It is pathetic that only after many aeons of human existence the dimensionality of man has been discovered and his proper status in _nature_ has been given by the definition of "time-binder." The old metaphysics, in spite of its being far from exact, accomplished a great deal. What prevented metaphysics from achieving more was its use of unmathematical method, or, to be more explicit, its failure to understand the importance of dimensions. Metaphysics used words and conceptions of multi-dimensional meanings which of necessity resulted in hopeless confusion, in "a talking" about words, in mere verbalism. An example will serve to make this clear. If we were to speak of a cow, a man, an automobile, and a locomotive as "pullers," and if we were not to use any other names in connection with them, what would happen? If we characterized these things or beings, by one common characteristic, namely, "to pull," havoc would be introduced into our conceptions and in practical life; we would try to milk an automobile or we would try to extract gasoline from a cow, or look for a screw in a man, or we would speculate about any or all of these things. Too obviously nonsensical-but exactly the same thing happens, in a much more subtle way, when we use such words as "life in a crystal" or "memory in animals"; we are thus mentally making a mistake no less nonsensical than the talk of "milking an automobile" would be. Laymen are baffled by the word dimension. They imagine that dimensions are applicable only to s.p.a.ce, which is three dimensional, but they are mistaken; a moving object is four-dimensional-that is, it has three dimensions as any object at rest, but, when the object is moving, a fourth dimension is necessary to give its _position_ at any one instant. We see, therefore, that a moving body has four dimensions, and so on. As a matter of fact, scientific psychology will very much need mathematics, but a special _humanized_ mathematics.

Can this be produced? It seems to me that it can.

It is a well known fact that experimental sciences bring us to face facts which require further theoretical elaboration; in this way experimental sciences are a permanent source of inspiration to mathematicians because new facts bring about the need of new methods of a.n.a.lysis.

In this book a new and experimental fact has been disclosed and a.n.a.lysed.

It is the fact that humanity is a time-binding cla.s.s of life where the time-binding capacity or the time-binding ENERGY is the highest function of humanity, including all the so-called mental, spiritual, will, etc., powers. In using the words mental, spiritual, and will powers, I deliberately accept and use them in the popular, ordinary sense without further a.n.a.lysing them.

Once the word and concept _Time_ enters, the ground for a.n.a.lysis and reasoning at once becomes very slippery. Mathematicians, physicists, etc., may feel that the expression is just a "well adapted one," and they may not be very much inclined to look closer into it or attentively to a.n.a.lyse it. Theologians and metaphysicians probably will speculate a great deal about it vaguely, with undefined terms and incoherent ideas with incoherent results; which will not lead us toward a scientific or true solution, but will keep us away from the discovery of truth.

In the meantime two facts remain facts: namely, mathematicians and physicists have almost all agreed with Minkowski "that s.p.a.ce by itself and time by itself, are mere shadows, and only a kind of blend of the two exists in its own right." The other fact-psychological fact-is that _time_ exists psychologically by itself, undefined and not understood. One chief difficulty is always that humans have to sit in judgment upon their own case. The psychological time as such, is our own human time; scientific time as such, is also our own human time. Which one of them is the best concept-which one more nearly corresponds to the truth about "time"? What is time (if any) anyway? Until now we have gone from "Cosmos" to "Bios,"

from "Bios" to "Logos," now we are confronted with the fact that "Logos"-Intelligence-and Time-binding are dangerously near to akin to each other, or may be identical. Do we in this way approach or go back to "Cosmos"? Such are the crucial questions which arise out of this new concept of Man. One fact must be borne in mind, that "the principles of dynamics appeared first to us, as experimental truths; but we have been obliged to use them as definitions. It is by definition that force is equal to the product of ma.s.s by acceleration, or that action is equal to reaction." (_The Foundation of Science_, by Henri Poincare); and mathematics also has its whole foundation in a few axioms, "self evident,"

but _psychological facts_. It must be noted that the time-binding energy-the higher or highest energies of man (one of its branches anyway, for sake of discrimination let us call it "_M_") when it works properly, that is, mathematically, does _not_ work _psychologically_ but works ABSTRACTLY: the higher the abstraction the less there is of the psychological element and the more there is, so to say, of the pure, impersonal time-binding energy (_M_). The definition of a man as a time-binder-a definition based on facts-suggests many reflections. One of them is the possibility that one of the functions of the time-binding energy in its pure form, in the highest abstraction (_M_), works automatically-machine-like, as it were, shaping _correctly_ the product of its activity, but whether _truly_ is another matter. Mathematics does not presume that its conclusions are true, but it does a.s.sert that its conclusions are correct; that is the inestimable value of mathematics.

