Mill's rejection of hypotheses formed by a mind whose operations have no discoverable continuity with the operations of things, or by things whose actions are independent of the operations of ideas, is forever sound. But his acceptance of the discontinuity between the acts of knowing and the operation of things, and the conclusion that these two conceptions of the origin and nature of hypotheses are the only alternatives, were the source of most of his difficulties.
The efforts of classic empiricism at the reform of logic have long been an easy mark for idealistic reformers. But it is interesting to observe that the idealistic logic from the beginning finds itself in precisely the same predicament regarding hypotheses;—they aretrifling or false. And in the end they are made, as in Mill, "accidents" of inference.
The part played by Kant's sense-material and the categories is almost the reverse of those of data and hypothesis in science. Sense material and the categories are the given elements from which objects are somehow made; in scientific procedure data and hypothesis are derived through logical observation and imagination from the content and operations of immediate experience. In Kant's account of the process by which objects are constructed we are nowhere in sight of any experimental procedure. Indeed, the real act of knowing, the selection and application of the category to the sense matter, is, as Kant in the end had to confess, "hidden away in the depths of the soul." Made in the presence of the elaborate machinery of knowing which Kant had constructed, this confession is almost tragic; and the tragic aspect grows when we find that the result of the "hidden" operation is merely a phenomenal object. That this should be the case, however, is not strange. A phenomenal object is the inevitable correlate of the "hidden" act of knowing whether in a "transcendental" or in an "empirical" logic. In vain do we call the act of knowing "constructive" and "synthetic" if its method of synthesis is hidden. A transcendental unity whose method is indefinable has no advantage over empirical association.
It was the dream of Kant as of Mill to replace the logics of sensationalism and rationalism with a "logic of things" and of "truth." But as Mill's things turned to states of consciousness, so Kant's are phenomenal.Their common fate proclaims their common failure—the failure to reëstablish continuity between the conduct of intelligence and other conduct.
One of the chief counts in Hegel's indictment of Kant's logic is that "it had no influence on the methods of science."15Hegel's explanation is that Kant's categories have no genesis; they are not constructed in and as part of logical operations. As given, ready-made, their relevance is a miracle. But if categories be "generated" in the process of knowing, says Hegel, they are indigenous, and their fitness is inevitable. In such statements Hegel raises expectations that we are at last to have a logic which squares with the procedure of science. But when we discover that instead of being "generated" out of all the material involved in the scientific problem Hegel's categories are derived from each other, misgivings arise. And when we further learn that this "genesis" is timeless, which means that, after all, the categories stand related to each other in a closed, eternal system of implication, we abandon hope of a scientific—i.e., experimental—logic.
Hegel also says it is the business of philosophy "to substitute categories or in more precise language adequate notions for the several modes of feeling, perception, desire, and will." The word "substitute" reveals the point at issue. If "to substitute" means that philosophy is a complete exchange of the modes of feeling, perception, desire, and will for a world of categories or notions, then, saying nothing of the range of values in such a world, the problem of the meaningof "adequate" is on our hands. What is the notion to be adequate to? But if "to substitute" means that the modes of feeling, perception, desire, and will, when in a specific situation of ambiguity and inhibition, go over into, take on, the modes of data and hypothesis in the effort to get rid of inhibiting conflict that is quite another matter. Here the "notion," as the scientific hypothesis, has a criterion for its adequacy. But if the notion usurps the place of feeling, perception, desire, and will, as many find, in the end, it does in Hegel's logic, it thereby loses all tests for the adequacy of its function and character as a notion.
In the development of the logical doctrines of Kant and Hegel by Lotze, Green, Sigwart, Bradley, Bosanquet, Royce, and others, there are indeed differences. But these differences only throw their common ground into bolder relief. This common ground is that, procedure by hypotheses, by induction, is, in the language of Professor Bosanquet, "a transient and external characteristic of inference."16And the ground of this verdict is essentially the same as Mill's, when he rejects hypotheses "made by the mind," namely, that such hypotheses are too subjective in their origin and nature to have objective validity. "Objective" idealism is trying, like Mill, to escape the subjectivism of the purely individual and "psychical" knower. But, being unable to reconstruct the finite knower, and being too sophisticated to make what it regards as Mill'snaïve appeal to "hypotheses found in things," it transfers the real process of inference to the "objective universal," and the process of all thought, including inference, is now defined as "the reproduction, by a universal presented in a content, of contents distinguished from the presented content which also are differences of the same universal."17
It need scarcely be said that in inference thus defined there is scant room for hypotheses. There is nothing "hypothetical," "experimental," or "tentative" in this process of reproduction by the objective universal as such. As little is there any possibility of error. If there is anything hypothetical, or any possibility of error, in inference, it is due to the temporal, finite human being in which, paradoxically enough, this process of "reproduction" goes on and to whom, at times, is given an "infinitesimal" part in the operation, while at other times he is said merely to "witness" it. But the real inference does not "proceed by hypotheses"; it is only the finite mind in witnessing the real logical spectacle or in its "infinitesimal" contribution to it that lamely proceeds in this manner.
