Math.
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 formedMath.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 writtenMath.instead ofMath.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 womanhas 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 passed 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 amusement. 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 assemblies 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 assist 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 contributedmore to the general progress of intelligence than this." The notation no longer merely records results, but now assists 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 formulæ striking enough to be recorded by investigators, so far as I have been able to discover. But just as the commercial life of the Phœnicians 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 platitudinous 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 Phœnicians, 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."
The earliest Egyptian mathematical writing that we know is that of Ahmes (2000 B. C.), but long before this the mural decorations of the temple wall involved many figures, the construction of which involved a certain amount of working knowledge of such operations as may be performed with the aid of a ruler and compass. The fact that these operations did not earlier lead to geometry, as ruler and compass work seems to have done in Japan in the nineteenth century (Smith and Mikami, index, "Geometry"), is probably due to the stage at which the development of Egyptian intelligence had arrived, feebly advanced on the road to higher abstract thinking. It is everywhere characteristic of Egyptian genius that little purely intellectual curiosity is shown. Even astronomical knowledge was limited to those determinations which had religious or magically practical significance, and its arithmetic and geometry never escaped these bounds as with the more imaginative Pythagoreans, where mystical interpretation seems to have been a consequence of rather than a stimulus to investigation. An old Egyptian treatise reads (Cantor, p. 63): "I hold the wooden pin (Nebi) and the handle of the mallet (semes), I hold the line in concurrence with the Goddess Sạfech. My glance follows the course of the stars. When my eye comes to the constellation of the great bear and the time of the number of the hour determined by me is fulfilled, I place the corner of the temple." This incantation method could hardly advance intelligence; but the methods of practical measuring were more effective. Here the rather happy device of using knottedcords, carried about by the Harpedonapts, or cord stretchers, was of some moment. Especially, the fact that the lengths 3, 4, and 5, brought into triangular form, served for an interesting connection between arithmetic and the right triangle, was not a little gain, later making possible the discovery of the Pythagorean theorem, although in Egypt the theoretical properties of the triangle were never developed. The triangle obviously must have been practically considered by the decorators of the temple and its builders, but the cord stretchers rendered clear its arithmetical significance. However, Ahmes' "Rules for attaining the knowledge of all dark things ... all secrets that are contained in objects" (Cantor,loc. cit., p. 22) contains merely a mixture of all sorts of mathematical information of a practical nature,—"rules for making a round fruit house," "rules for measuring fields," "rules for making an ornament," etc., but hardly a word of arithmetical and geometrical processes in themselves, unless it be certain devices for writing fractions and the like.
A characteristic of Greek social life is responsible both for the next phase of the development of mathematical thought and for the misapprehension of its nature by so many moderns. "When Archytas and Menaechmus employed mechanical instruments for solving certain geometrical problems, 'Plato,' says Plutarch, 'inveighed against them with great indignation and persistence as destroying and perverting all thegood that there is in geometry; for the method absconds from incorporeal and intellectual or sensible things, and besides employs again such bodies as require much vulgar handicraft: in this way mechanics was dissimilated and expelled from geometry, and being for a long time looked down upon by philosophy, became one of the arts of war.' In fact, manual labor was looked down upon by the Greeks, and a sharp distinction was drawn between the slaves who performed bodily work and really observed nature, and the leisured upper classes who speculated, and often only knew nature by hearsay. This explains much of the naïve dreamy and hazy character of ancient natural science. Only seldom did the impulse to make experiments for oneself break through; but when it did, a great progress resulted, as was the case of Archytas and Archimedes. Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but, whatever might have been his view of what ought to be in the case, he did actually introduce a large number of new inventions" (Jourdain,The Nature of Mathematics, pp. 18-19). Following the Greek lead, certain empirically minded modern thinkers construe geometry wholly from an intellectual point of view. History is read by them as establishing indubitably the proposition that mathematics is a matter of purely intellectual operations. But by so construing it, they have, in geometry, remembered solely the measuring and forgotten the land, and, in arithmetic, remembered the counting and forgotten the things counted.
Arithmetic experienced little immediate gain from its new association with geometry, which was destined to be of momentous import in its latter history, beyond the discovery of irrationals (which, however, were for centuries not accepted as numbers), and the establishment of the problem of root-taking by its association with the square, and interest in negative numbers.
The Greeks had only subtracted smaller numbers from larger, but the Arabs began to generalize the process and had some acquaintance with negative results, but it was difficult for them to see that these results might really have significance. N. Chuquet, in the fifteenth century, seems to have been the first to interpret the negative numbers, but he remained a long time without imitators. Michael Stifel, in the sixteenth century, still calls them "Numeri absurdi" as over against the "Numeri veri." However, their geometrical interpretation was not difficult, and they soon won their way into good standing. But the case of the imaginary is more striking. The need for it was first felt when it was seen that negative numbers have no square roots. Chuquet had dealt with second-degree equations involving the roots of negative numbers in 1484, but says these numbers are "impossible," and Descartes (Geom., 1637) first uses the word "imaginary" to denote them. Their introduction is due to the Italian algebrists of the sixteenth century. They knew that the real roots of certain algebraic equations of the third degree are represented as results of operations effected upon "impossible" numbers of the forma+b√-1 (whereaandbare real numbers) without it being possible in general to find an algebraic expression for the roots containing only real numbers. Cardan calculated with these "impossibles," using them to get real results [(5 + √-15) (5 - √-15) = 25 - (-15) = 40], but adds that it is a "quantitas quae vere est sophistica" and that the calculus itself "adeo est subtilis ut est inutilis." In 1629, Girard announced the theorem that every complete algebraic equation admits of as many roots, real or imaginary, as there are units in its degree, but Gauss first proved this in 1799, and finally, in hisTheory of Complex Quantity, in 1831.
Geometry, however, among the Greeks passed into a stage of abstraction in which lines, planes, etc., in the sense in which they are understood in our elementary texts, took the place of actually measured surfaces, and also took on the deductive form of presentation that has served as a model for all mathematical presentation since Euclid. Mensuration smacked too much of the exchange, and before the time of Archimedes is practically wholly absent. Even such theorems as "that the area of a triangle equals half the product of its base and its altitude" is foreign to Euclid (cf. Cajori, p. 39). Lines were merely directions, and points limitations from which one worked. But there was still dependence upon the things that one measures. Euclid's elements, "when examined in the light of strict mathematical logic, ... has been pronounced by C. S. Peirce to be 'Riddled with fallacies'" (Cajori, p. 37). Not logic, but observationof the figures drawn, that is, concrete symbolization of the processes indicated, saves Euclid from error.
