Chapter X. Conclusion

Chapter X. Conclusion“In Europe we know that an age is dying. Here it would be easy to miss the signs of coming changes, but I have little doubt that it will come. A realization of theaimlessnessof life lived to labor and to die, having achieved nothing but avoidance of starvation, and the birth of children also doomed to the weary treadmill, has seized the minds of millions.”Sir Auckland Geddes, British Ambassador to the U. S. 1920.In conclusion let me say very briefly, as I said in the beginning, that this little book has aimed to be only a sketch. The Problem of Life is old. I have endeavored to approach it afresh, with a new method, in a new spirit, from a new point of view. The literature of the subject is vast. It displays great knowledge and skill. Much of it is fitted to inform and to inspire such as really read with a genuine desire to understand. Its weakness is due to the absence of a true conception of what human beings are. That is what I miss in it and it is that lack of fundamental and central thought that I have striven to supply. If I have succeeded in that, I have no fear—all else will follow quickly, inevitably, as a matter of course. For a fundamental conception, once it is formed and expressed, has a strange power—the power of enlisting the thought and cooperation[pg 205]of many minds. And no conception can have greater power in our human world than atrueconception of the nature of Man. For that most important of truths the times are ripe; the world is filled with the saddest of memories, with gloom, forebodings and fear. Without the truth in this matter, there can be no rational hope—history must go on in its dismal course; butwiththe truth, there is not only hope but certitude that the old order has passed and that humanity's manhood dates from the present day. That I have here presented the truth in this matter—the true conception of the human class of life—I have personally no doubt; and I have no doubt that that conception is to be the base, the guide, the source of light, of a new civilization. Whether I am mistaken or not, time will decide. I feel as Buckle felt in writing hisHistory of Civilization:“Whether or not I have effected anything of real value ... is a question for competent judges to decide. Of this, at least, I feel certain, that whatever imperfections may be observed, the fault consists, not in the method proposed, but in the extreme difficulty of any single man putting into full operation all the parts of so vast a scheme. It is on this point, and on this alone, that I feel the need of great indulgence. But, as to the plan itself, I have no misgivings. Of defects in its execution I am not unconscious. I can only plead the immensity of the subject, the shortness of a single life and the imperfection of every single enterprise. I, therefore, wish this work to be estimated, not according to the[pg 206]finish of its separate parts, but according to the way in which those parts have been fused into a complete and symmetrical whole. This, in an undertaking of such novelty and magnitude, I have a right to expect, and I would moreover, add, that if the reader has met with opinions adverse to his own, he should remember, that his views are, perhaps, the same as those which I too once held, and which I have abandoned, because, after a wider range of study, I found them unsupported by solid proof, subversive of the interest of Man, and fatal to the progress of his knowledge. To examine the notions in which we have been educated, and to turn aside from those which will not bear the test, is a task so painful, that they who shrink from the sufferings should pause before they reproach those by whom the suffering is undergone.... Conclusions arrived at in this way are not to be overturned by stating that they endanger some other conclusions; nor can they be even affected by allegation against their supposed tendency. The principles which I advocate are based upon distinct arguments supported by well ascertained facts. The only points, therefore, to be ascertained, are, whether the arguments are fair, and whether the facts are certain. If these two conditions have been obeyed, the principles follow by an inevitable inference.”And why have I sought throughout to follow the spirit of mathematics? Because I have been dealing with ideas and have desired, above all things else, to be right and clear. Ideas have a character of their own—they are right or wrong independently of our hopes and passions and will. In the connection of ideas there is an unbreakable thread of destiny. That is why in hisMathematical PhilosophyProfessor Keyser has truly said:[pg 207]“Mathematics is the study of Fate—not fate in a physical sense, but in the sense of the binding thread that connects thought with thought and conclusions with their premises. Where, then, is our freedom? What do you love? Painting? Poetry? Music? The muses aretheirfates. Whoso loves them is free. Logic is the muse of Thought.”No doubt mathematics is truly impersonal in method; too impersonal maybe to please the sentimentalists before they take the time to think; mathematical analysis of life phenomena elevates our point of view above passion, above selfishness in any form, and, therefore, it is the only method which can tell us genuine truths about ourselves. Spinosa even in the 17th Century had well realized this fact and although imperfect in many ways, his was an effort in the right direction and this quoted conclusion may well be a conclusion for ourselves in the 20th century:“The truth might forever have remained hid from the human race, if mathematics, which looks not to the final cause of figures, but to their essential nature and the properties involved in it, had not set another type of knowledge before them.... When I turned my mind to this subject, I did not propose to myself any novel or strange aim, but simply to demonstrate by certain and indubitable reason, those things which agree best with practice. And in order that I might enquire into the matters of the science with the same freedom of mind with which we are wont to treat lines and surfaces in mathematics; I determined not to laugh or to weep over the actions of men, but simply to understand them; and to contemplate their affections and passions, such as love, hate, anger, envy, arrogance, pity and all other disturbances[pg 208]of soul not as vices of human nature, but as properties pertaining to it in the same way as heat, cold, storm, thunder pertain to the nature of the atmosphere. For these, though troublesome, are yet necessary, and have certain causes through which we may come to understand them, and thus, by contemplating them in their truth, gain for our minds much joy as by the knowledge of things that are pleasing to the senses.”If only this little book willinitiatethe scientific study of Man, I shall be happy; for then we may confidently expect a science and art that will know how to direct the energies of man to the advancement of human weal.What else? Many topics have not even been broached. Time-binding energy—what may it not achieve in course of the aeons to come? What light may it not yet throw upon such fundamental phenomena asSpace,Time,Infinity, and so on? What, if any, are the limits of Time-binding? In it are somehow involved all the higher functions of mind. Is Time identical with Intelligence? Is either of them the other's cause? Is Timeinthe Cosmos or is the latter in the former? Is the Cosmos intelligent? Many no doubt and marvelous are the fields which the scientific study of man will open for research.