Here, again, we have the same break in continuity between the finite, human act of knowing and the operations that constitute the real world. When the logic of the objective universal rejects imputations of harboring a despoiled psychical knower it has in mind, of course, the objective universal as knower, not the finite, human act. But, if the participations of the latter are all accidents of inference, as they are said to be, its advantageover a purely psychical knower, or "states of consciousness," is difficult to see. The rejection of metaphysical dualism is of no consequence if the logical operations of the finite, human being are only "accidents" of the real logical process. As already remarked, the metaphysical disjunction is merely a schematism of the more fundamental, logical disjunction.
As for tautology and miracle, the follower of Mill might well ask: how an association of particulars, whether mental states or things, could be more tautologous than a universal reproducing its own differences? And if the transition from particular to particular is a miracle in which the grace of God is disguised as "habit," why is not habit as good a disguise for Providence as universals? Moreover, by what miracle does the one all-inclusive universal becomeauniversal? And since perception always presents a number of universals, what determines which one shall perform the reproduction? Finally, since there are infinite differences of the universal that might be reproduced, what determines just which differences shall be reproduced? In this wise the controversy has gone on ever since the challenge of the old rationalistic logic by the nominalists launched the issue of empiricism and rationalism. All the charges which each makes against the other are easily retorted upon itself. Each side is resistless in attack, but helpless in defense.
In a conception of inference in which both data and hypothesis are regarded as the tentative, experimental results of the processes of perception, memory, andconstructive imagination engaged in the special task of removing conflict, ambiguity, and inhibition, and in which these processes are not conceived as the functions of a private mind nor of an equally private brain and nervous system, but as functions of interacting beings,—in such a conception there is no ground for anxiety concerning the simplicity of data, nor the objectivity of hypotheses. Simplicity and objectivity do not have to be secured through elaborate and labored metaphysical construction. The data are simple and the hypothesis objective in so far as they accomplish the work where unto they are called—the removal of conflict, ambiguity, and inhibition in conduct and affection.
In the experimental conception of inference it is clear that the principles of formal logic must play their rôle wholly inside the course of logical operations. They do not apply to relationsbetweenthese operations and "reality"; nor to "reality" itself. Formal identity and non-contradiction signify, in experimental logic, the complete correlativity of data and hypothesis. They mean thatinthe logical procedure data must not be shifted without a corresponding change in the hypothesis and conversely. The doctrine that "theoretically" there may be any number of hypotheses for "the same facts" is, when these multiple hypotheses are anything more than different names or symbols, nothing less than the very essence of formal contradiction. It doubtless makes little difference whether a disease be attributed to big or little, black or red, demons or whether the cause be represented by a, b, or c,etc. But where data and hypotheses are such as are capable of verification, i.e., of mutually checking up each other, a change in one without a corresponding modification of the other is the principle of all formal fallacies.18
With this conception of the origin, nature, and functions of logical operations little remains to be said of their truth and falsity. If the whole enterprise of logical operation, of the construction and verification of hypothesis, is in the interest of the removal of ambiguity, and inhibition in conduct, the only relevant truth or falsity they can possess must be determined by their success or failure in that undertaking. The acceptance of this view of truth and error, be it said again, depends on holding steadfastly to the conception of the operations of knowing asreal acts, which, though having a distinct character and function, are yet in closest continuity with other acts of which indeed they are but modifications and adaptations in order to meet the logical demand.
Here, perhaps, is the place for a word on truth and satisfaction. The satisfaction which marks the truth of logical operations—"intellectual satisfaction"—is the satisfaction which attends the accomplishment of their task, viz., the removal of ambiguity in conduct, i.e., in our interaction with other beings. It does not mean that this satisfaction is bound to be followed bywholly blissful consequences. All our troubles are not over when the distress of ambiguity is removed. It may be indeed that the verdict of the logical operation is that we must face certain death. Very well, we must have felt it to be "good to know the worst," or no inquiry would have been started. We should have deemed ignorance bliss and sat with closed eyes waiting for fate to overtake us instead of going forward to meet it and in some measure determine it. Death anticipated and accepted isrealitervery different from death that falls upon us unawares, however we may estimate that difference. If this distinction in thefociof satisfaction is kept clear it must do away with a large amount of the hedonistic interpretations of satisfaction in which many critics have indulged.
But hereupon some one may exclaim, as did a colleague recently: "Welcome to the ranks of the intellectualists!" If so, the experimentalist is bound to reply that he is as willing, and as unwilling, to be welcomed to the ranks of intellectualism as to those of anti-intellectualism. He wonders, however, how long the welcome would last in either. Among the intellectualists the welcome would begin to cool as soon as it should be discovered that the ambiguity to which logical operations are the response is not regarded by the experimentalist as a purely intellectual affair. It is an ambiguity in conduct with all the attendant affectional values that may be at stake.19It is, to be sure, the fact of ambiguity, and the effort to resolve it, that addsthe intellectual, logical character to conduct and to affectional values. But if the logical interest attempts entirely to detach itself it will soon be without either subject-matter or criterion. And if it sets itself up as supreme, we shall be forced to say that our quandaries of affection, our problems of life and death are merely to furnish occasions and material for logical operations.
On the other hand, the welcome of the anti-intellectualists is equally sure to wane when the experimentalist asserts that the doctrine that logical operations mutilate the wholeness of immediate experience overlooks the palpable fact that it is precisely these immediate experiences—the experiences of intuition and instinct—that get into conflict and inhibit and mutilate one another, and as a consequence are obliged to go into logical session to patch up the mutilation and provide new and better methods of coöperation.
