EVERY now and then there appear in the history of humanity gigantic thinkers who shape and mold our mental processes for centuries to come. In our own time we are witnessing such a turning of the page in human history. The birth of a new era is upon us; a host of men in all walks of life feel it unconsciously and work toward it. A few leading mathematicians have made these unconscious strivings of mankind conscious—without them we would feel our way but in the darkness, which is a slow, very slow process of guess work, whereas with their work our path is clear.
I hope the reader will understand the inherent difficulties which beset any attempt to give a general summary of a new epoch which is still making its own foundations.
In the space allotted for this writing only a very few of the most momentous points can be sketched, and I make no pretense to finality. The aim is to draw the attention of scientists and thinkers to the fact that something of grave importance for all our human future is going on, and to encourage inquiry and collaboration, thus accelerating the inevitable.
What I here call the inevitable is the coming of the empire of sound logic—a logic demanding scientific knowledge of human nature, adjusting human beliefs, institutions, doctrines and conduct to the essential facts and laws of human nature, and converting the pseudo-sciences of ethics, economics and government into genuine sciences for promoting human welfare.
The "Brotherhood of Man," of which we all dream, can be accomplished only and exclusively by the "Brotherhood of Doctrines."
It will be found that when what Professor Cassius Keyser calls the "Great Stupidity" has been eliminated by sound logic, all that is dismal, destructive, woeful and despairing will become constructive, hopeful and favorable to human weal.
Such an inquiry will show that there still persist many doctrines originally established by myth and magic; and, although at the first
The history of human thought may be roughly divided into three periods, each period having gradually evolved from its predecessor. The beginning of one period overlaps the other. As a base for my classification I shall take the relationship between the observer and the observed. This relationship is clearly fundamental because there can be no "observer" without something to observe, and also no "observed" without somebody making the observation. To put it otherwise there is no such thing as a "fact" free from the share of the observer's mind. In speaking about these periods I shall not take into account individual thinkers, because in many instances it may be found that certain thinkers (Plato, Lucretius, Leibnitz, etc.) in a given period were far ahead of their contemporaries, and that their theories or discoveries which had no great influence in their own time were prophetic expressions of the latest development of science, therefore I shall only speak summarily about those currents of thought which have immediately affected the fate of our "common humanity."
The first period may be called the Greek, or Metaphysical, or Pre-Scientific Period. In this period the observer was everything, the observed did not matter.
The second period may be called the Classical or Semi-Scientific—still reigning in most fields—where the observer was almost nothing and the only thing that mattered was the observed. This tendency gave rise to that which we may call gross empiricism and gross materialism.
The third period may be called the Mathematical, or Scientific Period. It began in 1854 with George Boole's The Laws of Thought. This work started an internal revolution in logic and also in mathematics which ultimately resulted in the last few years in the merging of both—the discovery that logic and mathematics are one. In this period mankind will understand (some understand it already) that all that man can know is a joint phenomenon of the observer and the observed.
We might otherwise call the three periods:
(1) The Absolutist Period.
(2) The Confused Absolutist-Relativist Period.
(3) The Relativist Period.
The general characteristics of the first two periods was that they both used traditional, insufficient, and often fallacious subject-predicate, Aristotelian logic which must result as it did, in a philosophical impasse. The confusion became so acute that hardly any
The Old Logic Hampers Everything
It may be proved also that the direct result of this faulty logic has hampered enormously the natural sciences and progress in all fields of human affairs. The history of mankind, despite all the beauty and culture in it, has been in greater measure a history of misery and periodical collapses, wars and revolutions.
The old complete, consistent "absolutism" leads obviously to blind fanatical theories. The mixture of absolute and relative concepts and words leads to confusion and bewildering paradoxes. Consistent "relativism" clarifies this whole hopeless mess and probably will lead toward some "absolute"—if such a thing exists.
In the new mathematical-scientific era the simple truth has been discovered that all we know is a joint phenomenon of the observer and the observed, which means that for science and life logic is as vital a factor as "facts" because, for human knowledge, there are no "facts" free from the share of the observer's mind.