Roman practical geometry seems to have come from the Etruscans, but the Roman here is as little inventive as in his arithmetical ventures, although the latter were stimulated somewhat by problems of inheritance and interest reckoning. Indeed, before the entrance of Arabic learning into Europe and the translation of Euclid from the Arabic in 1120, there is little or no advance over the Egyptian geometry of 600 B. C. Even the universities neglected mathematics. At Paris "in 1336 a rule was introduced that no student should take a degree without attending lectures on mathematics, and from a commentary on the first six books of Euclid, dated 1536, it appears that candidates for the degree of A. M. had to give an oath that they had attended lectures on these books. Examinations, when held at all, probably did not extend beyond the first book, as is shown by the nickname 'magister matheseos' applied to theTheorem of Pythagoras, the last in the first book.... At Oxford, in the middle of the fifteenth century, the first two books of Euclid were read" (Cajori,loc. cit., p. 136). But later geometry dropped out and not till 1619 was a professorship of geometry instituted at Oxford. Roger Bacon speaks of Euclid's fifth proposition as "elefuga," and it also gets the name of "pons asinorum" from its point of transition to higher learning. As late as the fourteenth century an English manuscript begins "Nowe sues here a Tretis of Geometri whereby you may knowe thehegte, depnes, and the brede of most what erthely thynges."
The first significant turning-point lies in the geometry of Descartes. Viete (1540-1603) and others had already applied algebra to geometry, but Descartes, by means of coördinate representation, established the idea of motion in geometry in a fashion destined to react most fruitfully on algebra, and through this, on arithmetic, as well as enormously to increase the scope of geometry. These discoveries are not, however, of first moment for our problem, for the ideas of mathematical entities remain throughout them the generalized processes that had appeared in Greece. It is worth noting, however, that in England mechanics has always been taught as an experimental science, while on the Continent it has been expanded deductively, as a development ofa prioriprinciples.
To develop the complete history of arithmetic and geometry would be a task quite beyond the limits of this paper, and of the writer's knowledge. In arithmetic we were able to observe a stage in which spontaneous behavior led to the invention of number names and methods of counting. Then, by certain speculative and "play" impulses, there arose elementary arithmetical problems which began to be of interest in themselves. Geometry here also comes into consideration, and, in connection with positional number symbols, begin thoseinteractions between arithmetic and geometry that result in the forms of our contemporary mathematics. The complex quantities represented by number symbols are no longer merely the necessary results of analyzing commercial relations or practical measurements, and geometry is no longer directly based upon the intuitively given line, point, and plane. If number relations are to be expressed in terms of empirical spatial positions, it is necessary to construct many imaginary surfaces, as is done by Riemann in his theory of functions, a construction representing the type of imagination which Poincaré has called the intuitional in contradistinction to the logical (Value of Science, Ch. I). And geometry has not only been led to the construction of many non-Euclidian spaces, but has even, with Peano and his school, been freed from the bonds of any necessary spatial interpretation whatsoever.
To trace in concrete detail the attainment of modern refinements of number theory would likewise exhibit nothing new in the building up of mathematical intelligence. We should find, here, a process carried out without thought of the consequences, there, an analogy suggesting an operation that might lead us beyond a difficulty that had blocked progress; here, a play interest leading to a combination of symbols out of which a new idea has sprung; there, a painstaking and methodical effort to overcome a difficulty recognized from the start. It is rather for us now to ask what it is that has been attained by these means, to inquire finally what are those things called "number" and "line"in the broad sense in which the terms are now used.
In so far as the cardinal number at least is concerned, the answer generally accepted by Dedekind, Peano, Russell, and such writers is this: the number is a "class of similar classes" (Whitehead and Russell,Prin. Math., Vol. II, p. 4). To the interpretation of this answer, Mr. Russell, the most self-consciously philosophical of these mathematicians, has devoted his full dialectic skill. The definition has at least the merit of being free from certain arbitrary psychologizing that has vitiated many earlier attempts at the problem. Mr. Russell claims for it "(1) that the formal properties which we expect cardinal numbers to have result from it; (2) that unless we adopt this definition or some more complicated and practically equivalent definition, it is necessary to regard the cardinal number of a class as indefinable" (loc. cit., p. 4). That the definition's terms, however, are not without obscurity appears in Mr. Russell's struggles with the zigzag theory, the no-class theory, etc., and finally in his taking refuge in the theory of "logical types" (loc. cit., Vol. III, Part V. E.), whereby the contradiction that subverted Frege and drove Mr. Russell from the standpoint of thePrinciples of Mathematicsis finally overcome.
The second of Mr. Russell's claims for his definition adds nothing to the first, for it merely asserts that unless we adopt some definition of the cardinal number from which its formal properties result, number is undefined. Any such definition would be,ipso facto, a practical equivalent of the first. We need only considerwhether or not the formal properties of numbers clearly follow from this definition.
Mr. Russell's own experience makes us hesitate. When he first adopted this definition from Frege, he was led to make the inference that the class of all possible classes might furnish a type for a greatest cardinal number. But this led to nothing but paradox and contradiction. The obvious conclusion was that something was wrong with the concept of class, and the obvious way out was to deny the possibility of any such all-inclusive class. Just why there should be such limitation, except that it enables one to escape the contradiction, is not clear from Mr. Russell's analysis (cf. Brown, "The Logic of Mr. Russell,"Journ. of Phil., Psych., and Sci. Meth., Vol. VIII, No. 4, pp. 85-89). Furthermore, to pass to the theory of types on this ground is to give up the value of the first claim for the definition (quoted above), since the formal properties of numbers now merely follow from the definition because the terms of the definition are reinterpreted from the properties of number, so that these properties will follow from it. The definition has become circular.
The real difficulty lies in the concept of the class. Dogmatic realism is prone to find here an entity for which, as it is obviously not a physical thing, a home must be provided in some region of "being." Hence arises the realm of subsistence, as for Plato the world of facts duplicated itself in a world of ideas. But the subsistent realm of the mathematician is even more astounding than the ideal realm of Plato, for the latterworld is a prototype of the world of things, while the world of the mathematician is peopled by all sorts of entities that never were on land or sea. The transfinite numbers of Cantor have, without doubt, a definite mathematical meaning, but they have no known representatives in the world of things, nor in the imagination of man, and in spite of the efforts of philosophers it may even be doubted whether an entity correlative to the mathematical infinite has ever been or can ever be specified.