At this point the weakness in Bergson's view of logical operations appears. Bergson, too, is impressed by the break in continuity between logical operations and the rest of experience. But with Mr. Bradley he believes this breach to be essentially incurable, because the mutilations and disjunctions are due to and introduced by logical operations. Just why the latter are introduced remains in the end a mystery. Both, to be sure, believe that logical operations are valuable for "practical" purposes,—for action. But, aside from the question ofhowoperations essentially mutilative can be valuable for action, immediate intuitional experience being already in unity with Reality, why should there be any practical need for logical operations—leastof all such as introduce disjunction and mutilation?
The admission of a demand for logical operations, whether charged to matter, the devil, or any other metaphysical adversary, is, of course, a confession that conflict and ambiguity are as fundamental in experience as unity and immediacy and that logical operations are therefore no less indigenous. The failure to see this implication is responsible for the paradox that in the logic of Creative Evolution the operations of intelligence are neither creative nor evolutional. They not only have no constructive part but are positively destructive and devolutional.
Since, moreover, these logical operations, like those of the objective universal, and like Mill's association of particulars, can only reproduce in fragmentary form what has already been done, it is difficult to see how they can meet the demands of action. For here no more than in Mill, or in the logic of idealism, is there any place for constructive hypotheses or any technique by which they can become effective. Whatever "Creative Evolution" may be, there is no place in its logic for "Creative Intelligence."
The prominence in current discussion of the logical reforms proposed by the "analytic logic" of the neo-realistic movement and the enthusiastic optimism of its representatives over the prospective results of these reforms for logic, science, and practical life are the warrant for devoting a special section to their discussion.
There are indeed some marked differences of opinionamong the expounders of the "new logic" concerning the results which it is expected to achieve. Some find that it clears away incredible accumulations of metaphysical lumber; others rejoice that it is to restore metaphysics, "once the queen of the sciences, to her ancient throne."
But whatever the difference among the representatives of analytical logic all seem agreed at the outset on two fundamental reforms which the "new logic" makes. These are: first, that analytic logic gets rid entirely of theactof knowing, the retention of which has been the bane of all other logics; second, in its discovery of "terms and relations," "sense-data and universals" as the simple elements not only of logic but of the world, it furnishes science at last with the simple neutral elements at large which it is supposed science so long has sought, and "mourned because it found them not."
Taking these in order, we are told that "realism frees logic as a study of objective fact from all accounts of the states and operations of mind." ... "Logic and mathematics are sciences which can be pursued quite independently of the study of knowing."20"The new logic believes that it deals with no such entities as thoughts, ideas, or minds, but with entities that merely are."20
The motive for the banishment of the act of knowing from logic is that as anactknowing is "mental," "psychological," and "subjective."21All other logicshave indeed realized this subjective character of theactof knowing, but have neither dared completely to discard it nor been able sufficiently to counteract its effects even with such agencies as the objective universal to prevent it from infecting logic with its subjectivity. Because logic has tolerated and attempted to compromise with this subjective act of knowing, say these reformers, it has been forced constantly into epistemology and has become a hybrid science. Had logic possessed the courage long ago to throw overboard this subjective Jonah it would have been spared the storms of epistemology and the reefs of metaphysics.
Analytic logic is the first attempt in the history of modern logical theory at a deliberate, sophisticated exclusion of the act of knowing from logic. Other logics, to be sure, have tried to neutralize the effects of its presence, but none has had the temerity to cast it bodily overboard. The experiment, therefore, is highly interesting.
We should note at the outset that in regarding the act of knowing as incurably "psychical" and "subjective" analytic logic accepts a fundamental premise of the logics of rationalism, empiricism, and idealism which it seeks to reform. It is true that it is the bold proposal of analytic logic to keep logic out of the pit of epistemology by excluding the act of knowing from logic. Nevertheless analytic logic still accepts the subjective character of this act; and if it excludes it from its logic it welcomes itin its psychology. This is a dangerous situation. Can the analytic logician prevent all osmosis between his logic and his psychology?22If not, and if the psychological act is subjective, woe then to his logic. Had the new logic begun with a bold challenge of the psychical character of the act of knowing, the prospect of a logic free from epistemology would have been much brighter.
With the desire to rid logic of the epistemological taint the "experimental logic" of the pragmatic movement has the strongest sympathy. But the proposal to effect this by the excision of the act of knowing appears to experimental logic to be a case of heroic but fatal surgery.Prima faciea logic with no act of knowing presents an uncanny appearance. What sort of logical operations are possible in such a logic and of what kind of truth and falsity are they capable?
Before taking up these questions in detail it is worth while to note the character of the entities that "merely are" with which analytic logic proposes exclusively to deal. In their general form they are "terms" and "propositions," "sense-data" and universals. We are struck at once by the fact that these entities bear the names of logical operations. They are, to be sure, disguised as entities and have been baptised in a highly dilute solution of objectivity called "subsistence." But this does not conceal their origin, nor does it obscurethe fact that if it is possible for any entities that "merely are" to have logical character those made from hypostatized processes of logical operations should be the most promising. They might be expected to retain some vestiges of logical character even after they have been torn from the process of inquiry and converted into "entities that merely are." Also it is not surprising that having stripped the act of knowing of its constituent operations analytic logic should feel that it can well dispense with the empty shell called "mind" and, as Professor Dewey says, "wish it on psychology." But if the analytic logician be also a philosopher and perchance a lover of his fellow-man, it is hard to see how he can have a good conscience over this disposition of the case.
