Mathematics and Time-Binding

THE purpose of this appendix is to give an expression of some new ideas which evolve directly out of the fact that humans are time-binders and which may serve as suggestions for the foundation of scientific psychology. The problem is of exceeding difficulty to give expression to in any form and therefore much more difficult to express in any exact or correct form, and so I beg the reader's patience in regard to the language because some of the ideas are in themselves correct and sometimes very suggestive in spite of the language used. I am particularly interested that mathematicians, physicists and metaphysicians should read it carefully, forgive me the form, and look into the suggestions, because scientific psychology if such a science is to exist, would by necessity have to be a branch of physics. I particularly beg the mathematicians and physicists not to discard this appendix with too hasty a judgment of "Oh! metaphysics," and also the metaphysicians not to do the same with an equally hasty judgment "Oh! mathematics." I hope that if this appendix is sympathetically understood, mathematicians and physicists will be moved to investigate the problem. If mathematicians and physicists would be more tolerant toward metaphysics and if metaphysicians would be moved to study mathematics, both would find tremendous fields to work in.

Some scientists are very pedantic and therefore intolerant in their pedantry and they may say "the fellow should learn first how to express himself and then ask our attention." My answer is that the problems involved are too pressing, too vital, too fundamental for humankind, to permit me to delay for perhaps long years before I shall be able to present the subject in a correct and satisfactory form, and also that the problems involved cover too vast a field for a single man to work it conclusively. It seems best to give the new ideas to the public in a suggestive form so that many people may be led to work on them more fully.

The old word "metaphysics" is an illegitimate child of ignorance and an unnecessary word in the scientific study of nature. Every phenomenon of nature can be classed and studied in physics or chemistry or mathematics; the problem, therefore, is not in any way supernatural or superphysical, but belongs rather to an unknown or an undeveloped branch of physics. The problem, therefore, may be not that of some new science, but rather that of a new branch of mathematics or physics, or chemistry, etc., or all combined.

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 space, 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 analysis.

In this book a new and experimental fact has been disclosed and analysed. It is the fact that humanity is a time-binding class 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 analysing them.

Once the word and concept Time enters, the ground for analysis 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 analyse 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 space 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 mass by acceleration, or that action is equal to reaction." (The Foundation of Science, by Henri Poincaré); 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-inder-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 assert that its conclusions are correct; that is the inestimable value of mathematics. This becomes a very comprehensive fact if we approach and analyse 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-bindinginvolved 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 (l, 2, . . . (n).

(2) Scientific truth (s), 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 l, 2. . n for the "psychological," "private," or "relative" truths, between which, for the moment, I will not discriminate.

sl,s2 . . . sn will be used for scientific truths, and finally for the absolute truth valid in infinity.

To make it easier to explain, I will illustrate 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-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 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 O, 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 "O"-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 O, and next to it very small number, which is a common tangible base for psychological as well as analytical 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 analytical-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 analytical 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=ekt, with the formula "P RT," 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 it "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 analyse 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 (I) 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 Poincaré (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 Poincaré was a physicist (as well as a mathematician) and, therefore, approaches the problems somewhat from a physicist's point of view, a circumstance 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:


"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. Classes. 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 constituents of matter. On the notion of cause.


"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. Space, 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.

"The Concept of Nature." Cambridge, 1920. Selection from contents:

Nature and thought. Time. The method of extensive abstraction. Space and motion. Objects. The ultimate physical concepts.

"Principia Mathematica" By A. N. Whitehead and Bertrand Russell. Cambridge, 1910-1913.

This monumental work stands alone. "As a work of constructive criticism it has never been surpassed. To every one and especially to philosophers and men of natural science, it is an amazing revelation of how the familiar terms with which they deal plunge their roots far into the darkness beneath the surface of common sense. It is a noble monument to the critical spirit of science and to the idealism of our time."

"Human Worth of Rigorous Thinking." C. J. Keyser.

(2) The physicist's point of view:


"The Foundations of Science." The Science Press, N. Y., 1913.

Selection from contents:

I. Science and hypothesis. Number and magnitude. Space. Force. Nature. II. The value of science. The mathematical sciences. The physical sciences. The objective value of science. III. Science and method. Science and the scientist. Mathematical reasoning. The new mechanics. Astronomic science.

(3) The human, civilizing, practical life, point of view:


"Science and Religion The Rational and the Super-rational." The Yale University Press.

"The New Infinity and the Old Theology." The Yale University Press.

"The Human Worth of Rigorous Thinking." Essays and Addresses. Columbia University Press, 1916.

Selection from contents:

The human worth of rigorous thinking. The human significance of mathematics. The walls of the world or concerning the figure and the dimensions of the Universe of space. The universe and beyond. The existence of the hypercosmic. The axiom of infinity: A new presupposition of thought. Research in American Universities. Mathematical productivity in the United States.

"Mathematical Philosophy, the Study of Fate and Freedom. Lectures for Educated Laymen." Forthcoming Book.

Selection from contents of general interest.

The mathematical obligations of philosophy. Humanistic and industrial education. Logic the muse of thought. Radiant aspects of an over-world.-Verifiers and falsifiers. Significance and nonsense.- Distinction of logical and psychological. A diamond test of harmony.-Distinction of doctrine and method. -Theoretical and practical doubt.-Mathematical philosophy in the role of critic. A world uncriticised- the garden of the devil. "Supersimian" Wisdom. Autonomous truth and autonomous falsehood. Other Varieties of truth and untruth. Mathematics as the study of fate and freedom. The prototype of reasoned discourse often disguised as in the Declaration of Independence, the Constitution of the United States, the Origin of Species, the Sermon on the Mount.-Nature of mathematical transformation. No transformation, no thinking. Transformation law essentially psychological, Relation function and transformation as three aspects of one thing. Its study, the common enterprise of science. The static and the dynamic worlds. The problem of time and kindred problems. Importation of time and suppression of time as the classic devices of sciences.- The nature of invariance. The ages-old problem of permanence and change. The quest of what abides in a fluctuant world as the binding thread of human history. The tie of comradeship among the enterprises of human spirit.-The concept of a group. The notion simply exemplified in many fields, is "Mind" a group. The philosophy of the cosmic year.-Limits and limit processes omnipresent as ideals and idealization, in all thought and human aspiration. Ideals the flint of reality.-Mathematical infinity, its dynamic and static aspects. Need of history of the Imperious concept. The role of infinity in a mighty poem.-Meaning of dimensionality. Distinction of imagination and conception. Logical existence and sensuous existence. Open avenues to unimaginable worlds.-The theory of logical types. A supreme application of it to definition of man, and the science of human welfare.-The psychology of mathematics and the mathematics of psychology. Both of them in their infancy. Consequent retardation of science. The symmetry of thought. The asymmetry of imagination.-Science and engineering. Science as engineering in preparation. Engineering as science in action. Mathematics the guide of the engineer. Engineering the guide of humanity. Humanity the civilizing or Time-Binding class of life. Qualities essential to engineering leadership. The ethics of the art. The engineer as educator, as scientist, as philosopher, as psychologist, as economist, as statesman, as mathematical thinker-as a Man.