[pg 209]Appendix I. Mathematics And Time-BindingThe 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 ofscientific 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[pg 210]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 waysupernatural orsuperphysical, but belongs rather to an unknown or an undeveloped branch of physics. The problem, therefore, may be not that of somenewscience, 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 innaturehas 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[pg 211]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 itspositionat 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 specialhumanizedmathematics. 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-bindingenergyis 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 conceptTimeenters, 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[pg 212]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 thattimeexists 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,”butpsychological 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, doesnotworkpsychologicallybut worksabstractly: 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[pg 213]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, shapingcorrectlythe product of its activity, but whethertrulyis 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-binding involved in it, no matter whattimeis (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 putpsychologicalaxioms into the time-binding apparatus, it will thrash out the resultscorrectly, but whether the results aretrueis 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 (α1, α2, ... αn)(2) Scientific truth (αs), 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,[pg 214](3) The absolute truth, which will be thefinal definitionof a phenomenon based upon the final knowledge ofprimal causation valid in infinity.For simplicity's sake I will use the signs α1, α2, ... αnfor the“psychological,”“private,”or“relative”truths, between which, for the moment, I will not discriminate.αs1, αs2, ... αsn, will be used for scientific truths, and finally αinfinityfor 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 theorganicchemistry correspond to radium in theinorganicchemistry; 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 theMrays of the time-binding energy; the“spiritual”capacities, theArays; the“will”powers, theBrays; and so on. Psychological truths will then be a function of all rays together, namelyABC...M... orf(ABC...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”—theMrays—and have a complete abstraction (which in the present could only be in mathematics), then the work ofMcould be compared to the work of an impersonal machine which always gives the samecorrectlyshaped productno matter what isthe 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 αinfinitytype but are of thef(ABC...) personal type and[pg 215]this may be why mathematics cannot account for psychological facts. If psychology is to be anexact scienceit 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. Thebaseof mathematics wasf(ABC...M...); thework, or the development, of mathematics isf(M); this is the reason for the“ghosts”in the background of mathematics. Thef(M) evolved from thisf(ABC...M...)basea 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 firstfiniteor“normal number”; then the old finites or the normal numbers would become very great numbers and the old very-great[pg 216]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 thestartwould 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 andtruein 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 commontangiblebase forpsychologicalas well asanalyticaltruths.This new philosophic mathematics would eliminate the concept of“infinitesimals”as such, which is anartificialconcept and is not as aconceptan element of Nature. The so-calledinfinitesimals are Nature's real, natural finites. In mathematics the infinitesimals were an analytical—an“M”—time-binding—necessity,[pg 217]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 alsotrueresults. Keeping in mindbothconceptions 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 afunctionalname 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 considerfactsalone. 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 wherecorrectformulas 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 growthy==ekt, with the formula“P RT,”we see that they are of the same type and thelaw of organic growthapplies to the humantime-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.[pg 218]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 toproduce, 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 ofchemicalenergy 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 thetransformation of organic atoms, and—what is very characteristic—the results aretime-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[pg 219]“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 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[pg 220]to make that effort toward clear thinking (which in the meantime will remain of thehighestimportance 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. 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.[pg 221]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. 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:Poincaré, Henri.“The Foundations of Science.”The Science Press, N. Y., 1913.Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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:Keyser, Cassius J.“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 rôle 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[pg 223]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 rôle 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.[pg 224]