General truths cannot be established by gross empiricism because it deals only and exclusively with particular observations, and this is why the orthodox tradition led automatically to doubt and unwarranted pessimism, so characteristic of that period. Obviously if there is such a thing as general knowledge, its foundation must be found outside of gross empiricism. Most probably such a thing does exist and its origin may be traced to the constitution of the human mind itself—to sound modern logic (mathematics).
Someone may ask, How about "intuitions," "emotions," etc.? The answer is simple and positive. It is a fallacy of the old schools to divide man into parcels, elements; all human faculties consist of an inter-connected whole. We choose to deal with logic because laws of thought are the only aspects of the whole which are tangible and invariant, the eternal laws of thought which can be handled rigorously. When the problems of these aspects are solved, the others, the vague ones, like "intuitions," "emotions," etc., will fall into line automatically. As Keyser has pointed out, it matters what an animal is; with man it matters not only what man is, but even more what we humans think man is. The tragedy of man has been and is that in creating his institutions and ethics he has never been conscious of this.
Already I have given a hint as to how the source of general knowledge can be found in the inherent constitution of the human mind. If I may, I shall give more hints. Let us imagine that in the
night, during our sleep, the universe, ourselves included, should "grow up," ten, one hundred, or "n" times. Is there any human possibility of detecting in the morning this remarkable event? It is a well proved fact that the answer must be negative. Man could not detect the change. His room had, let us say, ten steps in the evening before the change; it would have ten steps in the morning after the change. It is obvious that such metaphysical, so-called "absolute" space is not an absolute space; this example does away with absolute space. But it is easy to see that the number ten (or any other) has remained. Similar reasoning proves that, to the best of our knowledge today, all absolutes have gone except number, whatever number is. If we could succeed in squeezing out some wisdom, some general knowledge, from number, which is this "only absolute left," we should be entitled to expect that this wisdom would contain the germ of absolute knowledge. As a fact this is being done by a few leading mathematicians.
Modern mathematics deals formally with what can be said about anything or any property. As the reader can easily see, we are witnessing the birth of the wonder of wonders—the birth of what may be called "qualitative" mathematics. Here it may be explained why mathematics has this exclusive position among other sciences. It must be emphasized that it was not some special genius of the mathematicians, as such, that was responsible for it. With the birth of the rational being—man—rational activity began spontaneously (no matter how slowly) and this rational activity manifested itself in every line of human endeavor—no matter how slight this rational activity was. Today we know that all man can know is an abstraction. I use the term "abstraction" in the sense of Whitehead: "To be an abstraction does not mean that an entity is nothing. It merely means that its existence is only one factor of a more concrete element of nature." The process of constructing those abstractions is quite arbitrary. From the time man began he has been plunging into this process of constructing arbitrary abstractions—it was and is the very nature of his being.
Man Blundered into Mathematics
Obviously, in the beginning, man did not know anything about the universe or himself; he went ahead spontaneously. It is no wonder that some of his abstractions were false to fact; that some of them were devoid of meaning, and hence neither true nor false but strictly meaningless; and that some of them were correct. In this endless spontaneous process of constructing abstractions he started from that which was the nearest to him—namely himself—and ignorantly attributed his human faculties to all the universe
around him. He did not realize that he—man—was the latest product in the universe; he reversed the order and anthropomorphized all around him. He did not realize, and this is true even today in most cases, that by so doing he was building up a logic and a language unfit to deal with the actual universe, life, man included, and that by doing so he was building for himself a mental impasse,through his inconsistency and naive observation. In a few instances good luck was with him; he made a few abstractions which were at once the easiest to handle and were correct, that is, corresponding to actual facts in this actual universe.
These abstractions were numbers.
Let us see how numbers originated and what was their significance. Anyone may see that there is an actual difference between such groups as X or XX or as XXX, whatever the class is composed of, be it stones, figs, or snakes. And man could not miss for long the peculiar similarity between such X class of stones or such X class of snakes, etc., and here happened a fact of crucial significance for the future of man; he named these different classes by definite names; he called the class of all such classes X "one," the class of all such classes XX "two," XXX "three," etc., and number was born!