Turning now to the character of inference and of truth and falsity which are possible in a logic which excludes the operation of knowing and deals only with "entities that are," all the expounders seem to agree that in such a logic inference must be purely deductive. All alleged induction is either disguised deduction or a lucky guess. This raises apprehension at the start concerning the value of analytic logic for other sciences. But let us observe what deduction in analytic logic is.
We begin at once with a distinction which involves the whole issue.23We are asked to carefully distinguish"logical" deduction from "psychological" deduction. The latter is the vulgar meaning of the term, and is "the thinker's name for his own act of conforming his thought" to the objective and independent processes that constitute the real logical process. This act of conforming the mind is a purely "psychological" affair. It has no logical function whatever. In what the "conforming" consists is not clear. It seems to be merely the act of turning the "psychological" eye on the objective logical process. "One beholds it (the logical process) as one beholds a star, a river, a character in a play.... The novelist and the dramatist, like the mathematician and logician, are onlookers at the logical spectacle."24On the other hand, the term "conforming" suggests a task, with the possibilities of success and failure. Have we, then, two wholly independent possibilities of error—one merely "psychological," the other "logical"? The same point may be made even more obviously with reference to the term "beholding." The term is used as if beholding were a perfectly simple act, having no problems and no possibilities of mistakes—as if there could be no mis-beholding.25
But fixing our psychological eye on the "logicalspectacle," what does it behold? A universal generating an infinite series of identical instances of itself—i.e., instances which differ only in "logical position." If in a world of entities that "merely are" the term "generation" causes perplexity, the tension is soon relieved; for this turns out to be a merely subsistential non-temporal generation which, like Hegel's generation of the categories, in no way compromises a world of entities that "merely are."
Steering clear of the thicket of metaphysical problems that we here encounter, let us keep to the logical trail. First it is clear that logical operations are of the same reproductive repetitive type that we have found in the associational logic of empiricism, and in the logic of the objective universal. Indeed, after objective idealism has conceded that the finite mind merely "witnesses" or at most contributes only in an "infinitesimal" degree to the logical activity of the objective universal, what remains of the supposed gulf between absolute idealism and analytic realism?
It follows, of course, that there can be no place in analytic logic for "procedure by hypotheses." However, it is to the credit of some analytic logicians that they see this and frankly accept the situation instead of attempting to retain hypotheses by making them "accidents" or mere "auxiliaries" of inference. On the other hand, others find that the chief glory of analytic logic is precisely that it "gives thought wings"26forthe free construction of hypotheses. In his lectures on "Scientific Methods in Philosophy" Mr. Russell calls some of the most elemental and sacred entities of analytic logic "convenient fictions." This retention of hypotheses at the cost of cogency is of course in order to avoid a break with science. Those who see that there is no place in analytic logic for hypotheses are equally anxious to preserve their connections with science. Hence they boldly challenge the "superstition" that science has anything to do with hypotheses. Newton's "Hypotheses non fingo" should be the motto of every conscientious scientist who dares "trust his own perceptions and disregard the ukase of idealism." "The theory of mental construction is the child of idealism, now put out to service for the support of its parents." "Theory is no longer regarded in science as an hypothesis added to the observed facts," but a law which is "found in the facts."27The identity of this with Mill's doctrine of hypotheses as "found in things" is obvious.
As against the conception of hypotheses as "free," "winged," constructions of a psychical, beholding, gossiping mind we may well take our stand with those who would exclude such hypotheses from science. And this doubtless was the sort of mind and sort of hypotheses Newton meant when he said "Hypotheses non fingo."28But had Newton's mind really been of the character which he, as a physicist, had learned from philosophersto suppose it to be, and had he really waited to find his hypotheses ready-made in the facts, there never would have been any dispute about who discovered the calculus, and we should never have been interested in what Newton said about hypotheses or anything else. What Newton did is a much better source of information on the part hypotheses play in scientific method than what he said about them. The former speaks for itself; the latter is the pious repetition of a metaphysical creed made necessary by the very separation of mind from things expressed in the statement quoted.
Logically there is little to choose between hypotheses found ready-made in the facts and those which are the "winged" constructions of a purely psychical mind. Both are equally useless in logic and in science. One makes logic and science "trifling," the other makes them "miraculous." But if hypotheses be conceived not as the output of a cloistered psychical entity but as the joint product of all the beings and operations involved in the specific situation in which logical inquiry originates, and more particularly in all those involved in the operations of the inquiry itself (including all the experimental material and apparatus which the inquiry may require), we shall have sufficient continuity between hypotheses and things to do away with miracle, and sufficient reconstruction to avoid inference that is trifling.
It is, however, the second contribution of analytic logic that is the basis of the enthusiasm over its prospective value for other sciences. This is the discovery that terms and propositions, sense-data, and universals,are not only elements of logical operation but are the simple, neutral elements at large which science is supposed to have been seeking. "As the botanist analyzes the structures of the vegetable organism and finds chemical compounds of which they are built so the ordinary chemist analyzes these compounds into their elements, but does not analyze these. The physical chemist analyzes these elemental atoms, as now appears, into minuter componentswhich he in turn must leave to the mathematicians and logicians further to analyze."29
Again it is worth noting that this mutation of logical into ontological elements seems to differ only "in position" from the universal logicism of absolute idealism.
