every process a succession of steps or stages, the notion of series
and process plainly involves that of _number_, and must be rigorously
dissociated from the idea of infinity. At any one step, at any one term,
the number attained is determinate, hence finite. The fact that, by the
law of the series or of the process, _we_ may continue the operation _as
long as we please_, does not justify the application of the term
infinite to the operation itself; if any thing is infinite, it is the
will which continues the operation, which is absurd if said of human
wills.

Consequently, the attribute of infinity is not predicable either of
'diminution without limit,' 'augmentation without limit,' or 'endless
approximation to a fixed limit,' for these mathematical processes
continue only as we continue them, consist of steps successively
accomplished, and are limited by the very fact of this serial
incompletion.

"We can not forbear pointing out an important application of these
results to the Critical Philosophy. Kant bases each of his famous four
antinomies on the demand of pure reason for unconditioned totality in a
regressive series of conditions. This, he says, must be realized either
in an absolute first of the series, conditioning all the other members,
but itself unconditioned, or else in the absolute infinity of the series
without a first; but reason is utterly unable, on account of mutual
contradiction, to decide in which of the two alternatives the
unconditioned is found. By the principles we have laid down, however,
the problem is solved. The absolute infinity of a series is a
contradiction _in adjecto_. As every number, although immeasurably and
inconceivably great, is impossible unless _unity_ is given as its basis,
so every series, being itself a number, is impossible unless a _first
term_ is given as a commencement. Through a first term alone is the
unconditioned possible; that is, if it does not exist in a first term,
it can not exist at all; of the two alternatives, therefore, one
altogether disappears, and reason is freed from the dilemma of a
compulsory yet impossible decision. Even if it should be allowed that
the series has no first term, but has originated _ab aeterno_, it must
always at each instant have a _last term_; the series, as a whole, can
not be infinite, and hence can not, as Kant claims it can, realize in
its wholeness unconditioned totality. Since countless terms forever
remain unreached, the se