Here as everywhere else "le premier pas qui coûte" ("it is the first step that costs"); having created number the rest followed as a comparatively easy task. Man could not miss for long seeing, that if such a class X is added to such a class X he gets such a class XX, but the other day he had called such classes by names "one" and "two," so he concluded that "one and one make two"—mathematics was born—exact knowledge began.
Good luck combined with his human faculties has helped man to discover one of the eternal truths.
The creation of number was the most reasonable, the first truly scientific act done by man; in mathematics this reasonable being produced a perfect abstraction, the first perfect instrument by which to train his brain, his nerve currents, in the ideal way befitting the actual universe (not a fiction) and himself a part of it. Now it is easy to understand from this physiological point of view why mathematics has developed so soundly.
The opposite must be said about the other disciplines. They started with fictions mostly, and until this day they persist and, playing in vacuo, bring havoc into the life of man.
Mathematics started aright—the others did not!
The biggest triumph of human thought was, and forever will be, the discovery of new mathematical methods embracing larger and larger parts of the whole—these are the milestones of man's progress.
Mathematical Philosophy by Keyser is one of these milestones of everlasting significance. In this monumental work there are discoveries of the gravest importance. Keyser is one of the very few in the world, as far as I know, who is blazing a new trail in this field. Whoever is interested in human progress, and which of us is not, should read and re-read this book. The peculiarity of such work is, that the range of their bearing is so vast that it takes time and meditation to digest and appropriate their seemingly simple content. Neither must it be forgotten that the old traditional logic and its progeny, our language and habits, work against us.
The reader may be reassured that this "new" wisdom is much easier than the old one. Mathematics is nothing else than common sense refined and elevated to the rank of science—it is natural to man—it covers his "intuitions"; whereas the old logic was not equipped to deal with the living thought without an unnatural constraint; generally speaking, it rarely covered "intuitions" and common sense.
The reader will get the
first sharp mental shock by reading the title which tells us that mathematical
philosophy—that is, the only rigorous scientific philosophy—is the "study of
fate and freedom . . . it will become increasingly evident as we advance that
the work we are to be engaged in is fundamentally the study of fate and
freedom—logical fate and intellectual freedom. . . . Without more talk and
without danger of misunderstanding, we may, I believe, now speak of ideas as
constituting a world—the world of ideas. With that world all human beings as
humans have to deal—there is no escape; it is there and only there that
foundations are found—foundations for science, foundations for philosophy,
foundations for art, foundations for religion, for ethics, for government and
education; it is in the world of ideas and only there that human beings as
humans may find principles or bases for rational theories and rational conduct
of life, whether individual life or community life; choices differ but some
choice of principles we must make if we are to he really human—if, that is, we
are to be rational—and when we have made it, we are at once bound by a destiny
of consequences beyond the power of passion or will to control or modify;
another choice of principles is but the election of another destiny. The world
of ideas is, you see, the empire of fate.
"Is the human intellect,
then, a slave? No: it is free; but its freedom is not absolute; it is limited
by fact and by law—by the laws of thought, by the immutable character of ideas
and by their unchanging eternal relationships. Intellectual freedom is freedom
to think in accord with the laws of thought, in accord with the nature of
ideas, in accord with their inter-relations, which are unalterable. And
no variety of human freedom—no institution erected in its sacred name—if it
does not conform to the eternal conditions of intellectual freedom—can stand."
"Is the human intellect, then, a slave? No: it is free; but its freedom is not absolute; it is limited by fact and by law—by the laws of thought, by the immutable character of ideas and by their unchanging eternal relationships. Intellectual freedom is freedom to think in accord with the laws of thought, in accord with the nature of ideas, in accord with their inter-relations, which are unalterable. And no variety of human freedom—no institution erected in its sacred name—if it does not conform to the eternal conditions of intellectual freedom—can stand."