What are these simple elements into which the mathematician and logician are to analyze the crude elements of the laboratory? And how are these elements to be put into operation in the laboratory? Let us picture an analytic logician meeting a physical scientist at a moment when the latter is distressed over the unmanageable complexity of his elements. Will the logician say to the scientist: "Your difficulty is that you are trusting too much to your mundane apparatus. The kingdom of truth cometh not with such things. Forsake your microscopes, test tubes, refractors and resonators, and follow me, and you shall behold the truly simple elements of which you have dreamed."? And when the moment of revelation arrives and theexpectant scientist is solemnly told that the "simple elements" which he has sought so long are "terms and propositions," sense-data and universals, is it surprising that he does not seem impressed? Will he not ask: "What am I to do with these in the specific difficulties of my laboratory? Shall I say to the crude and complex elements of my laboratory operations: 'Be ye resolved into terms and propositions, sense-data and universals'; and will they forthwith obey this incantation and fall apart so that I may locate and remove the hidden source of my difficulty? Are you not mocking me and deceiving yourself with the old ontological argument? Your 'simple' elements—are they anything but the hypostatized process by which elements may be found?"30
The expounders as well as the critics of analytic logic have agreed that it reaches its most critical junction when it faces the problem of truth and error. There is no doubt that the logic of objective idealism, in other respects so similar to analytic logic, has at this point an advantage; for it retains just enough of the finite operation of knowing—an "infinitesimal" part will answer—to furnish the culture germs of error. But analytic logic having completely sterilized itself against this source of infection is in serious difficulty.
Here again it is Professor Holt who has the courage to follow—or shall we say "behold"?—his theory as it "generates" the doctrine that error is a given objective opposition of forces entirely independent ofany such thing as a process of inquiry and all that such a process presupposes. "All collisions between bodies, all inference between energies, all process of warming and cooling, of starting and stopping, of combining and separating, all counterbalancings, as in cantilevers and gothic vaultings, are contradictory forces which can be stated only in propositions that manifestly contradict each other."31But the argument proves too much. For in the world of forces to which we have here appealed there is no force which is not opposed by others and no particle which is not the center of opposing forces. Hence error is ubiquitous. In making error objective we have made all objectivity erroneous. We find ourselves obliged to say that the choir of Westminster Abbey, the Brooklyn bridge, the heads on our shoulders are all supported by logical errors!
Following these illustrations of ontological contradictions there is indeed this interesting statement: "Nature is so full of these mutually negative processes that we are moved to admiration when a few forces coöperate long enough to form what we call an organism."32The implication is, apparently, that as an "opposition" of forces is error, "coöperation" of forces is truth. But what is to distinguish "opposition" from "coöperation"? In the illustration it is clear that opposing forces—error—do not interfere with coöperative forces—truth. Where should we find more counterbalancing, more starting and stopping, warming and cooling, combining and separatingthan in an organism? And if these processes can be stated only in propositions that are "manifestly contradictory," are we to understand that truth has errors for its constituent elements? Such paradoxes have always delighted the soul of absolute idealism. But, as we have seen, only the veil of an infinitesimal finitude intervenes between the logic of the objective universal of absolute idealism and the objective logic of analytic realism.
It is, of course, this predicament regarding objective truth and error that has driven most analytic logicians to recall the exiled psychological, "mental" act of knowing. It had to be recalled to provide some basis of distinction between truth and error, but, this act having already been conceived as incurably "subjective," the result is only an exchange of dilemmas. For the reinstatement of this actipso factoreinstates the epistemological predicament to get rid of which it was first banished from logic.
Earnest efforts to escape this outcome have been made by attaching the act of knowing to the nervous system, and this is a move in the right direction. But so far the effort has been fruitless because no connection has been made between the knowing function of the nervous system and its other functions. The result is that the cognitive operation of the nervous system, as of the "psychical" mind, is that of a mere spectator; and the epistemological problem abides. An onlooking nervous system has no advantage over an "onlooking" mind. Onlooking, beholding may indeedbe a part of a genuine act of knowing. But in that act it is always a stimulus or response to other acts. It is one of them;—never a mere spectator of them. It is when the act of knowing is cut off from its connection with other acts and finds itself adrift that it seeks metaphysical lodgings. And this it may find either in an empty psychical mind or in an equally empty body.33
If, in reinstating the act of knowing as a function of the nervous system, neo-realism had recognized the logical significance of the fact that the nervous system of which knowing is a function is the same nervous system of which loving and hating, desiring and striving are functions and that the transition from these to the operations of inquiry and knowing is not a capricious jump but a transition motived by the loving and hating, desiring and striving—if this had been recognized the logic of neo-realism would have been spared its embarrassments over the distinction of truthand error. It would have seen that the passage from loving and hating, desiring and striving to inquiry and knowing is made in order to renew and reform specific desires and strivings which, through conflict and consequent equivocation, have become fruitless and vain; and it must have seen that the results of the inquiry are true or false as they succeed or fail in this reformation and renewal.
