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

1âM4 MATHEMATICAL PHILOSOPHY
structions with Ruler and Compasses and The History and
Transcendence of r.
A moment ago I referred to a remarkable postulate
system devised by the Italian mathematician, Pieri. It
was published in I899 in Memorie della R. Academia delle
Scienze di Torino under the title " Della Geometria elementare come sistema ipotetico-detuttivo; monografia del
punto e del mote "; an excellent abstract of it was published
in I905 by the late Louis Couturat in his Les Principes
des Mathématiques and was partly reproduced in I9I1 by
Professor J. W. Young in his admirable Lectures on Fundamental Concepts of Algebra and Geometry. I desire, in
passing, to recommend these books of Couturat and
Young as well worth your attention, provided you will
really read them-pondering what is said in themand not be content with merely glançing through them.
They handle, in excellent style, some important matters
which these lectures touch but lightly or not at all.
You can not fail to observe, if you will examine and
compare them-as 1 hope you will-that Veblen's system
and that of Pieri differ from Hilbert's in various ways.
For example, Veblen's system contains 12 postulates;
Hilbert's, 2z; Pieri's, 20; again, while Hilbert's system
contains, as we have seen, five undefined terms, or variables, Veblen's has but two-" point" and " between "'and Pieri's also has but two —" point" and "motion."
In studying the Italian's beautiful system, your understanding of it will be much facilitated by noticing that the
undefined term "motion " is used in the sense of a
unique and reciprocal correspondence-a one-to-one
transformation-between points, and not in the manin-the-street's sense of a physical time.coisuming change
of place.