This discovery of logical fate and freedom, its formulation and elaboration, is of such importance that, were it the only one in the book, the book would live forever. After some reflection, its practical bearing becomes evident in that all our talkings about "Brotherhood of Man" or "Democracy," etc., are beautiful words but meaningless so long as we do not inquire into the basic premises which underly those doctrines and investigate if the premises are true; because, if the premises should prove to be false, this "logical fate" would drive us to disasters. Sad experience is daily making it more evident that a scientific (not metaphysical) inquiry is imperative. As a fact we have not hitherto had the method by which to approach or handle human affairs in a truly scientific spirit, but once this method is discovered, we have no more excuses for continuing to welter in the old chaos.
There is, perhaps, nothing wrong with "human nature," but there is something basically wrong with our old premises and logic. As a fact, every human activity has at its foundation some doctrine as an inherent, unconditionally inseparable part of it. Because of this logical fate, the analyzing of these doctrines, which underly all human activities, becomes the most important—nay the all important—fact for all the future of man.
Keyser, to the best of my knowledge, is the discoverer of a new mathematical method whereby this can be accomplished; in a wonderfully precise and clear way he elaborates the theory of postulates and doctrinal functions. Most of what he has to say is either entirely new, or given in a new form; he illustrates his thesis continually with many examples so as to make it perfectly clear to the reader. By the discovery and elaboration of this logical fate which dominates our lives by the discovery and elaboration of the theory of the doctrinal function, Keyser goes to the very roots, not only of all actual, but all potential human knowledge; to the roots of all human problems and relations.
What is the importance of such theoretical works? Let me answer by an example:
The pyramids were built without the knowledge of exact sciences; quite true, but what was the waste in effort, the price in life and happiness which such ignorance brought upon the people! A Galileo, a Newton, a Leibnitz for instance, discovered some new facts, gave us some new definitions and formulated some, new methods of handling old problems and at each stage of civilization such discoveries and their logical consequences transformed deeply all our knowledge and therefore affected enormously our practical achievements. As a matter of fact, at the bottom of every "practical" achievement there is some theory, and it is not a paradox to say that
The theoretical discoveries and knowledge as expounded in Keyser's work will deeply transform all human activities, because they will enable man to revise uncriticised prejudices which, until now, we have accepted as truth. "Thought unexpressed is thought concealed, and concealed thought—light hid under a bushel—fades and perishes with the thinker. Expressed, however, it lives and grows, engendering its kind, adding its flame to the flame of other thought, and so that radiance which is 'all there is' increases and tends to abide."
Keyser's book deals with many interconnected ideas of universal interest of grave importance; they form a system which is bound to abide. A short list of his subjects is an evidence of this: Intellectual Freedom and Logical Fate—Mathematical Obligation of Philosophy—Humanistic and Industrial Education—Human Ethics not a Branch of Zoology—Postulates—The Model of Principles and Platforms—Criticism and the Sword of the Gadfly—Municipal Laws and the Laws of Thought—Basic Concepts—Propositional and Doctrinal Functions—Marriage of Matter and Form—Its Infinite Fertility—Doctrines as Offspring—Interpretations—One Doctrinal Function the Matrix of Infinitely Many Doctrines Identical in Form, Diverse in Content—Essential Discriminations—Distinction of Logical and Psychological—Postulate Properties—Truth and Criticism—Mathematical Philosophy in the Rôle of Critic—Autonomous Truth and Falsehood—the Prototype of Reasoned Discourse Often Disguised as in the Declaration of Independence, the Constitution of the United States, the Origin of Species, the Sermon on the Mount—Transformation—Involved in all Thinking—Its Study the Common Enterprise of Sciences—The Problem of Time and Kindred Problems—Invariance—The Ages-old Problem of Permanence and Change—The Group Concept—Variables and Limits—Mathematical Infinity—Hyperspaces—Open Avenues to Higher Worlds—Forms of Intellectual Emancipation—Mathematics of Psychology—Psychology of Mathematics—Science and Engineering—Change of Emphasis from Non-Human to Human Energies—Science as Engineering in Preparation—Engineering as Science in Action—Mathematics the Guide of the Engineer—Engineering the Guide of Humanity, etc., etc.