Mr. Russell now teaches that "classes are merely symbolic" (Sci. Meth. in Phil., p. 208), but this expression still needs elucidation. It does, to be sure, avoid the earlier difficulty of admitting "new and mysterious metaphysical entities" (loc. cit., p. 204), but the "feeling of oddity" that accompanies it seems not without significance. What can be meant by a merely symbolic class of similar classes themselves merely symbolical? I do not know, unless it is that we are to throw overboard the effort aimed at arbitrary and creative definition and proceed in simple inductive and interpretative fashion. With classes as entities abandoned, we are left, until we have passed to a new point of view as to arithmetical entities, in the position of the intelligent ignoramus who defined a stock market operation as buying what you can't get with money you never had, and selling what you never owned for more than it was ever worth.
The situation seems to be that we are now face to face with new generalizations. Just as number symbols arose to denote operations gone through in countingthings when attention is diverted from the particular characteristics of the things counted, and remained a symbol for those operations with things, so now we are becoming self-conscious of the character of the operations we have been performing and are developing new symbols to express possible operations with operations. The infinity of the number series expresses the fact that it is possible to continue the enumerating process indefinitely, and when we are asked by certain mathematicians to practise ourselves in such thoughts as that for infinite series a proper part can be the equal of the whole, where equality is defined through the establishment of one-one correspondence, we are really merely informed that among the group of symbols used to denote the concrete steps of an ever open counting process are groups of symbols that can be used to indicate operations that are of the same type as the given one in so far as the characteristic of being an open series is concerned. If there were anywhere an infinity of things to count, an unintelligible supposition, it would by no means be true that any selection of things from that series would be the equivalent of all things in the series, except in so far as equivalence meant that they could be arranged in the same type of series as that from which they were drawn.
Similarly the mathematical conception of the continuum is nothing but a formulation of the manner in which the cuts of a line or the numbers of a continuous series must be chosen so that there shall remain no possible cut or number of which the choice is not indicated. Correspondence is reached between elementsof such series when the corresponding elements can be reached by an identical process. It seems to me, however, a mistake toidentifythe number continuum with the linear continuum, for the latter must include the irrational numbers, whereas the irrational number can never represent a spatial position in a series. For example, the √2 is by nature a decimal involving an infinite, i.e., an ever increasing, number of digits to express it and, by virtue of the infinity of these digits, they can never be looked upon as all given. It is then truly a number, for it expresses a genuine numerical operation, but it is not a position, for it cannot be a determinate magnitude but merely a quantity approaching a determinate magnitude as closely as one may please. That is, without its complete expression, which would be analogous to the self-contradictory task of finding a greatest cardinal number, there can be no cut in the line which is symbolized by it. But the operations of translating algebraic expressions into geometrical ones and vice versa (operations which are so important in physical investigations) are facilitated by the notion of a one to one correspondence between number and space.
When we pass to the transfinite numbers, we have nothing in the Alephs but the symbols of certain groupings of operations expressible in ordinary number series. And the many forms of numbers are all simply the result of recognizing value in naming definite groups of operations of a lower level, which may itself be a complication of processes indicated by the simple numerical signs. To create such symbols is by nomeans illegitimate and no paradox results in any forms as long as we remember that our numbers are not things but are signs of operations that may be performed directly upon things or upon other operations.
For example, let us consider such a symbol as √-5. -5 signifies the totality of a counting process carried on in an opposite sense from that denoted by +5. To take the square root is to symbolize a number, the totality of an operation, such that when the operation denoted by multiplying it by itself is performed the result is 5. Consequently the √-5 is merely the symbol of these processes combined in such a way that the whole operation is to be considered as opposite in some sense to that denoted by √5. Hence, an easy method for the representation of such imaginaries is based on the principle of analytic geometry and a system of co-ordinates.
The nature of this last generalization of mathematics is well shown by Mr. Whitehead in his monumentalUniversal Algebra. The work begins with the definition of a calculus as "The art of manipulating substitutive signs according to fixed rules, and the deduction therefrom of true propositions" (loc. cit., p. 4). The deduction itself is really a manipulation according to rules, and the truth consists essentially in the results being actually derived from the premises according to rule. Following Stout, substitutive signs are characterized thus: "a word is an instrument for thinking about the meaning which it expresses; a substitutive sign is a means of not thinking about the meaning which it symbolizes." Mathematical symbolshave, then, become substitutive signs. But this is only possible because they were at an early stage of their history expressive signs, and the laws which connected them were derived from the relations of the things for which they stood. First it became possible to forget the things in their concreteness, and now they have become mere terms for the relations that had been generalized between them. Consequently, the things forgotten and the terms treated as mere elements of a relational complex, it is possible to state such relational complexes with the utmost freedom. But this does not mean that mathematics can be created in a purely arbitrary fashion. The mark of its origin is upon it in the need of exhibiting some existing situation through which the non-contradictory character of its postulates can be verified. The real advantage of the generalization is that of all generalizations in science, namely, that by looking away from practical applications (as appears in a historical survey) results are frequently obtained that would never have been attained if our labor had been consciously limited merely to those problems where the advantages of a solution were obvious. So the most fantastic forms of mathematics, which themselves seem to bear no relation to actual phenomena, just because the relations involved in them are the relations that have been derived from dealing with an actual world, may contribute to the solutions of problems in other forms of calculus, or even to the creation of new forms of mathematics. And these new forms may stand in a more intimate connection with aspects of the real world than the original mathematics.
In 1836-39 there appeared in theGelehrte Schriften der Universität Kasan, Lobatchewsky's epoch-making "New Elements of Geometry, with a Complete Theory of Parallels." After proving that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another," Euclid, finding himself unable to prove that in every other case they were not parallel, assumed it in an axiom. But it had never seemed obvious. Lobatchewsky's system amounted merely to developing a geometry on the basis of the contradictory axiom, that through a point outside a line an indefinite number of lines can be drawn, no one of which shall cut a given line in that plane. In 1832-33, similar results were attained by Johann Bolyai in an appendix to his father's "Tentamen juventutem studiosam in elementa matheseosos puræ ... introducendi" entitled "The Science of Absolute Space." In 1824 the dissertation of Riemann, under Gauss, introduced the idea of ann-ply extended magnitude, or a study ofn-dimensional manifolds and a new road was opened for mathematical intelligence.
