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.