An introduction to mathematics, by A. N. Whitehead.

IMAGINARY NUMBERS 91
algebraic laws of transformation; and it was
seen that a whole world of interesting results
could be attained, if only these symbols might
legitimately be used. Many mathematicians
were not then very clear as to the logic of
their procedure, and an idea gained ground
that, in some mysterious way, symbols which
mean nothing can by appropriate manipulation yield valid proofs of propositions.
Nothing can be more mistaken. A symbol
which has not been properly defined is not a
symbol at all. It is merely a blot of ink on
paper which has an easily recognized shape.
Nothing can be proved by a succession of
blots, except the existence of a bad pen or a
careless writer. It was during this epoch
that the epithet "imaginary" came to be
applied to V/(-1). What these mathematicians had really succeeded in proving were
a series of hypothetical propositions, of which
this is the blank form: If interpretations
exist for /(-1) and for the addition, subtraction, multiplication, and division of
v(-i1) which make the ordinary algebraic
rules (e.g. x+y=y+x, etc.) to be satisfied,
then such and such results follow. It was
natural that the mathematicians should not
always appreciate the big "If," which ought
to have preceded the statements of their results.
As may be expected the interpretation,