At first this new knowledge suggested all sorts of metaphysical hypotheses. If it is possible to build geometries ofn-dimensions or geometries in which the axiom of parallels is no longer true, why may it not be that the space in which we make our measurements and on which we base our mechanics is some one of these "non-Euclidian" spaces? And indeed many experiments were conducted in search of some clue that this might be the case. Such experiments in relation to"curved spaces" seemed particularly alluring, but all have turned out to be fruitless in results. Failure leads to investigation of the causes of failure. If our space had been some one of these spaces how would it have been possible for us to know this fact? The traditional definition of a straight line has never been satisfactory from a physical point of view. To define it as the shortest distance between two points is to introduce the idea of distance, and the idea of distance itself has no meaning without the idea of straight line, and so the definition moves in a vicious circle. On the metaphysical side, Lotze (Metaphysik, p. 249) and others (Merz,History of European Thought in the Nineteenth Century, Vol. II, p. 716) criticized these attempts, on the whole justly, but the best interpretation of the situation has been given by Poincaré.
Two lines of thought now lead to a recasting of our conceptions of the fundamental notions of geometry. On the one hand, that very investigation of postulates that had led to the discovery of the apparently strange non-Euclidian geometries was easily continued to an investigation of the simplest basis on which a geometry could be founded. Then by reaction it was continued with similar methods in dealing with algebra, and other forms of analysis, with the result that conceptions of mathematical entities have gradually emerged that represent a new stage of abstraction in the evolution of mathematics, soon to be discussed as the dominating conceptions in contemporary thought. On the other hand, there also developed the problem of the relations of these geometrical worlds to one another, which hasbeen primarily significant in helping to clear up the relations of mathematics in its "pure" and "applied" forms.
Geometry passed through a stage of abstraction like that examined in connection with arithmetic. Beginning with the discovery of non-Euclidian geometry, it has been becoming more and more evident that a line need not be a name for an aspect of a physical object such as the ridge-pole line of a house and the like, nor even for the more abstract mechanical characteristic of direction of movement;—although the persistency with which intuitionally minded geometers have sought to adapt such illustrations to their needs has somewhat obscured this fact. However, even a cursory examination of a modern treatise on geometry makes clear what has taken place. For example, Professor Hilbert begins hisGrundlagen der Geometrie, not with definition of points, lines, and planes, but with the assumption of three different systems of things (Dinge) of which the first, called points, are denoted A, B, C, etc., second, called straight lines (Gerade), are denoted a, b, c, etc., and the third, called planes, are denoted by α, β, γ, etc. The relations between these things then receive "genaue und vollständige Beschreibung" through the axioms of the geometry. And the fact that these "things" are called points, lines, and planes is not to give to them any of the connotations ordinarily associated with these words further than are determined by the axiom groups that follow. Indeed, other geometers are even more explicit on this point. Thus for Peano (I Principii di Geometria, 1889) theline is a mere class of entities, the relations amongst which are no longer concrete relations but types of relations. The plane is a class of classes of entities, etc. And an almost unlimited number of examples, about which the theorems of the geometry will express truths, can be exhibited, not one of which has any close resemblance to spatial facts in the ordinary sense.
Philosophers, it seems to me, have been slow to recognize the significance of the step involved in this last phase of mathematical thought. We have been so schooled in an arbitrary distinction between relations and concepts, that while long familiar with general ideas of concepts, we are not familiar with generalized ideas of relations. Yet this is exactly what mathematics is everywhere presenting. A transition has been made from relations to types of relations, so that instead of speaking in terms of quantitative, spatial and temporal relations, mathematicians can now talk in terms of symmetrical, asymmetrical, transitive, intransitive relational types and the like. These present, however, nothing but the empirical character that is common to such relations as that of father and son; debtor and creditor; master and servant; a is to the left of b, b of c; c of d; a is older than b, b than c, c than d, etc. Hence this is not abandonment of experience but a generalization of it, which results in a calculus potentially applicable not only to it but also to other subject-matter of thought. Indeed, if it were not for the possibility of this generalization, the almost unlimited applicability of diagrams, so useful in theclassroom, to illustrate everything from the nature of reality to the categorical imperative, as well as to the more technical usages of the psychological and social sciences, would not be understandable.
It would be a paradox, however, if starting out from processes of counting and measuring, generalizations had been attained that no longer had significance for counting or measuring, and the non-Euclidian hyper-dimensional geometries seem at first to present this paradox. But, as the outcome of our second line of thought proves, this is not the case. The investigation of the relations of different geometrical systems to each other has shown (cf. Brown, "The Work of H. Poincaré,"Journ. of Phil., Psy., and Sci. Meth., Vol. XI, No. 9, p. 229) that these different systems have a correspondence with one another so that for any theorem stated in one of them there is a corresponding theorem that can be stated in another. In other words, given any factual situation that can be stated in Euclidian geometry, the aspect treated as a straight line in the Euclidian exposition will be treated as a curve in the non-Euclidian, and a situation treated as three-dimensional by Euclid's methods can be treated as of any number of dimensions when the proper fundamental element is chosen, and vice versa, although of course the element will not be the line or plane in our empirical usage of the term. This is what Poincaré means by saying that our geometry is a free choice, but not arbitrary (The Value of Science, Pt. III, Ch. X, Sec. 3), for there are many limitations imposed by fact upon the choice, and usually there is someclear indication of convenience as to the system chosen, based on the fundamental ideal of simplicity.
It is evident, then, that geometry and arithmetic have been drawing closer together, and that to-day the distinction between them is somewhat hard to maintain. The older arithmetic had limited itself largely to the study of the relations involved in serial orders as suggested by counting, whereas geometry had concerned itself primarily with the relations of groups of such series to each other when the series, or groups of series, are represented as lines or planes. But partly by interaction in analytic geometry, and partly in the generalization of their own methods, both have come to recognize the fundamental character of the relations involved in their thought, and arithmetic, through the complex number and the algebraic unknown quantities, has come to consider more complex serial types, while geometry has approached the analysis of its series through interaction with number theory. For both, the content of their entities and the relations involved have been brought to a minimum. And this is true even of such apparently essentially intuitional fields as projective geometry, where entities can be substituted for directional lines and the axioms be turned into relational postulates governing their configurations.
