Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

242 MATHEMATICAL PHILOSOPHY
dom,-is conditioned, restricted, limited; but it is fundamentally limited by only one Law-the law which says to
Intellect, "Thou shalt not incur a contradiction in
terms." This law is the eternal guardian of intellectual
integrity; reverence for it,-the disposition to keep it,is the absolute invariant of intellectual life; disregard of
the law,-I do not mean inadvertent violation of it,means intellectual extinction: for intellect, disloyalty is
death. Incidentally, we thus glimpse another phase of the
truth, mentioned before, that mathematics is the study
of Fate and Freedom.
Examples illustrating the mathematical concept of a
variable are more numerous than the sands of the seashore
or indeed the stars of the heavens, even if the multitude
of the stars be infinite. In examining the following more
specific and more familiar specimens it should be borne
in mind that in mathematical discourse the range of a
variable is very frequently indicated without explicitly
stating the propositional function or functions necessarily
involved in determining the range; such statement is,
however, always possible and is often made. And now
some familiar specimens.
(i) Consider the finite cardinal numbers: 0, I, 2,
3, 4, 5,..; let x represent any one of them; here the
symbol x is a variable; its range is, not the endless row
as such, but the class of terms (numbers) in the row; the
range is the same as would be indicated if we said, let x
represent any one of the verifiers of the propositional
function-n is a cardinal number. What is here the
range of n?
(2) A variable's range may be finite or infinite. In (I)
the range, you note, is infinite. If we let x represent any
cardinal greater than I and less than Io, x's range is the