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

800 MATHEMATICAL PHILOSOPHY
definite concept or at best a vague and emotional intuition? Was his thought and use of it mystical, or logical
and analytical, or both? Did he regard his infinite as
a fact or as an hypothesis, and why? Was it time? An
extension in time? Space? An extension in space? Was
it matter or mind or both? Was it physical or spiritual?
Concrete or abstract? Did he define it and, if so, in
what terms? Or did he take it as a primitive, and, if so,
did he do it consciously? Did he think of it as magnitude
or as multitude or as both? Had he but one infinite or
many of them? If many, were they coordinate or hierarchical? If the latter, was the hierarchy crowned or
summitless? Was his infinite subordinate in his thought
or central and dominant? Did he employ it consistently
or confusedly? Was its function poetic or scientific or
both? What was its relation to the modern concept of
mathematical infinity?
It has seemed to me that I could best serve you in this
hour by sketching what I conceive should be an important chapter in such a critical work. The sketch, which
will be very imperfect, is offered, not as a model, but only
as a concrete suggestion. I have selected for the purpose
the philosophy of Lucretius, in which, as you are no doubt
aware, the notion-or some notion-of infinity is very
conspicuous. The question is: what notion and what is
its significance there?
It will facilitate the discussion if we first remind ourselves of the meaning of mathematical infinity and, in
connection therewith, note one or two distinctions and
make the acquaintance of two important technical terms
-equivalence of classes, and denumerability. In the lecture on the nature of mathematical transformation, we
met the notion of an infinite class of terms, or objects of