Nevertheless, geometry like arithmetic, has remained true to the need that gave it initial impulse. As in the beginning it was only a method of dealing with a concrete situation, so in the end it is nothing but such a method, although, as in the case of arithmetic, from ever closer contact with the situation in question, ithas been led, by refinements that thoughtful and continual contact bring, to dissect that situation and give heed to aspects of it which were undreamed of at the initial moment. In a sense, then, there are no such things as mathematical entities, as scholastic realism would conceive them. And yet, mathematics is not dealing with unrealities, for it is everywhere concerned with real rational types and systems where such types may be exemplified. Or we can say in a purely practical way that mathematical entities are constituted by their relations, but this phrase cannot here be interpreted in the Hegelian ontological sense in which it has played so great and so pernicious a part in contemporary philosophy. Such metaphysical interpretation and its consequences are the basis of paradoxical absolutisms, such as that arrived at by Professor Royce (World and the Individual, Vol. II, Supplementary Essay). The peculiar character of abstract or pure mathematics seems to be that its own operations on a lower level constitute material which serves for the subject-matter with which its later investigations deal. But mathematics is, after all, not fundamentally different from the other sciences. The concepts of all sciences alike constitute a special language peculiarly adapted for dealing with certain experience adjustments, and the differences in the development of the different sciences merely express different degrees of success with which such languages have been formulated with respect to making it possible to predict concerning not yet realized situations. Some sciences are still seeking their terms and fundamentalconcepts, others are formulating their first "grammar," and mathematics, still inadequate, yearly gains both in vocabulary and flexibility.
But if we are to conceive mathematical entities as mere terminal points in a relational system, it is necessary that we should become clear as to just what is meant by relation, and what is the connection between relations and quantities. Modern thought has shown a strong tendency to insist, somewhat arbitrarily, on the "internal" or "external" character of relations. The former emphasis has been primarily associated with idealistic ontology, and has often brought with it complex dialectic questions as to the identity of an individual thing in passing from one relational situation to another. The latter insistence has meant primarily that things do not change with changing relations to other things. It has, however, often implied the independent existence, in some curiously metaphysical state, of relations that are not relating anything, and is hardly less paradoxical than the older view. In the field of physical phenomena, it seems to triumph, while the facts of social life, on the other hand, lend some countenance to the view of the "internalists." Like many such discussions, the best way around them is to forget their arguments, and turn to a fresh and independent investigation of the facts in question.
As I write, the way is paved for me by Professor Cohen (Journ. of Phil., Psy., and Sci. Meth., Vol. XI,No. 23, Nov. 5, 1914, pp. 623-24), who outlines a theory of relations closely allied to that which I have in mind. Professor Cohen writes: "Like the distinction between primary and secondary qualities, the distinction between qualities and relations seems to me a shifting one because the 'nature' of a thing changes as the thing shifts from one context to another.... To Professors Montague and Lovejoy the 'thing' is like an old-fashioned landowner and the qualities are its immemorial private possessions. A thing may enter into commercial relations with others, but these relations are extrinsic. It never parts with its patrimony. To me, the 'nature' of a thing seems not to be so private or fixed. It may consist entirely of bonds, stocks, franchises, and other ways in which public credit or the right to certain transactions is represented.... At any rate, relations or transactions may be regarded as wider or more primary than qualities or possessions. The latter may be defined as internal relations, that is, relationswithinthe system that constitutes the 'thing.' The nature of a thing contains an essence, i.e., a group of characteristics which, in any given system or context, remain invariant, so that if these are changed the things drop out of our system ... but the same thing may present different essences in different contexts. As a thing shifts from one context to another, it acquires new relations and drops old ones, and in all transformations there is a change or readjustment of the line between the internal relations which constitute theessence and the external relations which are outside the inner circle...."
Before continuing, however, I wish to make certain interpretations of these statements for which, of course, Professor Cohen is not responsible, and with which he would not be wholly in agreement. My general attitude will be shown by the first comment. Concepts are only means of denoting fragments of experience directly or indirectly given. If we then try to speak of a "nature of a thing" two interpretations of this expression are possible. The "thing" as such is only a bit of reality which some motive, that without undue extension of the term can be called practical, has led us to treat as more or less isolable from the rest of reality. Its nature, then, may consist of either its relations to other practically isolated realities or things, its actual effective value in its environment (and hence shift with the environment as Professor Cohen points out), or may consist of its essence, the "relations within the system," considered from the point of view of the potentialities implied by these for various environments. In the first sense the nature may easily change with change in environment, but if it changes in the second sense, as Professor Cohen remarks, it "drops out of our system." This I should interpret as meaning that we no longer have that thing, but some other thing selected from reality by a different purpose and point of view. I should not say with Professor Cohen that "the same thing may present different essences in different contexts." Every reality is more than one thing—man is an aggregate of atoms,a living being, an animal, and a thinker, and all of these are different things in essence, although having certain common characteristics. All attribution of "thingship" is abstraction, and all particular things may be said to participate in higher, i.e., more abstract, levels of thingship. Hence the effort to retain a thingship through a changing of essence seems to me but the echo of the motive that has so long deduced ontological monism from the logical fact that to conceive any two things is at least to throw them into a common universe of discourse. Consequently I should part company from Professor Cohen on this one point (which is perhaps largely a matter of definition, though here not unimportant) and distinguish merely the nature of a thing asactualand aspotential. Of these the former alone changes with the environment, while the latter changes only as the thing ceases to be by passing into some other thing. In other words, if the example does not do violence to Professor Cohen's thought, I can quite understand this paper as a stimulator of criticism, or as a means of kindling a fire. Professor Cohen would, I suspect, take this to mean that the same thing—this paper—must be looked upon as having two different essences in two different contexts, for "the same thing may possess two different essences in different contexts," whereas I should prefer to interpret the situation as meaning that there are before me three (and as many more as may be) different things having three different essences: first, the paper as a physical object having a considerable number of definite properties; second,written words, which are undoubtedly in one sense mere structural modifications of the physical object paper (i.e., coloring on it by ink, etc.), but whose reality for my purpose lies in the power of evoking ideas acquired by things as symbols (things, indeed, but things whose essence lies in the effects they produce upon a reader rather than in their physical character); and third, the chemical and combustion producing properties of the paper. Now it is simpler for me to consider the situation as one in which three things have a common point in thingship, i.e., an abstract element in common, than to think of "athing" shifting contexts and thereby changing its essence.
