Let us use the
following example to clear this up. We say that the
equation x²
= ‒ 1 has the solution ± √‒1. There was a time when one said that this equation had
no solution. Now this statement, whether agreeing or
disagreeing with the one which told us the solutions, certainly
hasn't its multiplicity. But we can easily
give it that multiplicity by saying that an equation
x² +
ax + b = 0 hasn't got a solution but
comes α near to the nearest solution
which is β. Analogously we can say either
“A straight line always intersects a circle; sometimes
in real, sometimes in complex points”, or,
“A straight line either intersects a circle, or it
doesn't and is α far from
from doing so.
These
two statements mean exactly the
same. They will be more or less satisfactory according
to the way a man wishes to look at it. He may wish to
make the difference between intersecting and not intersecting as
inconspicuous as possible. Or on the other hand he may
wish to stress it; and either tendency may be justified, say,
by his particular practical purposes. But this may not
be the reason at all why he prefers one form of expression to the
other. Which form he prefers, and whether he has a
preference at all, often depends on general, deeply rooted
48.
tendencies of
his thinking.