This becomes a very comprehensive fact if we approach and a.n.a.lyse the mathematical processes as some branch (_M_) of the time-binding process, which they are; then this process at once becomes impersonal and cosmic, because of the time-binding involved in it, no matter what _time_ is (if there is such a thing as time).

Is the succession of cosmos, bios, logos, time-binding taking us right back to cosmos again? Now if we put _psychological_ axioms into the time-binding apparatus, it will thrash out the results _correctly_, but whether the results are _true_ is another question.

To be able to talk about these problems I have to introduce three new definitions, which are introduced only for practical purposes. It may happen that after some rewording these definitions may become scientific.

I will try to define "truth" and for this purpose I will divide the concept "truth" into three types:

(1) Psychological, or private, or relative truth, by which I will mean such conceptions of the truth as any one person possesses, but different from other types of truth (a1, a2, ... an)

(2) Scientific truth (as), by which I will mean a psychological truth when it is approved by the time-binding faculties or apparatus in the present stage of our development. This scientific truth represents the "bound-up-time" in our present knowledge; and finally,

(3) The absolute truth, which will be the _final definition_ of a phenomenon based upon the final knowledge of _primal causation valid in infinity_.

For simplicity's sake I will use the signs a1, a2, ... an for the "psychological," "private," or "relative" truths, between which, for the moment, I will not discriminate.

as1, as2, ... asn, will be used for scientific truths, and finally ainfinity for the absolute truth valid in infinity.

To make it easier to explain, I will ill.u.s.trate the suggestions by an example. Let us suppose that the human time-binding capacities or energies in the _organic_ chemistry correspond to radium in the _inorganic_ chemistry; being of course of different dimensions and of absolutely different character. It may happen, for it probably is so, that the complex time-binding energy has many different stages of development and different kinds of "rays" _A_, _B_, _C_, ... _M_....

Let us suppose that the so-called mental capacities are the _M_ rays of the time-binding energy; the "spiritual" capacities, the _A_ rays; the "will" powers, the _B_ rays; and so on. Psychological truths will then be a function of all rays together, namely _A_ _B_ _C_ ... _M_ ... or _f_ (_A_ _B_ _C_ ... _M_ ...), the character of any "truth" in question will largely depend upon which of these elements prevail.

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-pa.s.sionless. 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 ainfinity 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 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, s.p.a.ce, 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 _a.n.a.lytical_ truths.

This new philosophic mathematics would eliminate the concept of "infinitesimals" as such, which is an _artificial_ concept and is not as a _concept_ an element of Nature. The so-called _infinitesimals are Nature's real, natural finites_. In mathematics the infinitesimals were an a.n.a.lytical-an "_M_"-time-binding-necessity, because of our starting point.

I repeat once again that this transposition of our starting point would not affect the normal mathematics for normal purposes; it would build rather a new philosophic mathematics rigorously correct where a.n.a.lytical facts would be also psychological facts. This new mathematics would not only give correct results but also _true_ results. Keeping in mind _both_ conceptions of time, the scientific time and the psychological time, we may see that the human capacity of "Time-binding" is a very practical one and that this time-binding faculty is a _functional_ name and definition for what we broadly mean by human "intelligence"; which makes it obvious that time (in any understanding of the term) is somehow very closely related to intelligence-the mental and spiritual activities of man. _All we know about _"time"_ will explain to us a great deal about Man, and all we know about Man will explain to us a great deal about time_, if we consider _facts_ alone. The "ghosts" in the background will rapidly vanish and become intelligible facts for philosophic mathematics. The most vital importance, nevertheless, is that taking zero as the limit and the next to it very small magnitude for the real starting point, it will give us a mathematical science from a natural base where _correct_ formulas will be also true formulas and will correspond to psychological truths.

We have found that man is an exponential function where time enters as an exponent. If we compare the formula for organic growth _y==e__kt_, with the formula "_P R__T_," we see that they are of the same type and the _law of organic growth_ applies to the human _time-binding energy_. We see, too, that the time-binding energy is also "_alive_" and multiplying in larger and larger families. The formula for the decomposing of radium is the same-only the exponent is negative instead of positive. This fact is indeed very curious and suggestive. Procreation, the organic growth, is also some function of time. I call "time-linking" for the sake of difference. Whether the energy of procreation or that of "time-linking"

can be accounted for in units of chemical energy taken up in food, I do not know. Not so with the mind-this "time-binding," higher exponential energy, "able to direct basic powers." If we a.n.a.lyse this energy, free from any speculation, we will find that this higher energy which is somehow directly connected with "time"-no matter what time is-is able to _produce_, by transformation or by drawing on other sources of energy, new energies unknown to nature. Thus the solar energy transformed into coal is, for instance, transformed into the energy of the drive of a piston, or the rotary energy in a steam engine, and so on. It is obvious that no amount of _chemical_ energy in food can account for such an energy as the time-binding energy. There is only one supposition left, namely, that the time-binding apparatus has a source for its tremendous energy in the _transformation of organic atoms_, and-what is very characteristic-the results are _time_-binding energies.