But now my divergence from Professor Cohen becomes more marked. He continues with the following example (p. 622): "Our neighbor M. is tall, modest, cheerful, and we understand a banker. His tallness, modesty, cheerfulness, and the fact that he is a banker we usually regard as his qualities; the fact that he is our neighbor is a relation which he seems to bear to us. He may move his residence, cease to be our neighbor, and yet remain the same person with the same qualities. If, however, I become his tailor, his tallness becomes translated into certain relations of measurement; if I become his social companion, his modesty means that he will stand in certain social relations with me, etc." In other words, we are illustrating the doctrine that "qualities are reducible to relations" (cf. p. 623). This doctrine I cannot quite accept without modification, for I cannot tell what it means. Without any presuppositions as to subjectivity orconsciousness (cf. p. 623, (a).) there are in the world as I know it certain colored objects—let the expression be taken naïvely to avoid idealistico-realistic discussion which is here irrelevant. Now it is as unintelligible to me that the red flowers and green leaves of the geraniums before my windows should be reducible to mere relations in any existential sense, as it would be to ask for the square root of their odor, though of course it is quite intelligible that the physical theory and predictions concerning green and red surfaces (or odors) should be stated in terms of atomic distances and ether vibrations of specific lengths. The scientific conception is, after all, nothing more than an indication of how to take hold of things and manipulate them to get foreseen results, and its entities are real things only in the sense that they are the practically effective keynotes of the complex reality. Accordingly, instead of reducing qualities to relations, it seems to me a much more intelligible view to consider relations as abstract ways of taking qualities in general, as qualities thought of in their function of bridging a gap or making a transition between two bits of reality that have previously been taken as separate things. Indeed, it is just because things are not ontologically independent beings (but rather selections from genuinely concatenated existence) that relations become important as indications of the practical significance of qualitative continuities which have been neglected in the prior isolation of the thing. Thus, instead of an existential world that is "a network of relations whose intersections are called terms"(p. 622), I find more intelligible a qualitatively heterogeneous reality that can be variously partitioned into things, and that can he abstractly replaced by systems of terms and relations that are adequate to symbolize their effective nature in particular respects. There is a tendency for certain attributes to maintain their concreteness (qualitativeness) in things, and for others to suggest the connection of things with other things, and so to emphasize a more abstract aspect of experience. Thus then arises a temporary and practical distinction that tends to be taken as opposition between qualities and relations. As spatial and temporal characteristics possess their chief practical value in the connection of things, so they, like Professor Cohen's neighbor-character, are ordinarily assumed abstractly as mere relations, while shapes, colors, etc., and Professor Cohen's "modesty, tallness, cheerfulness," may be thought of more easily without emphasis on other things and so tend to be accepted in their concreteness as qualities, but how slender is the dividing-line Professor Cohen's easy translation of these things into relations makes clear.
Taken purely intellectualistically, there would be first a fiction of separation in what is really already continuous and then another fiction to bridge the gap thus made. This would, of course, be the falsification against which Bergson inveighs. But this interpretation is to misunderstand the nature of abstraction. Abstraction does not substitute an unreal for a real, but selects from reality a genuine characteristic of it which is adequate for a particular purpose. Thus toconceive time as a succession of moments is not to falsify time, but to select from processes going on in time a characteristic of them through which predictions can be made, which may be verified and turned into an instrument for the control of life or environment. A similar misunderstanding of abstraction, coupled with a fuller appreciation than Bergson evinces of the value of its results, has led to the neo-realistic insistence on turning abstractions into existent entities of which the real world is taken to be an organized composite aggregate.
The practice of turning qualities into merely conscious entities has done much to obscure the status of scientific knowing, for it has left mere quantity as the only real character of the actual world. But once take a realistic standpoint, and quantity is no more real than quality. For primitive man, the qualitative aspect of reality is probably the first to which he gives heed, and it is only through efforts to get along with the world in its qualitative character that its quantitative side is forced upon the attention. Then so-called "exact" science is born, but it does not follow that qualities henceforth become insignificant. They are still the basis of all relations, even of those that are most directly construed as quantitative. Quality and quantity are only different aspects of the world which the status of our practical life leads us to take separately or abstractly. "Thing" is no less an abstraction, in which we disregard certain continuities with the rest of the world because we are so constituted that the demands of living make it expedient to do so.Things once given, further abstractions become possible, among which are those leading to mathematical thinking, in which higher abstractions are made, guided always by the "generating problem" (cf. Karl Schmidt,Jour. of Phil., Psy., and Sci. Meth., Vol. X, No. 3, 1913, pp. 64-75).
The controlling factors for the progress of scientific thought are inventions that lead the scientist into closer contact with his data, and direct attention to complexities which would otherwise have escaped observation. This end is best fulfilled by conceiving entities that under some point of view are practically isolable from the context in which they occur. Only too often philosophic thought has confused this practical segregation with ontological separation, and so been obliged to introduce metaphysical and external relations to bring these entities together again in a real world, when in reality they have never been separated from one another and hence not from the real world. Furthermore, the conceptual model, built on the lines of a calculus of mathematics, is often considered the truthpar excellenceafter the analogy of a camera's portrait. Progress in science, however, shows that these models have to be continually rebuilt. Each seems to lead to further knowledge that necessitates its reconstruction, so that truth takes on an ideal value as an ultimate but unattained, if not unattainable, goal, while existing science becomes reduced toworking hypotheses. From a positivistic point of view, however, the goal is not only practically unattainable, but it is irrational, for there seems to be every evidence that it expresses something contrary to the nature of the real. Yet scientific theory is not wholly arbitrary. We cannot construe nature as constituted of any sorts of entities that may suit our whim. And this is because science itself recognizes that its entities are not really isolated, but are endowed with all sorts of properties that serve to connect them with other entities. They are only symbols of critical points of reality which, conceived in a certain way, make the behavior of the whole intelligible. Indeed, the only significant sense in which they are true for the scientist is that they indicate real connections that might otherwise have been overlooked, and this is only possible from the fact that reality has the characteristics that they present and that, with their relations, they give an approximate presentation of what is actually presented just as a successful portrait painter considers the individuality of the eyes, nose, mouth, etc., although he does not imply that a face is compounded of these separate features as a house is built of boards.
The atomic theory, for example, has undoubtedly been of the greatest service to chemistry, and atoms undoubtedly denote a significant resting-place in the analysis of the physical world. Yet in the light of electron theories, it is becoming more and more evident that atoms are not ultimate particles, and are not even all alike (Becker, "Isostasy and Radioactivity,"Sci.,Jan. 29, 1915) when they represent a single substance. Again, while there is as yet no evidence to suggest that the electron must itself be considered as divisible (unless it be the distinction between the positive and negative electron), there are suggestions that electrons may themselves arise and pass away (cf. Moore,Origin, and Nature of Life, p. 39). "A wisely positivistic mind," writes Enriques (Problems of Science, p. 34), "can see in the atomic hypothesis only a subjective representation,"34and, we might add, "in any other hypothesis." He continues (pp. 34-36): "robbing the atom of the concrete attributes inherent in its image, we find ourselves regarding it as a mere symbol. The logical value of the atomic theory depends, then, upon the establishment of a proper correspondence between the symbols which it contains and the reality which we are trying to represent.