Chapter X. Conclusion“In Europe we know that an age is dying. Here it would be easy to miss the signs of coming changes, but I have little doubt that it will come. A realization of theaimlessnessof life lived to labor and to die, having achieved nothing but avoidance of starvation, and the birth of children also doomed to the weary treadmill, has seized the minds of millions.”Sir Auckland Geddes, British Ambassador to the U. S. 1920.In conclusion let me say very briefly, as I said in the beginning, that this little book has aimed to be only a sketch. The Problem of Life is old. I have endeavored to approach it afresh, with a new method, in a new spirit, from a new point of view. The literature of the subject is vast. It displays great knowledge and skill. Much of it is fitted to inform and to inspire such as really read with a genuine desire to understand. Its weakness is due to the absence of a true conception of what human beings are. That is what I miss in it and it is that lack of fundamental and central thought that I have striven to supply. If I have succeeded in that, I have no fear—all else will follow quickly, inevitably, as a matter of course. For a fundamental conception, once it is formed and expressed, has a strange power—the power of enlisting the thought and cooperation[pg 205]of many minds. And no conception can have greater power in our human world than atrueconception of the nature of Man. For that most important of truths the times are ripe; the world is filled with the saddest of memories, with gloom, forebodings and fear. Without the truth in this matter, there can be no rational hope—history must go on in its dismal course; butwiththe truth, there is not only hope but certitude that the old order has passed and that humanity's manhood dates from the present day. That I have here presented the truth in this matter—the true conception of the human class of life—I have personally no doubt; and I have no doubt that that conception is to be the base, the guide, the source of light, of a new civilization. Whether I am mistaken or not, time will decide. I feel as Buckle felt in writing hisHistory of Civilization:“Whether or not I have effected anything of real value ... is a question for competent judges to decide. Of this, at least, I feel certain, that whatever imperfections may be observed, the fault consists, not in the method proposed, but in the extreme difficulty of any single man putting into full operation all the parts of so vast a scheme. It is on this point, and on this alone, that I feel the need of great indulgence. But, as to the plan itself, I have no misgivings. Of defects in its execution I am not unconscious. I can only plead the immensity of the subject, the shortness of a single life and the imperfection of every single enterprise. I, therefore, wish this work to be estimated, not according to the[pg 206]finish of its separate parts, but according to the way in which those parts have been fused into a complete and symmetrical whole. This, in an undertaking of such novelty and magnitude, I have a right to expect, and I would moreover, add, that if the reader has met with opinions adverse to his own, he should remember, that his views are, perhaps, the same as those which I too once held, and which I have abandoned, because, after a wider range of study, I found them unsupported by solid proof, subversive of the interest of Man, and fatal to the progress of his knowledge. To examine the notions in which we have been educated, and to turn aside from those which will not bear the test, is a task so painful, that they who shrink from the sufferings should pause before they reproach those by whom the suffering is undergone.... Conclusions arrived at in this way are not to be overturned by stating that they endanger some other conclusions; nor can they be even affected by allegation against their supposed tendency. The principles which I advocate are based upon distinct arguments supported by well ascertained facts. The only points, therefore, to be ascertained, are, whether the arguments are fair, and whether the facts are certain. If these two conditions have been obeyed, the principles follow by an inevitable inference.”And why have I sought throughout to follow the spirit of mathematics? Because I have been dealing with ideas and have desired, above all things else, to be right and clear. Ideas have a character of their own—they are right or wrong independently of our hopes and passions and will. In the connection of ideas there is an unbreakable thread of destiny. That is why in hisMathematical PhilosophyProfessor Keyser has truly said:[pg 207]“Mathematics is the study of Fate—not fate in a physical sense, but in the sense of the binding thread that connects thought with thought and conclusions with their premises. Where, then, is our freedom? What do you love? Painting? Poetry? Music? The muses aretheirfates. Whoso loves them is free. Logic is the muse of Thought.”No doubt mathematics is truly impersonal in method; too impersonal maybe to please the sentimentalists before they take the time to think; mathematical analysis of life phenomena elevates our point of view above passion, above selfishness in any form, and, therefore, it is the only method which can tell us genuine truths about ourselves. Spinosa even in the 17th Century had well realized this fact and although imperfect in many ways, his was an effort in the right direction and this quoted conclusion may well be a conclusion for ourselves in the 20th century:“The truth might forever have remained hid from the human race, if mathematics, which looks not to the final cause of figures, but to their essential nature and the properties involved in it, had not set another type of knowledge before them.... When I turned my mind to this subject, I did not propose to myself any novel or strange aim, but simply to demonstrate by certain and indubitable reason, those things which agree best with practice. And in order that I might enquire into the matters of the science with the same freedom of mind with which we are wont to treat lines and surfaces in mathematics; I determined not to laugh or to weep over the actions of men, but simply to understand them; and to contemplate their affections and passions, such as love, hate, anger, envy, arrogance, pity and all other disturbances[pg 208]of soul not as vices of human nature, but as properties pertaining to it in the same way as heat, cold, storm, thunder pertain to the nature of the atmosphere. For these, though troublesome, are yet necessary, and have certain causes through which we may come to understand them, and thus, by contemplating them in their truth, gain for our minds much joy as by the knowledge of things that are pleasing to the senses.”If only this little book willinitiatethe scientific study of Man, I shall be happy; for then we may confidently expect a science and art that will know how to direct the energies of man to the advancement of human weal.What else? Many topics have not even been broached. Time-binding energy—what may it not achieve in course of the aeons to come? What light may it not yet throw upon such fundamental phenomena asSpace,Time,Infinity, and so on? What, if any, are the limits of Time-binding? In it are somehow involved all the higher functions of mind. Is Time identical with Intelligence? Is either of them the other's cause? Is Timeinthe Cosmos or is the latter in the former? Is the Cosmos intelligent? Many no doubt and marvelous are the fields which the scientific study of man will open for research.[pg 209]Appendix I. Mathematics And Time-BindingThe 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 ofscientific 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[pg 210]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 waysupernatural orsuperphysical, but belongs rather to an unknown or an undeveloped branch of physics. The problem, therefore, may be not that of somenewscience, 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 innaturehas 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[pg 211]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 itspositionat 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 specialhumanizedmathematics. 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-bindingenergyis 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 conceptTimeenters, 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[pg 212]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 thattimeexists 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,”butpsychological 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, doesnotworkpsychologicallybut worksabstractly: 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[pg 213]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, shapingcorrectlythe product of its activity, but whethertrulyis 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-binding involved in it, no matter whattimeis (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 putpsychologicalaxioms into the time-binding apparatus, it will thrash out the resultscorrectly, but whether the results aretrueis 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 (α1, α2, ... αn)(2) Scientific truth (αs), 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,[pg 214](3) The absolute truth, which will be thefinal definitionof a phenomenon based upon the final knowledge ofprimal causation valid in infinity.For simplicity's sake I will use the signs α1, α2, ... αnfor the“psychological,”“private,”or“relative”truths, between which, for the moment, I will not discriminate.αs1, αs2, ... αsn, will be used for scientific truths, and finally αinfinityfor 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 theorganicchemistry correspond to radium in theinorganicchemistry; 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 theMrays of the time-binding energy; the“spiritual”capacities, theArays; the“will”powers, theBrays; and so on. Psychological truths will then be a function of all rays together, namelyABC...M... orf(ABC...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”—theMrays—and have a complete abstraction (which in the present could only be in mathematics), then the work ofMcould be compared to the work of an impersonal machine which always gives the samecorrectlyshaped productno matter what isthe 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 αinfinitytype but are of thef(ABC...) personal type and[pg 215]this may be why mathematics cannot account for psychological facts. If psychology is to be anexact scienceit 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. Thebaseof mathematics wasf(ABC...M...); thework, or the development, of mathematics isf(M); this is the reason for the“ghosts”in the background of mathematics. Thef(M) evolved from thisf(ABC...M...)basea 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 firstfiniteor“normal number”; then the old finites or the normal numbers would become very great numbers and the old very-great[pg 216]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 thestartwould 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 andtruein 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 commontangiblebase forpsychologicalas well asanalyticaltruths.This new philosophic mathematics would eliminate the concept of“infinitesimals”as such, which is anartificialconcept and is not as aconceptan element of Nature. The so-calledinfinitesimals are Nature's real, natural finites. In mathematics the infinitesimals were an analytical—an“M”—time-binding—necessity,[pg 217]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 alsotrueresults. Keeping in mindbothconceptions 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 afunctionalname 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 considerfactsalone. 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 wherecorrectformulas 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 growthy==ekt, with the formula“P RT,”we see that they are of the same type and thelaw of organic growthapplies to the humantime-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.[pg 218]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 toproduce, 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 ofchemicalenergy 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 thetransformation of organic atoms, and—what is very characteristic—the results aretime-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[pg 219]“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 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[pg 220]to make that effort toward clear thinking (which in the meantime will remain of thehighestimportance 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. 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.[pg 221]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. 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:Poincaré, Henri.“The Foundations of Science.”The Science Press, N. Y., 1913.Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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:Keyser, Cassius J.“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 rôle 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[pg 223]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 rôle 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.[pg 224]

