or each first element of one of the pairs in Ts) into the transform
T(x) in R (which is the appropriate second element of a pair in the set S which
constitutes Ts). The transformer T and the transformation Ts are inherently
different things, and there can be no doubt that our science is inadequately
developed when we apply the same name and the same symbol to the two things.
The worst of it is that, when the word "function" is used, this one word sometimes
means a transform T(x), sometimes means the transformer T, and sometimes
means the transformation Ts. Perhaps an assertion involving the word "func-
tion" will help us to see why we must make a rather serious study of terminology
before we can be intelligent readers and listeners. It is the function (see the
nonmathematical meanings given in a dictionary) of a function (transformer)
to carry an element of the domain D of the function (transformer or transforma-
tion) into the function (transform) in the range R of the function (transformer
or transformation). Commenting upon this matter from the point of view of
mathematical logic, Professor Rosser remarked to the author that some of our
terminological difficulties are due to the fact that the already overburdened
old word "function" was used as a name for the set S of ordered pairs. It is
possible that terminology will slowly improve, but meanwhile we can be com-
forted by the fact that the bad terminology rarely if ever actually injures us.
We can be irked by the fact that a "diameter" of a circle is sometimes a line
segment (a point set) and is sometimes a number (the length of the line segment),
but we are rarely if ever injured and there seems to be no overwhelming demand
for improvement of the terminology.
3.2 Limits When we were infants learning to walk and to talk, and
perhaps even after that, we heard many statements that we could not
comprehend. When an explorer tells us that he found a complete set of
normalized Legendre polynomials in an ancient cave in Peru and that the
carbon test shows that the set is 24,500 years old, it may be difficult for us
to learn what he is talking about and whether he is telling a truth.
Moreover, statements involving erudite technical terminology are not the
only ones that can be troublesome. Sometimes we must do considerable
working and thinking before we can fully understand statements that