"Now, if we go back to the time when the atomic theory was accepted by modern chemistry, we see that the plain atomic formulæ contain only the representation of the invariable relations in the combination of simple bodies, in weight and volume; these last being taken in relation to a well-defined gaseous state.
"But, once introduced into science, the atomic phraseology suggested the extension of the meaning of the symbols, and the search in reality for facts in correspondence with its more extended conception.
"The theory advances, urged on, as it were, by itsmetaphysical nature, or, if you wish, by the association of ideas which the concrete image of the atom carries with it.
"Thus for the plain formulæ we have substituted, in the chemistry of carbon compounds, structural formulæ, which come to represent, thanks to the disposition or grouping of atoms in a molecule, structural relations of the second degree, that is to say, relations inherent in certain chemical transformations with respect to which some groups of elements have in some way an invariant character. And here, because the image of a simple molecule upon a plane does not suffice to explain, for example, the facts of isomerism, we must resort to the stereo-chemical representation of Van't Hoff.
"Must we further recall the kinetic theory of gases, the facts explained by the breaking up of molecules into ions, the hypothesis suggested, for example, by Van der Waals by the view that an atom has an actual bulk? Must we point to a physical phenomenon of quite a different class, for example, to the coloring of the thin film forming the soap-bubbles which W. Thomson has taken as the measure of the size of a molecule?
"Such a résumé of results shows plainly that we cannot help the progress of science by blocking the path of theory and looking only at its positive aspects, that is to say, at the collection of facts that it explains. The value of a theory lies rather in the hypothesis which it can suggest, by means of the psychological representation of the symbols.
"We shall not draw from all this the conclusion that the atomic hypothesis ought to correspond to the extremely subtle sensations of a being resembling a perfected man. We shall not even reason about the possibility of those imaginary sensations, in so far as they are conceived simply as an extension of our own. But we shall repeat, in regard to the atomic theory, what an illustrious master is said to have remarked as to the unity of matter: if on first examination a fact seems possible which contradicts the atomic view of things, there is a strong probability that such a fact will be disproved by experience.
"Does not such a capacity for adaptation to facts, thus furnishing a model for them, perhaps denote thepositivereality of a theory?"
And the above principles are as true of mathematical concepts as of chemical. Everywhere it is "capacity of adaptation to facts" that is the criterion of a branch of mathematics, except, of course, that in mathematics the facts are not always physical facts. Mathematics has successfully accomplished a generalization whereby its own methods furnish the material for higher generalizations. The imaginary number and the hyper-dimensional or non-Euclidian geometries may be absurd if measured by the standard of physical reality, but they nevertheless have something real about them in relation to certain mathematical processes on a lower level. There is no philosophic paradox about modern arithmetic or geometry, once it is recognized that they are merely abstractions of genuine features of simpler and more obviously practical manipulationsthat are clearly derived from the dealing of a human being with genuine realities.
In the light of these considerations, I cannot help feeling that the frequent attempts of mathematicians with a philosophical turn of mind, and philosophers who are dipping into mathematics, to derive geometrical entities from psychological considerations are quite mistaken, and are but another example of those traditional presuppositions of psychology which, Professor Dewey has pointed out (Jour. of Phil., Psy., and Sci. Meth., XI, No. 19, p. 508), were "bequeathed by seventeenth-century philosophy to psychology, instead of originating within psychology" ... that "were wished upon it by philosophy when it was as yet too immature to defend itself."
Henri Poincaré (Science and Hypothesis, Ch. IV,The Value of Science, Ch. IV) and Enriques (Problems of Science, Ch. IV, esp. B—The Psychological Acquisition of Geometrical Concepts) furnish two of the most familiar examples of this sort of philosophizing. Each isolates special senses, sight, touch, or motion, and tries to show how a being merely equipped with one or the other of these senses might arrive at geometrical conceptions which differ, of course, from space as represented by our familiar Euclidian geometry. Then comes the question of fusing these different sorts of experience into a single experience of which geometry may be an intelligible transcription. Enriques finds a parallel between the historical development and the psycho-genetic development of the postulates of geometry (loc. cit., p. 214seq.). "The threegroups of ideas that are connected with the concepts that serve as the basis for the theory of continuum (Analysis situs), of metrical, and of projective geometry, may be connected, as to their psychological origin, with three groups of sensations: with the general tactile-muscular sensations, with those of special touch, and of sight, respectively." Poincaré even evokes ancestral experience to make good his case (Sci. and Hyp., Ch. V, end). "It has often been said that if individual experience could not create geometry, the same is not true of ancestral experience. But what does that mean? Is it meant that we could not experimentally demonstrate Euclid's postulate, but that our ancestors have been able to do it? Not in the least. It is meant that by natural selection our mind hasadapteditself to the conditions of the external world, that it has adopted the geometrymost advantageousto the species: or in other words, themost convenient."
Now undoubtedly there may be a certain modicum of truth in these statements. As implied by the last quotation from Poincaré, the modern scientist can hardly doubt that the fact of the adaptation of our thinking to the world we live in is due to the fact that it is in that world that we evolved. As is implied by both writers, if one could limit human contact with the world to a particular form of sense response, thought about that world would take place in different terms from what it now does and would presumably be less efficient. But these admissions do not imply that any light is thrown upon the nature of mathematical entities by such abstractions. Russell (Scientific Method inPhilosophy) is in the curious position of raising arithmetic to a purely logical status, but playing with geometry and sensation after the manner of Poincaré, to whom he gives somewhat grudging praise on this account.
The psychological methods upon which all such investigations are based are open to all sorts of criticisms. Chiefly, the conceptions on which they are based, even if correct, are only abstractions. There is not the least evidence for the existence of organisms with a single differentiated sense organ, nor the least evidence that there ever was such an organism. Indeed, according to modern accounts of the evolution of the nervous system (cf. G. H. Parker,Pop. Sci. Month., Feb., 1914) different senses have arisen through a gradual differentiation of a more general form of stimulus receptor, and consequently, the possibility of the detachment of special senses is the latter end of the series and not the first. But, however this may be, the mathematical concepts that we are studying have only been grasped by a highly developed organism, man, but they had already begun to be grasped by him in an early stage of his career before he had analyzed his experience and connected it with specific sense organs. It may of course be a pleasant exercise, if one likes that sort of thing, to assume with most psychologists certain elementary sensations, and then examine the amount of information each can give in the light of possible mathematical interpretations, but to do so is not to show that a being so scantily endowed would ever have acquired a geometry of the type inquestion, or any geometry at all. Inferences of the sort are in the same category with those from hypothetical children, that used to justify all theories of the pedagogue and psychologist, or from the economic man, that still, I fear, play too great a part in the world of social science.