This supposition is almost a certainty because it seems to be the only possible supposition to account for that energy. This supposition, which seems to be the only supposition, would bring us to face striking facts, namely, the transformation of organic atoms, which means a direct drawing upon the cosmic energy; and this cosmic energy-time-and intelligence are somehow connected-if not indeed equivalent. Happily these things can be verified in scientific laboratories. Radium was discovered only a few years ago and is still very scarce, but the results for science and life are already tremendous because scientific methods were applied in the understanding and use of it. We did not use any zoological or theological methods, but just direct, correct and scientific methods. There is no scarcity in "human radium," but, to my knowledge, physicists have never attempted to study this energy from that point of view. I am confident that, if once they start, there will be results in which all the so-called "supernatural, spiritual, psychic" phenomena, such as are not fakes, will become scientifically understood and will be consciously utilized. Now they are mostly wasted or only played with. It may happen that the science of Man-as the science of time-binding-will disclose to us the inner and final secrets-the final truth-of nature, valid in infinity.

It is very difficult to give in such a book as this an adequate list of the literature which may help to orient the reader in a general way in the great advance science has made in the last few years. This book is a pioneer book in its own way, and so there are no books dealing directly with its subject. There are two branches of science and one art which are fundamental for the further development of the subject; these two sciences are (1) Mathematical philosophy and (2) Scientific biology, the art is the art of creative engineering.

In mathematical philosophy there are to my knowledge only four great mathematical writers who treat the subject as a distinct science. They are two English scientists, Bertrand Russell and A. N. Whitehead; one Frenchman, Henri Poincare (deceased); and one American, Professor C. J.

Keyser. Messrs. Russell and Whitehead approach the problems from a purely logical point of view and therein lies the peculiar value of their work.

Henri Poincare was a physicist (as well as a mathematician) and, therefore, approaches the problems somewhat from a physicist's point of view, a circ.u.mstance giving his philosophy its particular value. Professor Keyser approaches the problems from both the logical and the warmly human points of view; in this is the great human and practical value of his work.

These four scientists are unique in their respective elaborations and elucidations of mathematical philosophy. It is not for me to advise the reader what selections to make, for if a thorough knowledge of the subject is desired the reader should read all these books, but not all readers are willing to make that effort toward clear thinking (which in the meantime will remain of the _highest_ importance in science). Some readers will wish to select for themselves and to facilitate their selection I will lay out a "Menu" of this intellectual feast by giving in some cases the chapter heads.

For many temporary reasons I was not able, before going into print, to give a fuller list of the writings of those four unique men; but there is no stroke of their pen but which should be read with great attention-besides which there is a very valuable literature about their work.

(1) The purely mathematical foundation:

RUSSELL, BERTRAND.

"The Principles of Mathematics." Cambridge University, 1903.

(I am not giving any selections from the contents of this book because this book should, without doubt, be read by every one interested in mathematical philosophy.)

"The Problems of Philosophy." H. Holt & Co., N. Y., 1912.

"Our Knowledge of the External World, as a Field for Scientific Method in Philosophy." Chicago, 1914.

"Introduction to Mathematical Philosophy." Macmillan, N. Y.

Selection from contents: Definition of number. The Definition of order.

Kinds of relations. Infinite cardinal numbers. Infinite series and ordinals. Limits and continuity. The axiom of infinity and logical types.

Cla.s.ses. Mathematics and logic.

"Mysticism and Logic." Longmans Green & Co. 1919. N. Y.

Selection from contents: Mathematics and the metaphysicians. On scientific method in philosophy. The ultimate const.i.tuents of matter. On the notion of cause.

WHITEHEAD, ALFRED N.

"An Introduction to Mathematics." Henry Holt & Co. 1911. N. Y.

"The Organization of Thought Educational and Scientific." London, 1917.

Selections from contents: The principles of mathematics in relation to elementary teaching. The organization of thought. The anatomy of some scientific ideas. s.p.a.ce, time, and relativity.

"An Enquiry Concerning the Principles of Natural Knowledge." Cambridge, 1919.

Selection from contents: The traditions of science. The data of science.

The method of extensive abstraction. The theory of objects.