Chapter X. Conclusion“In Europe we know that an age is dying. Here it would be easy to miss the signs of coming changes, but I have little doubt that it will come. A realization of theaimlessnessof life lived to labor and to die, having achieved nothing but avoidance of starvation, and the birth of children also doomed to the weary treadmill, has seized the minds of millions.”Sir Auckland Geddes, British Ambassador to the U. S. 1920.In conclusion let me say very briefly, as I said in the beginning, that this little book has aimed to be only a sketch. The Problem of Life is old. I have endeavored to approach it afresh, with a new method, in a new spirit, from a new point of view. The literature of the subject is vast. It displays great knowledge and skill. Much of it is fitted to inform and to inspire such as really read with a genuine desire to understand. Its weakness is due to the absence of a true conception of what human beings are. That is what I miss in it and it is that lack of fundamental and central thought that I have striven to supply. If I have succeeded in that, I have no fear—all else will follow quickly, inevitably, as a matter of course. For a fundamental conception, once it is formed and expressed, has a strange power—the power of enlisting the thought and cooperation[pg 205]of many minds. And no conception can have greater power in our human world than atrueconception of the nature of Man. For that most important of truths the times are ripe; the world is filled with the saddest of memories, with gloom, forebodings and fear. Without the truth in this matter, there can be no rational hope—history must go on in its dismal course; butwiththe truth, there is not only hope but certitude that the old order has passed and that humanity's manhood dates from the present day. That I have here presented the truth in this matter—the true conception of the human class of life—I have personally no doubt; and I have no doubt that that conception is to be the base, the guide, the source of light, of a new civilization. Whether I am mistaken or not, time will decide. I feel as Buckle felt in writing hisHistory of Civilization:“Whether or not I have effected anything of real value ... is a question for competent judges to decide. Of this, at least, I feel certain, that whatever imperfections may be observed, the fault consists, not in the method proposed, but in the extreme difficulty of any single man putting into full operation all the parts of so vast a scheme. It is on this point, and on this alone, that I feel the need of great indulgence. But, as to the plan itself, I have no misgivings. Of defects in its execution I am not unconscious. I can only plead the immensity of the subject, the shortness of a single life and the imperfection of every single enterprise. I, therefore, wish this work to be estimated, not according to the[pg 206]finish of its separate parts, but according to the way in which those parts have been fused into a complete and symmetrical whole. This, in an undertaking of such novelty and magnitude, I have a right to expect, and I would moreover, add, that if the reader has met with opinions adverse to his own, he should remember, that his views are, perhaps, the same as those which I too once held, and which I have abandoned, because, after a wider range of study, I found them unsupported by solid proof, subversive of the interest of Man, and fatal to the progress of his knowledge. To examine the notions in which we have been educated, and to turn aside from those which will not bear the test, is a task so painful, that they who shrink from the sufferings should pause before they reproach those by whom the suffering is undergone.... Conclusions arrived at in this way are not to be overturned by stating that they endanger some other conclusions; nor can they be even affected by allegation against their supposed tendency. The principles which I advocate are based upon distinct arguments supported by well ascertained facts. The only points, therefore, to be ascertained, are, whether the arguments are fair, and whether the facts are certain. If these two conditions have been obeyed, the principles follow by an inevitable inference.”And why have I sought throughout to follow the spirit of mathematics? Because I have been dealing with ideas and have desired, above all things else, to be right and clear. Ideas have a character of their own—they are right or wrong independently of our hopes and passions and will. In the connection of ideas there is an unbreakable thread of destiny. That is why in hisMathematical PhilosophyProfessor Keyser has truly said:[pg 207]“Mathematics is the study of Fate—not fate in a physical sense, but in the sense of the binding thread that connects thought with thought and conclusions with their premises. Where, then, is our freedom? What do you love? Painting? Poetry? Music? The muses aretheirfates. Whoso loves them is free. Logic is the muse of Thought.”No doubt mathematics is truly impersonal in method; too impersonal maybe to please the sentimentalists before they take the time to think; mathematical analysis of life phenomena elevates our point of view above passion, above selfishness in any form, and, therefore, it is the only method which can tell us genuine truths about ourselves. Spinosa even in the 17th Century had well realized this fact and although imperfect in many ways, his was an effort in the right direction and this quoted conclusion may well be a conclusion for ourselves in the 20th century:“The truth might forever have remained hid from the human race, if mathematics, which looks not to the final cause of figures, but to their essential nature and the properties involved in it, had not set another type of knowledge before them.... When I turned my mind to this subject, I did not propose to myself any novel or strange aim, but simply to demonstrate by certain and indubitable reason, those things which agree best with practice. And in order that I might enquire into the matters of the science with the same freedom of mind with which we are wont to treat lines and surfaces in mathematics; I determined not to laugh or to weep over the actions of men, but simply to understand them; and to contemplate their affections and passions, such as love, hate, anger, envy, arrogance, pity and all other disturbances[pg 208]of soul not as vices of human nature, but as properties pertaining to it in the same way as heat, cold, storm, thunder pertain to the nature of the atmosphere. For these, though troublesome, are yet necessary, and have certain causes through which we may come to understand them, and thus, by contemplating them in their truth, gain for our minds much joy as by the knowledge of things that are pleasing to the senses.”If only this little book willinitiatethe scientific study of Man, I shall be happy; for then we may confidently expect a science and art that will know how to direct the energies of man to the advancement of human weal.What else? Many topics have not even been broached. Time-binding energy—what may it not achieve in course of the aeons to come? What light may it not yet throw upon such fundamental phenomena asSpace,Time,Infinity, and so on? What, if any, are the limits of Time-binding? In it are somehow involved all the higher functions of mind. Is Time identical with Intelligence? Is either of them the other's cause? Is Timeinthe Cosmos or is the latter in the former? Is the Cosmos intelligent? Many no doubt and marvelous are the fields which the scientific study of man will open for research.[pg 209]Appendix I. Mathematics And Time-BindingThe 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 ofscientific 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[pg 210]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 waysupernatural orsuperphysical, but belongs rather to an unknown or an undeveloped branch of physics. The problem, therefore, may be not that of somenewscience, 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 innaturehas 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[pg 211]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 itspositionat 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 specialhumanizedmathematics. 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-bindingenergyis 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 conceptTimeenters, 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[pg 212]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 thattimeexists 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,”butpsychological 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, doesnotworkpsychologicallybut worksabstractly: 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[pg 213]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, shapingcorrectlythe product of its activity, but whethertrulyis 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-binding involved in it, no matter whattimeis (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 putpsychologicalaxioms into the time-binding apparatus, it will thrash out the resultscorrectly, but whether the results aretrueis 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 (α1, α2, ... αn)(2) Scientific truth (αs), 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,[pg 214](3) The absolute truth, which will be thefinal definitionof a phenomenon based upon the final knowledge ofprimal causation valid in infinity.For simplicity's sake I will use the signs α1, α2, ... αnfor the“psychological,”“private,”or“relative”truths, between which, for the moment, I will not discriminate.αs1, αs2, ... αsn, will be used for scientific truths, and finally αinfinityfor 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 theorganicchemistry correspond to radium in theinorganicchemistry; 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 theMrays of the time-binding energy; the“spiritual”capacities, theArays; the“will”powers, theBrays; and so on. Psychological truths will then be a function of all rays together, namelyABC...M... orf(ABC...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”—theMrays—and have a complete abstraction (which in the present could only be in mathematics), then the work ofMcould be compared to the work of an impersonal machine which always gives the samecorrectlyshaped productno matter what isthe 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 αinfinitytype but are of thef(ABC...) personal type and[pg 215]this may be why mathematics cannot account for psychological facts. If psychology is to be anexact scienceit 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. Thebaseof mathematics wasf(ABC...M...); thework, or the development, of mathematics isf(M); this is the reason for the“ghosts”in the background of mathematics. Thef(M) evolved from thisf(ABC...M...)basea 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 firstfiniteor“normal number”; then the old finites or the normal numbers would become very great numbers and the old very-great[pg 216]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 thestartwould 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 andtruein 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 commontangiblebase forpsychologicalas well asanalyticaltruths.This new philosophic mathematics would eliminate the concept of“infinitesimals”as such, which is anartificialconcept and is not as aconceptan element of Nature. The so-calledinfinitesimals are Nature's real, natural finites. In mathematics the infinitesimals were an analytical—an“M”—time-binding—necessity,[pg 217]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 alsotrueresults. Keeping in mindbothconceptions 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 afunctionalname 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 considerfactsalone. 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 wherecorrectformulas 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 growthy==ekt, with the formula“P RT,”we see that they are of the same type and thelaw of organic growthapplies to the humantime-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.[pg 218]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 toproduce, 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 ofchemicalenergy 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 thetransformation of organic atoms, and—what is very characteristic—the results aretime-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[pg 219]“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 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[pg 220]to make that effort toward clear thinking (which in the meantime will remain of thehighestimportance 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. 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.[pg 221]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. 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:Poincaré, Henri.“The Foundations of Science.”The Science Press, N. Y., 1913.Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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:Keyser, Cassius J.“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 rôle 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[pg 223]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 rôle 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.[pg 224]