The real nature of intelligence as it appears in the development of mathematics is something quite other than that of sensory analysis. Intelligence is fundamentally skill, and although skill may be acquired in connection with some sort of sensory contact of an organism and environment, it is only determined by that contact in the sense that if the sensory conditions were different the needs of the organism might be different, and the kind and degree of skill it could attain would be other than under the conditions at first assumed. Whenever the beginnings of mathematics appear with primitive people, we find a stage of development that calls for the exercise of skill in dealing with certain practical situations. Hence we found early in our investigations that it was impossible to affirm a weak intelligence from limited achievements in counting, just as it would be absurd to assume the feeble intelligence of a philosopher from his inability to manipulate a boomerang. The instance merely suggests a kind of skill that he has never been led to acquire.
Yet it is possible to distinguish intellectual skill, orbetter skills, from physical or athletic prowess. Primarily, it is directed at the formation and use of concepts, and the concept is only a symbol that can be substituted for experiences. A well-built concept is a part of a system of concepts where relations have taken the place of real connections in such a fashion that, forgetting the actuality, it is possible to present situations that have never occurred or at least are not immediately given at the time and place of the presentation, and to substitute them for actual situations in such a fashion that these may be expediently met, if or when such situations present themselves. An isolated concept, that is, one not a part of any system, is as mythical an entity as any savage ever dreamed. Indeed, it would add much to the clearness of our thinking if we could limit the use of "intelligence" to skill in constructing and using different systems of concepts, and speak concretely of mathematical intelligence, philosophical intelligence, economic intelligence, historic intelligence, and the like. The problem of creative intelligence is, after all, the problem of the acquisition of certain forms of skill, and while the general lines are the same for all knowledge (because the instruments are everywhere symbolic presentations, or concepts), in each field the situation studied makes different types of difficulties to be overcome and suggest different methods of attaining the object.
In mathematics, the formal impulse to reduce the content of fundamental concepts to a minimum, and to stress merely relations has been most successful. We saw its results in such geometries as Hilbert's andPeano's, where the empty name "entity" supplants the more concrete "point," and the "1" of arithmetic has the same character. In the social sciences, however, such examples as the "political" and the "economic" man are signal failures, while, perhaps, the "atom" and the "electron" approach the ideal in physics and chemistry. In mathematics, all further concepts can be defined by collections of these fundamental entities constituted in certain specified ways. And it is worth noting that both factually and logically a collection of entities so defined is not a mere aggregate, but possesses a differentiated character of its own which, although the resultant of its constitution, is not a property of any of its elements. A whole number is thus a collection of 1s, but the properties of the whole number are something quite different from that of the elements through which it is constituted, just as an atom may be composed of electrons and yet, in valency, possess a property that is not the direct analogue of any property possessed by electrons not so organized.
Natural science, however, considers such building up of its fundamental entities into new entities as a process taking place in time rather than as consequent upon change of form of the whole rendering new analytic forms expedient. Hence it points to the occurrence of genuine novelties in the realm of objective reality. Mathematics, on the other hand, has generalized its concepts beyond the facts implied in spatial and temporal observations, so that while significant in both fields by virtue of the nature of its abstractions,its novelties are the novelties of new conceptual formations, a distinguishing of previously unnoted generalizations of relations existent in the realm of facts. But the fact that time has thus passed beyond its empirical meaning in the mathematical realm is no ground for giving mathematics an elevated position as a science of eternal realities, of subsistent beings, or the like. The generalization of concepts to cover both spatial and temporal facts does not create new entities for which a home must be provided in the partition of realities. Metaphysicians should not be the "needy knife grinders" of M. Anatole France (cf.Garden of Epicurus, Ch. "The Language of the Metaphysicians"). Nevertheless, the success of abstraction for mathematical intelligence has been immense.
No significant thinking is wholly the work of an individual man. Ideas are a product of social coöperation in which some have wrested crude concepts from nature, others have refined them through usage, and still others have built them into an effective system. The first steps were undoubtedly taken in an effort to communicate, and progress has been in part the progress of language. The original nature of man may have as a part those reactions which we call curiosity, but, as Auguste Comte long ago pointed out (Lévy-Bruhl,A. Comte, p. 67), these reactions are among the feeblest of our nature and without the pressure of practical affairs could hardly have advanced the race beyond barbarism. Science was the plaything of the Greek, the consolation of the Middle Ages, and only for the modern has it become an instrument in suchfashion as to mark an epoch in the still dawning discovery of mind.
Man is, after all, rational only because through his nervous system he can hold his immediate responses in check and finally react as a being that has had experiences and profited by them. Concepts are the medium through which these experiences are in effect preserved; they express not merely a fact recorded but also the significance of a fact, not merely a contact with the world but also an attitude toward the future. It may be that the mere judgment of fact, a citation of resemblances and differences, is the basis of scientific knowledge, but before knowledge is worthy of the name, these facts have undergone an ideal transformation controlled by the needs of successful prediction and motivated by that self-conscious realization of the value of control which has raised man above the beasts of the field.
The realm of mathematics, which we have been examining, is but one aspect of the growth of intelligence. But in theory, at least, it is among the most interesting, since in it are reached the highest abstractions of science, while its empirical beginnings are not lost. But its processes and their significance are in no way different in essence from those of the other sciences. It marks one road of specialization in the discovery of mind. And in these terms we may read all history. To quote Professor Woodbridge (Columbia University Quarterly, Dec., 1912, p. 10): "We may see man rising from the ground, startled by the first dim intimationthat the things and forces about him are convertible and controllable. Curiosity excites him, but he is subdued by an untrained imagination. The things that frighten him, he tries to frighten in return. The things that bless him, he blesses. He would scare the earth's shadow from the moon and sacrifice his dearest to a propitious sky. It avails not. But the little things teach him and discipline his imagination. He has kicked the stone that bruised him only to be bruised again. So he converts the stone into a weapon and begins the subjugation of the world, singing a song of triumph by the way. Such is his history in epitome—a blunder, a conversion, a conquest, and a song. That sequence he will repeat in greater things. He will repeat it yet and rejoice where he now despairs, converting the chaos of his social, political, industrial, and emotional life into wholesome force. He will sing again. But the discovery of mind comes first, and then, the song."