word "infinity" I will understand the meaning as "limitless." The base of
the whole of mathematics or rather the starting point of mathematics was
"psychological truths," axioms concerning normal numbers, and magnitudes
that were tangible for the senses. Here to my mind is to be found the
kernel of the whole trouble. The _base_ of mathematics was _f_ (_A_ _B_
_C_ ... _M_ ...); the _work_, or the development, of mathematics is _f_
(_M_); this is the reason for the "ghosts" in the background of
mathematics. The _f_ (_M_) evolved from this _f_ (_A_ _B_ _C_ ... _M_ ...)
_base_ a wonderful abstract theory absolutely correct for the normal, the
very small, and for the very great numbers. But the rules which govern the
small numbers, the normal, or psychological numbers, and the great
numbers, are not the same. As a matter of fact, in the meantime, the
physical world, the psychological world, is composed exclusively of very
great numbers and of very small magnitudes (atoms, electrons, etc.). It
seems to me that, if we want really to understand the world and man, we
shall have to start from the beginning, from 0, then take the next very
small number as the first _finite_ or "normal number"; then the old
finites or the normal numbers would become very great numbers and the old
very-great numbers would become the very great of the second order and so
on. Such transposed mathematics would become psychological and philosophic
mathematics and mathematical philosophy would become philosophic
mathematics. The immediate and most vital effect would be, that the
_start_ would be made not somewhere in the middle of the magnitudes but
from the beginning, or from the limit "zero," from the "0"--from the
intrinsic "to be or not to be"--and the next to it would be the very first
small magnitude, the physical and therefore psychological continuum (I use
the words physical continuum in the way Poincare used them) would become a
mathematical continuum in this new philosophic mathematics. This new
branch of philosophic, psychological mathematics would be absolutely
rigorous, correct and _true_ in addition to which, maybe, it would change
or enlarge and make humanly tangible for the layman, the concept of
numbers, continuum, infinity, space, time and so on. Such a mathematics
would be the mathematics for the time-binding psychology. Mathematical
philosophy is the highest philosophy in existence; nevertheless, it could
be changed to a still higher ord