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 a.n.a.lytic 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 a.n.a.lytic 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 a.n.a.lytic 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 of any 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."[31] But 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 ill.u.s.trations 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 cooperate long enough to form what we call an organism."[32] The implication is, apparently, that as an "opposition" of forces is error, "cooperation" of forces is truth. But what is to distinguish "opposition" from "cooperation"? In the ill.u.s.tration it is clear that opposing forces--error--do not interfere with cooperative forces--truth.

Where should we find more counterbalancing, more starting and stopping, warming and cooling, combining and separating than 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 const.i.tuent 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 a.n.a.lytic realism.

It is, of course, this predicament regarding objective truth and error that has driven most a.n.a.lytic 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 act _ipso facto_ reinstates 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 indeed be 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 embarra.s.sments over the distinction of truth and error. It would have seen that the pa.s.sage 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 ent.i.tized 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 elementary _pet.i.tio_.

Doubtless it will be urged again that the act of knowing 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 [Greek: logos] with no [Greek: legein], a science with no _scire_; 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 is _ipso facto_ established. 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.

INTELLIGENCE AND MATHEMATICS

HAROLD CHAPMAN BROWN

Herbart is said to have given the deathblow to faculty psychology. Man no longer appears endowed with volition, pa.s.sion, 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 subst.i.tuted 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; hence they have retained the faculty of intelligence knit into certain indefinables such as implication, relation, cla.s.s, 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 cla.s.sificatory 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 quant.i.ty 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, the foregoing 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 cert.i.tude 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 ent.i.ties to which its humble progenitors and relatives are denied the entree. 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.

I

BEGINNINGS OF ARITHMETIC AND GEOMETRY

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 this reason 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 a.s.sumption is as ridiculous as to a.s.sume 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 pa.s.sing 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 as _soul_; nor would his idea of the abstract number 5 be much clearer.

When he says _five_, he uses, in many cases at least, the same word that serves him when he wishes to say _hand_; and his mental concept when he says _five_ is 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 a.s.sistance furnished by the derivation of the word. But the number is still for a long time a certain number _of 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 skin and 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 pa.s.sed 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 pa.s.sed 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 p.r.o.noun, 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 cla.s.sification 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 repet.i.tion, 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 repet.i.tion with the record of the number of the repet.i.tion 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 c.u.mbersome 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 (_j.a.panese Mathematics_, pp. 18-19), "it"--the abacus--"has never been replaced, for the tschotu is used everywhere in Russia to-day, and when one pa.s.ses over into Persia the same type of abacus is common in all the bazaars. In China the swan-pan is universally used for the purposes of computation, and in j.a.pan 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 Phoenicians 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 j.a.panese mathematics (cf. Smith and Mikami, Ch. III). The earliest instrument of reckoning in j.a.pan 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 relatively large (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:

_ __ ___ ____

,

,

,

,

,

,

,

,

.

For the sake of clearness, tens, hundreds, etc., were expressed in the even place by horizontal instead of vertical lines and vice versa; thus 1267 would be formed

__ -

.

The rods were arranged on a sort of chessboard called the swan-pan. Much later the lines were transferred to paper, and a circle used to denote the vacant square. The use of squares, however, rendered it unnecessary to arrange the even places differently from the odd, so numbers like 38057 came to be written

+-------+-------+-------+-------+------+

___

__

+-------+-------+-------+-------+------+

instead of

+-------+-------+-------+-------+------+

-

-

__

-

-

-

-

-

-

+-------+-------+-------+-------+------+

as in the earlier notation.

Somewhere in the course of these early mathematical activities the process has changed from the more or less spontaneous operating that led primitive man to the first enunciation of arithmetical ideas, and has become a self-conscious striving for the solution of problems. This change had already taken place before the historical origins of arithmetic are met. Thus, the treatise of Ahmes (2000 B. C.) contains the curious problem: 7 persons each have 7 cats; each cat eats 7 mice; each mouse eats 7 ears of barley; from each ear 7 measures of corn may grow; how much grain has been saved? Such problems are, however, half play, as appears in a Leonardo of Pisa version some 3000 years later: 7 old women go to Rome; each woman has 7 mules; each mule, 7 sacks; each sack contains 7 loaves; with each loaf are 7 knives; each knife is in 7 sheaths. Similarly in Diophantus' epitaph (330 A. D.): "Diophantus pa.s.sed 1/6 of his life in childhood, 1/12 in youth, and 1/7 more as a bachelor; 5 years after his marriage, was born a son who died 4 years before his father at 1/2 his age." Often among peoples such puzzles were a favorite social amus.e.m.e.nt. Thus Braymagupta (628 A. D.) reads, "These problems are proposed simply for pleasure; the wise man can invent a thousand others, or he can solve the problems of others by the rules given here. As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in a.s.semblies of the people if he proposes algebraic problems, and still more if he solves them"

(Cajori, _Hist. of Math._, p. 92).

The limitation of these early methods is that the notation merely records and does not aid computation. And this is true even of such a highly developed system as was in use among the Romans. If the reader is unconvinced, let him attempt some such problem as the multiplication of CCCXVI by CCCCLXVIII, expressing it and carrying it through in Roman numerals, and he will long for the abacus to a.s.sist his labors. It was the positional arithmetic of the Arabians, of which the origins are obscure, that made possible the development of modern technique. Of this discovery, or rediscovery from the Hindoos, together with the zero symbol, Cajori (_Hist. of Math._, p. 11) has said "of all mathematical discoveries, no one has contributed more to the general progress of intelligence than this." The notation no longer merely records results, but now a.s.sists in performing operations.

The origins of geometry are even more obscure than those of arithmetic.

Not only is geometry as highly developed as arithmetic when it first appears in occidental civilization, but, in addition, the problems of primitive peoples seem to have been such that they have developed no geometrical formulae striking enough to be recorded by investigators, so far as I have been able to discover. But just as the commercial life of the Phoenicians early forced them self-consciously to develop arithmetical calculation, so environmental conditions seem to have forced upon the Egyptians a need for geometrical considerations.

It is almost plat.i.tudinous to quote Herodotus' remark that the invention of geometry was necessary because of the floods of the Nile, which washed away the boundaries and changed the contours of the fields. And as Proclus Diadochus adds (_Procli Diadochi, in primum Euclidis elementorum librum commentarii_--quoted Cantor, I, p. 125): "It is not surprising that the discovery of this as well as other sciences has sprung from need, because everything in the process of beginning proceeds from the incomplete to the complete. There takes place a suitable transition from sensible perception to thoughtful consideration and rational knowledge. Just as with the Phoenicians, for the sake of business and commerce, an exact knowledge of numbers had its beginning, so with the Egyptians, for the above-mentioned reasons, was geometry contrived."