Chapter X. Conclusion“In Europe we know that an age is dying. Here it would be easy to miss the signs of coming changes, but I have little doubt that it will come. A realization of theaimlessnessof life lived to labor and to die, having achieved nothing but avoidance of starvation, and the birth of children also doomed to the weary treadmill, has seized the minds of millions.”Sir Auckland Geddes, British Ambassador to the U. S. 1920.In conclusion let me say very briefly, as I said in the beginning, that this little book has aimed to be only a sketch. The Problem of Life is old. I have endeavored to approach it afresh, with a new method, in a new spirit, from a new point of view. The literature of the subject is vast. It displays great knowledge and skill. Much of it is fitted to inform and to inspire such as really read with a genuine desire to understand. Its weakness is due to the absence of a true conception of what human beings are. That is what I miss in it and it is that lack of fundamental and central thought that I have striven to supply. If I have succeeded in that, I have no fear—all else will follow quickly, inevitably, as a matter of course. For a fundamental conception, once it is formed and expressed, has a strange power—the power of enlisting the thought and cooperation[pg 205]of many minds. And no conception can have greater power in our human world than atrueconception of the nature of Man. For that most important of truths the times are ripe; the world is filled with the saddest of memories, with gloom, forebodings and fear. Without the truth in this matter, there can be no rational hope—history must go on in its dismal course; butwiththe truth, there is not only hope but certitude that the old order has passed and that humanity's manhood dates from the present day. That I have here presented the truth in this matter—the true conception of the human class of life—I have personally no doubt; and I have no doubt that that conception is to be the base, the guide, the source of light, of a new civilization. Whether I am mistaken or not, time will decide. I feel as Buckle felt in writing hisHistory of Civilization:“Whether or not I have effected anything of real value ... is a question for competent judges to decide. Of this, at least, I feel certain, that whatever imperfections may be observed, the fault consists, not in the method proposed, but in the extreme difficulty of any single man putting into full operation all the parts of so vast a scheme. It is on this point, and on this alone, that I feel the need of great indulgence. But, as to the plan itself, I have no misgivings. Of defects in its execution I am not unconscious. I can only plead the immensity of the subject, the shortness of a single life and the imperfection of every single enterprise. I, therefore, wish this work to be estimated, not according to the[pg 206]finish of its separate parts, but according to the way in which those parts have been fused into a complete and symmetrical whole. This, in an undertaking of such novelty and magnitude, I have a right to expect, and I would moreover, add, that if the reader has met with opinions adverse to his own, he should remember, that his views are, perhaps, the same as those which I too once held, and which I have abandoned, because, after a wider range of study, I found them unsupported by solid proof, subversive of the interest of Man, and fatal to the progress of his knowledge. To examine the notions in which we have been educated, and to turn aside from those which will not bear the test, is a task so painful, that they who shrink from the sufferings should pause before they reproach those by whom the suffering is undergone.... Conclusions arrived at in this way are not to be overturned by stating that they endanger some other conclusions; nor can they be even affected by allegation against their supposed tendency. The principles which I advocate are based upon distinct arguments supported by well ascertained facts. The only points, therefore, to be ascertained, are, whether the arguments are fair, and whether the facts are certain. If these two conditions have been obeyed, the principles follow by an inevitable inference.”And why have I sought throughout to follow the spirit of mathematics? Because I have been dealing with ideas and have desired, above all things else, to be right and clear. Ideas have a character of their own—they are right or wrong independently of our hopes and passions and will. In the connection of ideas there is an unbreakable thread of destiny. That is why in hisMathematical PhilosophyProfessor Keyser has truly said:[pg 207]“Mathematics is the study of Fate—not fate in a physical sense, but in the sense of the binding thread that connects thought with thought and conclusions with their premises. Where, then, is our freedom? What do you love? Painting? Poetry? Music? The muses aretheirfates. Whoso loves them is free. Logic is the muse of Thought.”No doubt mathematics is truly impersonal in method; too impersonal maybe to please the sentimentalists before they take the time to think; mathematical analysis of life phenomena elevates our point of view above passion, above selfishness in any form, and, therefore, it is the only method which can tell us genuine truths about ourselves. Spinosa even in the 17th Century had well realized this fact and although imperfect in many ways, his was an effort in the right direction and this quoted conclusion may well be a conclusion for ourselves in the 20th century:“The truth might forever have remained hid from the human race, if mathematics, which looks not to the final cause of figures, but to their essential nature and the properties involved in it, had not set another type of knowledge before them.... When I turned my mind to this subject, I did not propose to myself any novel or strange aim, but simply to demonstrate by certain and indubitable reason, those things which agree best with practice. And in order that I might enquire into the matters of the science with the same freedom of mind with which we are wont to treat lines and surfaces in mathematics; I determined not to laugh or to weep over the actions of men, but simply to understand them; and to contemplate their affections and passions, such as love, hate, anger, envy, arrogance, pity and all other disturbances[pg 208]of soul not as vices of human nature, but as properties pertaining to it in the same way as heat, cold, storm, thunder pertain to the nature of the atmosphere. For these, though troublesome, are yet necessary, and have certain causes through which we may come to understand them, and thus, by contemplating them in their truth, gain for our minds much joy as by the knowledge of things that are pleasing to the senses.”If only this little book willinitiatethe scientific study of Man, I shall be happy; for then we may confidently expect a science and art that will know how to direct the energies of man to the advancement of human weal.What else? Many topics have not even been broached. Time-binding energy—what may it not achieve in course of the aeons to come? What light may it not yet throw upon such fundamental phenomena asSpace,Time,Infinity, and so on? What, if any, are the limits of Time-binding? In it are somehow involved all the higher functions of mind. Is Time identical with Intelligence? Is either of them the other's cause? Is Timeinthe Cosmos or is the latter in the former? Is the Cosmos intelligent? Many no doubt and marvelous are the fields which the scientific study of man will open for research.