But once more, it must steadily be kept in view that while the loving and hating, desiring and striving, which the logical operations are reforming and renewing, are functions of the nervous system, they are not functions of the nervous system alone, else the door of subjectivism again closes upon us. Loving and hating, desiring and striving have their "objects." Hence any reformation of these functions involves no less a reformation of their objects. When therefore we say that truth and error are relevant to desires and strivings, this means relevant to them as including their objects, not as entitized processes (such are the pitfalls of language) inclosed in a nervous system or mind. With this before us the relevance of truth and error to desires and strivings can never be made the basis for the charge of subjectivism. The conception of desires as peculiarly individual and subjective is a survival of the very isolation which is the source of the difficulty with truth and error. Hence the appeal to this isolation, made alike by idealism and realism, in charging instrumental logic with subjectivism is an elementarypetitio.
Doubtless it will be urged again that the act ofknowing is motived by an independent desire and striving of its own. This is of course consonant with the neo-realistic atomism, however inconsonant it may be with the conception of implication which it employs. If we take a small enough, isolated segment of experience we can find meaning for this notion, as we may for the idea that the earth is flat and that the sun moves around the earth. But as consequences accrue we find as great difficulties with the one as with the other. If the course of events did not bring us to book, if we could get off with a mere definition of truth and error we might go on piling up subsistential definitional logics world without end. But sublime adventurers, logically unregenerate and uninitiated, will go on sailing westward to the confusion and confounding of all definitional systems that leave them out of account.
The conclusion is plain. If logic is to have room in its household for both truth and error, if it is to avoid the old predicament of knowledge that is trifling or miraculous, tautologous or false, if it is to have no fear of the challenge of other sciences or of practical life, it must be content to take for its subject-matter the operations of intelligence conceived as real acts on the same metaphysical plane and in strictest continuity with other acts. Such a logic will not fear the challenge of science, for it is precisely this continuity that makes possible experimentation, which is the fundamental characteristic of scientific procedure. Science without experiment is indeed a strange apparition. It is a λόγος with no λέγειν, a sciencewith noscire; and this spells dogmatism. How necessary such continuity is to experimentation is apparent when we recall that there is no limit to the range of operations of every sort which scientific experiment calls into play; and that unless there be thoroughgoing continuity between the logical demand of the experiment and all the materials and devices employed in the process of the experiment, the operations of the latter in the experiment will be either miraculous or ruinous.
Finally, if this continuity of the operations of intelligence with other operations be essential to science, its relation to "practical" life isipso factoestablished. For science is "practical" life aware of its problems and aware of the part that experimental—i.e., creative—intelligence plays in the solution of those problems.
Herbart is said to have given the deathblow to faculty psychology. Man no longer appears endowed with volition, passion, desire, and reason; and logic, deprived of its hereditary right to elucidate the operations of inherent intelligence, has the new problem of investigating forms of intelligence in the making. This is no inconsequential task. "If man originally possesses only capacities which after a given amount of education will produce ideas and judgments" (Thorndike,Educational Psychology, Vol. I, p. 198), and if these ideas and judgments are to be substituted for a mythical intelligence it follows that tracing their development and observing their functioning renders clearer our conception of their nature and value and brings us nearer that exact knowledge of what we are talking about in which the philosopher at least aspires to equal the scientist, however much he may fall below his ideal.
For contemporary thought concerning the mathematical sciences this altered point of view generates peculiarly pressing problems. Mathematicians have weighed the old logic and found it wanting. They have builded themselves a new logic more adequate to their ends. But they have not whole-heartedly recognized the change that has come about in psychology; hencethey have retained the faculty of intelligence knit into certain indefinables such as implication, relation, class, term, and the like, and have transported the faculty from the human soul to a mysterious realm of subsistence whence it radiates its ghostly light upon the realm of existence below. But while they reproach the old logic, often bitterly, their new logic merely furnishes a more adequate show-case in which already attained knowledge may be arranged to set off its charms for the observer in the same way that specimens in a museum are displayed before an admiring world. This statement is not a sweeping condemnation, however, for such a setting forth is not useless. It resembles the classificatory stage of science which, although not itself in the highest sense creative, often leads to higher stages by bringing under observation relations and facts that might otherwise have escaped notice. And in the realm of pure mathematics, the new logic has undoubtedly contributed in this manner to such discoveries. Danger appears when the logician attains Cartesian intoxication with the beauty of logico-mathematical form and tries to infer from the form itself the real nature of the formed material. The realm of subsistence too often has armed Indefinables with metaphysical myths whose attack is valiant when the doors of reflection are opened. It may be possible, however, to arrive at an understanding of mathematics without entering the kingdom of these warriors.
It is the essence of science to make prediction possible. The value of prediction lies in the fact that through this function man can control his environment,or, at worst, fortify himself to meet its vagaries. To attain such predictions, however, the world need not be grasped in its full concreteness. Hence arise processes of abstraction. While all other symptoms remain unnoticed, the temperature and pulse may mark a disease, or a barometer-reading the weather. The physicist may work only in terms of quantity in a world which is equally truly qualitative. All that is necessary is to select the elements which are most effective for prediction and control. Such selection gives the principle that dominates all abstractions. Progress is movement from the less abstract to the more abstract, but it is progress only because the more abstract is as genuinely an aspect of the concrete starting-point as anything is. Moreover, the outcome of progress of this sort cannot be definitely foreseen at the beginnings. The simple activities of primitive men have to be spontaneously performed before their value becomes evident. Only afterwards can they be cultivated for the sake of their value, and then only can the self-conscious cultivation of a science begin. The process remains full not only of perplexities, but of surprises; men's activities lead to goals far other than those which appear at the start. These goals, however, never deny the method by which the start is made. Developed intelligence is nothing but skill in using a set of concepts generated in this manner. In this sense the histories of all human endeavors run parallel.
