If it were possible to isolate completely from the other rays the “mental” process—the “logos”—the _M_ rays—and have a complete abstraction (which in the present could only be in mathematics), then the work of _M_ could be compared to the work of an impersonal machine which always gives the same _correctly_ shaped product _no matter what is_ the material put into it.
It is a fact that mathematics is correct—impersonal—passionless. Again, as a matter of fact, all the basic axioms which underlie mathematics are "psychological axioms"; therefore it may happen that these "axioms" are not of the αinfinity type but are of the _f_ (_A_ _B_ _C_ ...) personal type and this may be why mathematics cannot account for psychological facts. If psychology is to be an _exact science_ it must be mathematical in principle. And, therefore, mathematics must find a way to embrace psychology. Here I will endeavor to outline a way in which this can be done. To express it correctly is more than difficult: I beg the mathematical reader to tolerate the form and look for the sense or even the feelings in what I attempt to express. To make it less shocking to the ear of the pure mathematician, I will use for the “infinitesimals” the words “very small numbers,” for the “finite” the words “normal numbers” and for the “transfinite” the words “very great numbers.” Instead of using the word “number” I will sometimes use the word “magnitude” and under the 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 Poincaré 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 order in the way indicated here and become philosophic or psychological mathematics. This new science, of course, would not change the ordinary mathematics for ordinary purposes. It would be a special mathematics for the study of Man dealing only with the “natural finites” (the old infinitesimals) and great numbers of different orders (including the normal numbers), but starting from a real, common base—from 0, and next to it very small number, which is a common _tangible_ base for _psychological_ as well as _analytical_ truths.