“In Europe we know that an age is dying. Here it would be easy to miss the signs of coming changes, but I have little doubt that it will come. A realization of theaimlessnessof life lived to labor and to die, having achieved nothing but avoidance of starvation, and the birth of children also doomed to the weary treadmill, has seized the minds of millions.”Sir Auckland Geddes, British Ambassador to the U. S. 1920.

“In Europe we know that an age is dying. Here it would be easy to miss the signs of coming changes, but I have little doubt that it will come. A realization of theaimlessnessof life lived to labor and to die, having achieved nothing but avoidance of starvation, and the birth of children also doomed to the weary treadmill, has seized the minds of millions.”Sir Auckland Geddes, British Ambassador to the U. S. 1920.

In conclusion let me say very briefly, as I said in the beginning, that this little book has aimed to be only a sketch. The Problem of Life is old. I have endeavored to approach it afresh, with a new method, in a new spirit, from a new point of view. The literature of the subject is vast. It displays great knowledge and skill. Much of it is fitted to inform and to inspire such as really read with a genuine desire to understand. Its weakness is due to the absence of a true conception of what human beings are. That is what I miss in it and it is that lack of fundamental and central thought that I have striven to supply. If I have succeeded in that, I have no fear—all else will follow quickly, inevitably, as a matter of course. For a fundamental conception, once it is formed and expressed, has a strange power—the power of enlisting the thought and cooperation[pg 205]of many minds. And no conception can have greater power in our human world than atrueconception of the nature of Man. For that most important of truths the times are ripe; the world is filled with the saddest of memories, with gloom, forebodings and fear. Without the truth in this matter, there can be no rational hope—history must go on in its dismal course; butwiththe truth, there is not only hope but certitude that the old order has passed and that humanity's manhood dates from the present day. That I have here presented the truth in this matter—the true conception of the human class of life—I have personally no doubt; and I have no doubt that that conception is to be the base, the guide, the source of light, of a new civilization. Whether I am mistaken or not, time will decide. I feel as Buckle felt in writing hisHistory of Civilization:

“Whether or not I have effected anything of real value ... is a question for competent judges to decide. Of this, at least, I feel certain, that whatever imperfections may be observed, the fault consists, not in the method proposed, but in the extreme difficulty of any single man putting into full operation all the parts of so vast a scheme. It is on this point, and on this alone, that I feel the need of great indulgence. But, as to the plan itself, I have no misgivings. Of defects in its execution I am not unconscious. I can only plead the immensity of the subject, the shortness of a single life and the imperfection of every single enterprise. I, therefore, wish this work to be estimated, not according to the[pg 206]finish of its separate parts, but according to the way in which those parts have been fused into a complete and symmetrical whole. This, in an undertaking of such novelty and magnitude, I have a right to expect, and I would moreover, add, that if the reader has met with opinions adverse to his own, he should remember, that his views are, perhaps, the same as those which I too once held, and which I have abandoned, because, after a wider range of study, I found them unsupported by solid proof, subversive of the interest of Man, and fatal to the progress of his knowledge. To examine the notions in which we have been educated, and to turn aside from those which will not bear the test, is a task so painful, that they who shrink from the sufferings should pause before they reproach those by whom the suffering is undergone.... Conclusions arrived at in this way are not to be overturned by stating that they endanger some other conclusions; nor can they be even affected by allegation against their supposed tendency. The principles which I advocate are based upon distinct arguments supported by well ascertained facts. The only points, therefore, to be ascertained, are, whether the arguments are fair, and whether the facts are certain. If these two conditions have been obeyed, the principles follow by an inevitable inference.”