Where the empirical bases of a science are continually in the foreground, as in physics or chemistry, theforegoing formulation of procedure is intelligible and acceptable to most men. Mathematics seem, however, to stand peculiarly apart. Many, with Descartes, have delighted in them "on account of the certitude and evidence of their reasonings" and recognized their contribution to the advancement of mechanical arts. But since the days of Kant even this value has become a problem, and many a young philosophic student has the question laid before him as to why it is that mathematics, "a purely conceptual science," can tell us anything about the character of a world which is, apparently at least, free from the idiosyncrasies of individual mind. It may be that mathematics began in empirical practice, such philosophers admit, but they add that, somehow, in its later career, it has escaped its lowly origin. Now it moves in the higher circles of postulated relations and arbitrarily defined entities to which its humble progenitors and relatives are denied the entrée. Parvenus, however, usually bear with them the mark of history, and in the case of this one, at least, we may hope that the history will be sufficient to drag it from the affectations of its newly acquired set and reinstate it in its proper place in the workaday world. For the sake of this hope, we shall take the risk of being tedious by citing certain striking moments of mathematical progress; and then we shall try to interpret its genuine status in the world of working truths.
The most primitive mathematical activity of man is counting, but here his first efforts are lost in the obscurity of the past. The lower races, however, yield us evidence that is not without value. Although the savage mind is not identical with the mind of primitive man, there is much in the activities of undeveloped races that can throw light upon the behavior of peoples more advanced. We must be careful in our inferences, however. Among the Australians and South Americans there are peoples whose numerical systems go little, or not at all, beyond the first two or three numbers. "It has been inferred from this," writes Professor Boas (Mind of Primitive Man, pp. 152-53), "that the people speaking these languages are not capable of forming the concept of higher numbers.... People like the South American Indians, ... or like the Esquimo ... are presumably not in need of higher numerical expressions, because there are not many objects that they have to count. On the other hand, just as soon as these same people find themselves in contact with civilization, and when they acquire standards of value that have to be counted, they adopt with perfect ease higher numerals from other languages, and develop a more or less perfect system of counting.... It must be borne in mind that counting does not become necessary until objects are considered in such generalized form that their individualities are entirely lost sight of. For thisreason it is possible that even a person who owns a herd of domesticated animals may know them by name and by their characteristics, without even desiring to count them."
And there is one other false interpretation to be avoided. Man does not feel the need of counting and then develop a system of numerals to meet the need. Such an assumption is as ridiculous as to assume prehistoric man thinking to himself: "I must speak," and then inventing voice culture and grammar to make speaking pleasant and possible. Rather, when powers of communication are once attained, presumably in their beginnings also without forethought, man being still more animal than man, there were gradually dissociated communications of a kind approaching what numbers mean to us. But the number is not yet a symbol apart from that of the things numbered. Picture writing, re-representing the things meant, preceded developmentally any kind of symbolization representing the number by mere one-one correspondence with non-particularized symbols. It is plausible, although I have no anthropological authority for the statement, that the prevalence of finger words as number symbols (cf. infra) is originally a consequence of the fact that our organization makes the hand the natural instrument of pointing.
The difficulty of passing from concrete representations to abstract symbols has been keenly stated by Conant (The Number Concept, pp. 72-73), although his terminology is that of an old psychology and the limitations implied for the primitive mind are limitations of practice rather than of capacity as Mr. Conant seems to believe. "An abstract conception is something quite foreign to the essentially primitive mind, as missionaries and explorers have found to their chagrin. The savage can form no mental concept of what civilized man means by such a word assoul; nor would his idea of the abstract number 5 be much clearer. When he saysfive, he uses, in many cases at least, the same word that serves him when he wishes to sayhand; and his mental concept when he saysfiveis a hand. The concrete idea of a closed fist, of an open hand with outstretched fingers, is what is uppermost in his mind. He knows no more and cares no more about the pure number 5 than he does about the law of conservation of energy. He sees in his mental picture only the real, material image, and his only comprehension of the number is, "these objects are as many as the fingers on my hand." Then, in the lapse of the long interval of centuries which intervene between lowest barbarism and highest civilization, the abstract and concrete become slowly dissociated, the one from the other. First the actual hand picture fades away, and the number is recognized without the original assistance furnished by the derivation of the word. But the number is still for a long time a certain numberof objects, and not an independent concept."
An excellent fur trader's story, reported to me by Mr. Dewey, suggests a further impulse to count besides that given by the need of keeping a tally, namely, the need of making one thing correspond to another in a business transaction. The Indian laid down one skinand the trader two dollars; if he proposed to count several skins at once and pay for all together, the former replied "too much cheatem." The result, however, demanded a tally either by the fingers, a pebble, or a mark made in the sand, and as the magnitude of such transactions grows the need of a specific number symbol becomes ever more acute.