Such a book is bound to make a strong appeal to intelligent people. All intelligent persons will find some of their burning questions answered. For instance, parents are asking, Why should our children study mathematics? What is the educational value of mathematics? Scientists are asking, How is mathematical science related to the other cardinal enterprises of Man? Sociologists must ask afresh, What is Man? and how can mathematical thinking help to make the social sciences genuine sciences? Engineers would like to know, How can we humanize engineering? and so on?
The layman, the "practical" man, the man on the street, says, What is that to me? The answer is positive and weighty. Our life is entirely dependent on the established doctrines of ethics, sociology, political economy, government, law, medical science, etc. This affects everyone consciously or unconsciously, the man in the street in the first place, because he is the most defenseless.
As a fact most of the so-called scientists rejected logic entirely because the old logic is misleading, and they are entirely ignorant of this "new" logic, though it is seventy years old. I explained before that science is a joint phenomenon of logic and facts, and there can be no escape from the conclusion that such scientists as ignore sound logic are not scientists at all but merely clerks in scientific offices; and yet the people listen to them and are too often hypnotized by their nonsensical conclusions so misleading and immeasurably harmful.
How about the next generation, their future welfare and happiness? If they are taught false logic and false doctrines, mental cripples are produced, destined for a life of misery. Is this what parents want for their beloved ones? What of the teachers—the men and women, who in the literal sense are the builders of the next generations? What do they know about the latest progress of knowledge? Or are they still in the dark ages of ignorance? In the light of these questions, the man on the street has sufficient reason to be vitally interested in this subject.
The new sciences are not strictly "popular." Scientists who have spent their lives in the studies of classical texts, and who are not capable of following up a little piece of sound reasoning, and even some mathematicians and engineers who have learned technique without bothering to inquire into its meaning or justification, are bound to resent these views. The layman must understand the reasons for such opposition.
True knowledge cannot be concealed for long; but if ignorance, dullness, apathy retard its application this will mean one or two more generations of misery. It may take a still more terrible World War to whip mankind into the realization that man should use his brain and the knowledge already at hand.
The writings of Keyser, besides their great scientific value, have another quality not easily found in other scientific writers, namely, an unexcelled style of their own, making his writings not only jewels of thought but jewels of style and language as well.
To sum up. A diagram may help to visualize the power of one of the discoveries of Keyser:
|Old non-scientific assumptions, postulates, beliefs.
|Old non-scientific ideas, non-co-ordinations, wars, revolutions.
|New scientific assumptions, postulates, premises, truths.
|New scientific co-ordinated systems, ideas, ideals.
This diagram makes it evident that—
(1) Any change in (A) the old premises, postulates, necessarily involves changes in (B): it explains why the World War, having exposed many old, hidden fallacies (A), must affect our social, economic, political and other relations, and that, therefore, no return to the old (B) is possible.
(2) It is impossible to start with old (often false) premises (A) and reach new ideals (D) and convince all; because in such case Logical Destiny is against the would-be reformer, whoever he may be; because inconsistencies (E) arise, which prevent the general acceptance of the high-sounding, logically unsound doctrines. For example, we may preach "Brotherhood of Man" and still practice the "Wolfhood" of man.
(3) A new, better civilization (D) must start with new, truer, scientific premises, postulates (C); then, and then only, Logical Destiny will again be our ally, instead of enemy (E).
(4) In the old civilization everyone blames everyone else for everything; Nations blame Nations, Religions blame Religions, Labor blames Capital, Capital blames Labor, etc. Logical destiny proves that no one is to be blamed. In false premises are the roots of guilt—all the rest, the consequences, are but the outgrowth of them. This understanding at once abolishes ALL REASONS FOR BITTERNESS in individual life, community life, international life: it proves that a "League of Sound Logic" is the best "League of Nations" because effective under the subtle inevitable laws of Logical Fate—Unified Doctrines Will Unify Man.
So it may be hoped that those who most earnestly believe in the "Brotherhood of Man" will be re-inspired and be the most eager to investigate and understand and assist in the establishment of the "Brotherhood of Doctrines," because there and only there will be found the foundations of the higher aspects of the ideals for Brotherhood among men.