“Whether or not I have effected anything of real value ... is a question for competent judges to decide. Of this, at least, I feel certain, that whatever imperfections may be observed, the fault consists, not in the method proposed, but in the extreme difficulty of any single man putting into full operation all the parts of so vast a scheme. It is on this point, and on this alone, that I feel the need of great indulgence. But, as to the plan itself, I have no misgivings. Of defects in its execution I am not unconscious. I can only plead the immensity of the subject, the shortness of a single life and the imperfection of every single enterprise. I, therefore, wish this work to be estimated, not according to the[pg 206]finish of its separate parts, but according to the way in which those parts have been fused into a complete and symmetrical whole. This, in an undertaking of such novelty and magnitude, I have a right to expect, and I would moreover, add, that if the reader has met with opinions adverse to his own, he should remember, that his views are, perhaps, the same as those which I too once held, and which I have abandoned, because, after a wider range of study, I found them unsupported by solid proof, subversive of the interest of Man, and fatal to the progress of his knowledge. To examine the notions in which we have been educated, and to turn aside from those which will not bear the test, is a task so painful, that they who shrink from the sufferings should pause before they reproach those by whom the suffering is undergone.... Conclusions arrived at in this way are not to be overturned by stating that they endanger some other conclusions; nor can they be even affected by allegation against their supposed tendency. The principles which I advocate are based upon distinct arguments supported by well ascertained facts. The only points, therefore, to be ascertained, are, whether the arguments are fair, and whether the facts are certain. If these two conditions have been obeyed, the principles follow by an inevitable inference.”

And why have I sought throughout to follow the spirit of mathematics? Because I have been dealing with ideas and have desired, above all things else, to be right and clear. Ideas have a character of their own—they are right or wrong independently of our hopes and passions and will. In the connection of ideas there is an unbreakable thread of destiny. That is why in hisMathematical PhilosophyProfessor Keyser has truly said:

“Mathematics is the study of Fate—not fate in a physical sense, but in the sense of the binding thread that connects thought with thought and conclusions with their premises. Where, then, is our freedom? What do you love? Painting? Poetry? Music? The muses aretheirfates. Whoso loves them is free. Logic is the muse of Thought.”

“Mathematics is the study of Fate—not fate in a physical sense, but in the sense of the binding thread that connects thought with thought and conclusions with their premises. Where, then, is our freedom? What do you love? Painting? Poetry? Music? The muses aretheirfates. Whoso loves them is free. Logic is the muse of Thought.”

No doubt mathematics is truly impersonal in method; too impersonal maybe to please the sentimentalists before they take the time to think; mathematical analysis of life phenomena elevates our point of view above passion, above selfishness in any form, and, therefore, it is the only method which can tell us genuine truths about ourselves. Spinosa even in the 17th Century had well realized this fact and although imperfect in many ways, his was an effort in the right direction and this quoted conclusion may well be a conclusion for ourselves in the 20th century:

“The truth might forever have remained hid from the human race, if mathematics, which looks not to the final cause of figures, but to their essential nature and the properties involved in it, had not set another type of knowledge before them.... When I turned my mind to this subject, I did not propose to myself any novel or strange aim, but simply to demonstrate by certain and indubitable reason, those things which agree best with practice. And in order that I might enquire into the matters of the science with the same freedom of mind with which we are wont to treat lines and surfaces in mathematics; I determined not to laugh or to weep over the actions of men, but simply to understand them; and to contemplate their affections and passions, such as love, hate, anger, envy, arrogance, pity and all other disturbances[pg 208]of soul not as vices of human nature, but as properties pertaining to it in the same way as heat, cold, storm, thunder pertain to the nature of the atmosphere. For these, though troublesome, are yet necessary, and have certain causes through which we may come to understand them, and thus, by contemplating them in their truth, gain for our minds much joy as by the knowledge of things that are pleasing to the senses.”

“The truth might forever have remained hid from the human race, if mathematics, which looks not to the final cause of figures, but to their essential nature and the properties involved in it, had not set another type of knowledge before them.... When I turned my mind to this subject, I did not propose to myself any novel or strange aim, but simply to demonstrate by certain and indubitable reason, those things which agree best with practice. And in order that I might enquire into the matters of the science with the same freedom of mind with which we are wont to treat lines and surfaces in mathematics; I determined not to laugh or to weep over the actions of men, but simply to understand them; and to contemplate their affections and passions, such as love, hate, anger, envy, arrogance, pity and all other disturbances[pg 208]of soul not as vices of human nature, but as properties pertaining to it in the same way as heat, cold, storm, thunder pertain to the nature of the atmosphere. For these, though troublesome, are yet necessary, and have certain causes through which we may come to understand them, and thus, by contemplating them in their truth, gain for our minds much joy as by the knowledge of things that are pleasing to the senses.”

If only this little book willinitiatethe scientific study of Man, I shall be happy; for then we may confidently expect a science and art that will know how to direct the energies of man to the advancement of human weal.

What else? Many topics have not even been broached. Time-binding energy—what may it not achieve in course of the aeons to come? What light may it not yet throw upon such fundamental phenomena asSpace,Time,Infinity, and so on? What, if any, are the limits of Time-binding? In it are somehow involved all the higher functions of mind. Is Time identical with Intelligence? Is either of them the other's cause? Is Timeinthe Cosmos or is the latter in the former? Is the Cosmos intelligent? Many no doubt and marvelous are the fields which the scientific study of man will open for research.