The first obstacle, then, to overcome—and it has already been successfully passed by many primitive peoples—is the need of fortuitous attainment of a numerical symbol, which is not the mere repeated symbol of the things numbered. Significantly, this symbol is usually derived from the hand, suggesting gestures of tallying, and not from the words of already developed language. Consequently, number words relate themselves for the most part to the hand, and written number symbols, which are among the earliest writings of most peoples, tend to depict it as soon as they have passed beyond the stage mentioned above of merely repeating the symbol of the things numbered. W. C. Eells, in writing of the Number Systems of the North American Indians (Am. Math. Mo., Nov., 1913; pp. 263-72), finds clear linguistic evidence for a digital origin in about 40% of the languages examined. Of the non-digital instances, 1 was sometimes connected with the first personal pronoun, 2 with roots meaning separation, 3, rarely, meaning more, or plural as distinguished from the dual, just as the Greek uses a plural as well as a dual in nouns and verbs, 4 is often the perfect, complete right. It is often a sacred number and the base of a quarternary system. Conant(loc. cit.p. 98) also gives a classification of the meanings of simple number words for more advanced languages; and even in them the hand is constantly in evidence, as in 5, the hand; 10, two hands, half a man, when fingers and toes are both considered, or a man, when the hands alone are considered; 20, one man, two feet. The other meanings hang upon the ideas of existence, piece, group, beginning, for 1; and repetition, division, and collection for higher numerals.
A peculiar difficulty lies in the fact that when once numbering has become a self-conscious effort, the collection of things to be numbered frequently tends to exceed the number of names that have become available. Sometimes the difficulty is met by using a second man when the fingers and toes of the first are used up, sometimes by a method of repetition with the record of the number of the repetition itself added to the numerical significance of the whole process. Hence arise the various systems of bases that occur in developed mathematics. But the inertia to be overcome in the recognition of the base idea is nowhere more obvious than in the retention by the comparatively developed Babylonian system of a second base of 60 to supplement the decimal one for smaller numbers. Among the American Indians (Eells,loc. cit.) the system of bases used varies from the cumbersome binary scale, that exercised such a fascination over Leibniz (Opera,III, p. 346), through the rare ternary, and the more common quarternary to the "natural" quinary, decimal, and vigesimal systems derived from the use of the fingers and toes in counting.The achievement of a number base and number words, however, does not always open the way to further mathematical development. Only too often a complexity of expression is involved that almost immediately cuts off further progress. Thus the Youcos of the Amazon cannot get beyond the number three, for the simplest expression for the idea in their language is "pzettarrarorincoaroac" (Conant,loc. cit., pp. 145, 83, 53). Such names as "99, tongo solo manani nun solo manani" (i.e., 10, understood, 5 plus 4 times, and 5 plus 4) of the Soussous of Sierra Leone; "399, caxtolli onnauh poalli ipan caxtolli onnaui" (15 plus 4 times 20 plus 15 plus 4) of the Aztec; "29, wick a chimen ne nompah sam pah nep e chu wink a" (Sioux), make it easy to understand the proverb of the Yorubas of Abeokuta, "You may be very clever, but you can't tell 9 times 9."
Almost contemporaneously with the beginnings of counting various auxiliary devices were introduced to help out the difficult task. In place of many men, notched sticks, knotted strings, pebbles, or finger pantomime were used. In the best form, these devices resulted in the abacus; indeed, it was not until after the introduction of arabic numerals and well into the Renaissance period that instrumental arithmetic gave way to graphical in Europe (D. E. Smith,Rara Arithmetica, under "Counters"). "In eastern Europe," say Smith and Mikami (Japanese Mathematics, pp. 18-19), "it"—the abacus—"has never been replaced, for the tschotü is used everywhere in Russia to-day, and when one passes over into Persia the same type ofabacus is common in all the bazaars. In China the swan-pan is universally used for the purposes of computation, and in Japan the soroban is as strongly entrenched as it was before the invasion of western ideas."
Given, then, the idea of counting, and a mechanical device to aid computation, it still remains necessary to obtain some notation in which to record results. At the early dawn of history the Egyptians seem to have been already possessed of number signs (cf. Cantor,Gesch. de. Math., p. 44) and the Phœnicians either wrote out their number words or used a few simple signs, vertical, horizontal, and oblique lines, a process which the Arabians perpetuated up to the beginning of the eleventh century (Fink, p. 15); the Greeks, as early as 600 B. C., used the initial letters of words for numbers. But speaking generally, historical beginnings of European number signs are too obscure to furnish us good material.
Our Indians have few number symbols other than words, but when they occur (cf. Eells,loc. cit.) they usually take the form of pictorial presentation of some counting device such as strokes, lines dotted to suggest a knotted cord, etc. Indeed, the smaller Roman numerals were probably but a pictorial representation of finger symbols. However, a beautiful concrete instance is furnished us in the Japanese mathematics (cf. Smith and Mikami, Ch. III). The earliest instrument of reckoning in Japan seems to have been the rod, Ch'eou, adapted from the Chinese under the name of Chikusaku (bamboo rods) about 600 A. D. At first relativelylarge (measuring rods?), they became reduced to about 12 cm., but from their tendency to roll were quickly replaced by the sangi (square prisms, about 7 mm. thick and 5 cm. long) and the number symbols were evidently derived from the use of these rods: