Contemporary Thought in Arithmetic and Geometry
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 those interactions 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 the Principles of Mathematics is 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 consider whether 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 latter world 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 counting things 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 elements of such series when the corresponding elements can be reached by an identical process. It seems to me, however, a mistake to identify the 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 no means 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 monumental Universal 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 symbols have, 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 the Gelehrte 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 an n-ply extended magnitude, or a study of n-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 of n-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 has been 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 his Grundlagen 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) the line 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 the classroom, 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 some clear 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, it has 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 fundamental concepts, 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.