Appendix I. Mathematics And Time-BindingThe 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 ofscientific 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[pg 210]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 waysupernatural orsuperphysical, but belongs rather to an unknown or an undeveloped branch of physics. The problem, therefore, may be not that of somenewscience, 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 innaturehas 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[pg 211]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 itspositionat 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 specialhumanizedmathematics. 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-bindingenergyis 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 conceptTimeenters, 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[pg 212]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 thattimeexists 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,”butpsychological 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, doesnotworkpsychologicallybut worksabstractly: 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[pg 213]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, shapingcorrectlythe product of its activity, but whethertrulyis 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-binding involved in it, no matter whattimeis (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 putpsychologicalaxioms into the time-binding apparatus, it will thrash out the resultscorrectly, but whether the results aretrueis 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 (α1, α2, ... αn)(2) Scientific truth (αs), 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,[pg 214](3) The absolute truth, which will be thefinal definitionof a phenomenon based upon the final knowledge ofprimal causation valid in infinity.For simplicity's sake I will use the signs α1, α2, ... αnfor the“psychological,”“private,”or“relative”truths, between which, for the moment, I will not discriminate.αs1, αs2, ... αsn, will be used for scientific truths, and finally αinfinityfor 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 theorganicchemistry correspond to radium in theinorganicchemistry; 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 theMrays of the time-binding energy; the“spiritual”capacities, theArays; the“will”powers, theBrays; and so on. Psychological truths will then be a function of all rays together, namelyABC...M... orf(ABC...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”—theMrays—and have a complete abstraction (which in the present could only be in mathematics), then the work ofMcould be compared to the work of an impersonal machine which always gives the samecorrectlyshaped productno matter what isthe 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 αinfinitytype but are of thef(ABC...) personal type and[pg 215]this may be why mathematics cannot account for psychological facts. If psychology is to be anexact scienceit 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. Thebaseof mathematics wasf(ABC...M...); thework, or the development, of mathematics isf(M); this is the reason for the“ghosts”in the background of mathematics. Thef(M) evolved from thisf(ABC...M...)basea 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 firstfiniteor“normal number”; then the old finites or the normal numbers would become very great numbers and the old very-great[pg 216]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 thestartwould 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 andtruein 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 commontangiblebase forpsychologicalas well asanalyticaltruths.This new philosophic mathematics would eliminate the concept of“infinitesimals”as such, which is anartificialconcept and is not as aconceptan element of Nature. The so-calledinfinitesimals are Nature's real, natural finites. In mathematics the infinitesimals were an analytical—an“M”—time-binding—necessity,[pg 217]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 alsotrueresults. Keeping in mindbothconceptions 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 afunctionalname 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 considerfactsalone. 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 wherecorrectformulas 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 growthy==ekt, with the formula“P RT,”we see that they are of the same type and thelaw of organic growthapplies to the humantime-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.[pg 218]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 toproduce, 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 ofchemicalenergy 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 thetransformation of organic atoms, and—what is very characteristic—the results aretime-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[pg 219]“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 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[pg 220]to make that effort toward clear thinking (which in the meantime will remain of thehighestimportance 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. 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.[pg 221]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. 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:Poincaré, Henri.“The Foundations of Science.”The Science Press, N. Y., 1913.Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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:Keyser, Cassius J.“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 rôle 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[pg 223]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 rôle 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.

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 ofscientific 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[pg 210]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 waysupernatural orsuperphysical, but belongs rather to an unknown or an undeveloped branch of physics. The problem, therefore, may be not that of somenewscience, 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 innaturehas 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[pg 211]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 itspositionat 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 specialhumanizedmathematics. 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-bindingenergyis 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 conceptTimeenters, 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[pg 212]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 thattimeexists 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,”butpsychological 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, doesnotworkpsychologicallybut worksabstractly: 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[pg 213]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, shapingcorrectlythe product of its activity, but whethertrulyis 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-binding involved in it, no matter whattimeis (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 putpsychologicalaxioms into the time-binding apparatus, it will thrash out the resultscorrectly, but whether the results aretrueis 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 (α1, α2, ... αn)

(2) Scientific truth (αs), 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 thefinal definitionof a phenomenon based upon the final knowledge ofprimal causation valid in infinity.

For simplicity's sake I will use the signs α1, α2, ... αnfor the“psychological,”“private,”or“relative”truths, between which, for the moment, I will not discriminate.

αs1, αs2, ... αsn, will be used for scientific truths, and finally αinfinityfor 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 theorganicchemistry correspond to radium in theinorganicchemistry; 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 theMrays of the time-binding energy; the“spiritual”capacities, theArays; the“will”powers, theBrays; and so on. Psychological truths will then be a function of all rays together, namelyABC...M... orf(ABC...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”—theMrays—and have a complete abstraction (which in the present could only be in mathematics), then the work ofMcould be compared to the work of an impersonal machine which always gives the samecorrectlyshaped productno matter what isthe 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 αinfinitytype but are of thef(ABC...) personal type and[pg 215]this may be why mathematics cannot account for psychological facts. If psychology is to be anexact scienceit 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. Thebaseof mathematics wasf(ABC...M...); thework, or the development, of mathematics isf(M); this is the reason for the“ghosts”in the background of mathematics. Thef(M) evolved from thisf(ABC...M...)basea 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 firstfiniteor“normal number”; then the old finites or the normal numbers would become very great numbers and the old very-great[pg 216]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 thestartwould 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 andtruein 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 commontangiblebase forpsychologicalas well asanalyticaltruths.

This new philosophic mathematics would eliminate the concept of“infinitesimals”as such, which is anartificialconcept and is not as aconceptan element of Nature. The so-calledinfinitesimals are Nature's real, natural finites. In mathematics the infinitesimals were an analytical—an“M”—time-binding—necessity,[pg 217]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 alsotrueresults. Keeping in mindbothconceptions 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 afunctionalname 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 considerfactsalone. 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 wherecorrectformulas 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 growthy==ekt, with the formula“P RT,”we see that they are of the same type and thelaw of organic growthapplies to the humantime-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.[pg 218]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 toproduce, 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 ofchemicalenergy 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 thetransformation of organic atoms, and—what is very characteristic—the results aretime-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[pg 219]“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 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[pg 220]to make that effort toward clear thinking (which in the meantime will remain of thehighestimportance 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. 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.[pg 221]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. 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.

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. 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.

“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.

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. 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.

“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:Poincaré, Henri.“The Foundations of Science.”The Science Press, N. Y., 1913.Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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.

Poincaré, Henri.“The Foundations of Science.”The Science Press, N. Y., 1913.Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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.

“The Foundations of Science.”The Science Press, N. Y., 1913.

Selection from contents: Science and hypothesis. Number and magnitude.[pg 222]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:Keyser, Cassius J.“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 rôle 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[pg 223]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 rôle 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.

Keyser, Cassius J.“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 rôle 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[pg 223]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 rôle 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.

“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 rôle 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[pg 223]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 rôle 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.

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