The problem of the nature, office and test of ideas is not exhausted in the matter of physical conceptions we have discussed in the preceding chapter. Mathematical ideas are indispensable instruments of physical research, and no account of the method of the latter is complete that does not take into account the applicability of mathematical conceptions to natural existence. Such ideas have always seemed to be the very type of pure conceptions, of thought in its own nature unadulterated with material derived from experience. To a constant succession of philosophers, the rôle of mathematics in physical analysis and formulation has seemed to be a proof of the presence of an invariant rational element within physical existence, which is on that account something more than physical; this rôle of conceptions has been the stumbling block of empiricists in trying to account for science on an empirical basis.

The significance of mathematics for philosophy is not confined to this seemingly superphysical phase of the physical world, and a superempirical factor in knowledge of it. Mathematical conceptions as expressions of pure thought have also seemed to provide the open gateway to a realm of essence that is independent of existence, physical or mental—a self-subsisting realm of ideal and eternal objects which are the objects of the highest—that is, the most assured—knowledge. As was earlier noted, the Euclidean geometry was undoubtedly the pattern for the development of a formally rational logic; it was also a marked factor in leading Plato to his doctrine of a world of supersensible and superphysical ideal objects. The procedure of mathematics has, moreover, always been the chief reliance of those who have asserted that the demonstrated validity of all reflective thinking depends upon rational truths immediately known without any element of inference entering in. For mathematics was supposed to rest upon a basis of first truths or axioms, self-evident in nature, and needing only that the eye of reason should fall upon them to be recognized for what they are. The function of indemonstrables, of axioms and definitions, in mathematical deduction has been the ground for the distinction between intuitive and discursive reason, just as deductions have been taken to be the convincing proof that there is a realm of pure essences logically connected with one another:—universals having internal bonds with one another.

The theory that conceptions are definitions of consequences of operations needs therefore to be developed with reference to mathematical ideas both for its own sake, and for its bearing upon the philosophic issues which are basic to the logic of rationalism and to the metaphysics of essences and universals or invariants. We shall begin with mathematical concepts in their physical sense, and then consider them as they are developed apart from existential application. Although Descartes defined natural existence as extension, the classic tradition that only sense and imagination, among the organs of mind, refer to physical existence caused him to feel bound to offer justification for the doctrine that natural phenomena can be scientifically stated by purely mathematical reasoning without need of recourse to experimentation. His proof of the existence of God served the purpose of justifying this application of mathematical conceptions in physics. With Spinoza, the correspondence between physical existence and ideas did not need to be substantiated by God because it was God. This correspondence when modified to give thought such a priority as to include existence within itself became the animating motif of Post-Kantian idealistic systems.

Newton, being a man of science rather than a professed philosopher, made such assumptions as he thought scientific procedure demanded and its conclusions warranted. The skepticism of Hume (anticipated, however, by Berkeley as far as the Newtonian metaphysics of mathematical space and time were concerned) was, as is notorious, the chief factor in leading Kant to regard space and time as a priori forms of all perceptual experience. One of the grounds for Kant’s conviction that his doctrine was inconvertible was because he thought it had the support of Newtonian physics and was necessary to give that physics a firm foundation.

The consideration important for our special purpose is, however, the fact that Newton with respect to the doctrine of space, time and motion (involved in all conception of things dealt with in the universal physics of nature) frankly deserted the empirical method he professed to use in respect to the properties of the ultimate fixed substances. At the same time, he regarded the physical and the mathematical as complementary conceptions of two sets of properties of fixed forms of immutable Being. He assumed, in addition to atoms having mass, inertia and extension, the existence of empty immaterial space and time in which these substances lived, moved and had their being. The combination of the properties of these two kinds of Being provided the union of the empirically observed properties of phenomena with those that were rational and mathematical:—a union so complete and so intimate that it conferred upon the Newtonian system that massive solidity and comprehensiveness which seemed to render his system in its essential framework the last word possible of the science of nature.

Definition of space, time and motion from “the relation they bear to sense” is according to him “a vulgar prejudice.” As well as any contemporary physicist, he knew that phenomena of space, time and motion in their perceived forms are found in a frame of reference which is relative to an observer. In escape from the relativity of observable traits of the spatial and temporal motions of bodies, he assumed the existence of a fixed container of empty space in which bodies are located and an equably flowing time, empty in itself, in which changes take place. From this assumption, it followed that atoms have an intrinsically measurable motion of their own, independent of any connection with an observer. Absolute space, time and motion were thus the immutable frame within which all particular phenomena take place.

The assumption of these rational absolutes was also required by his basic metaphysics of fixed substances having their own inherent and unchangeable (or essential) properties of mass, extension and inertia. The sole ground of assurance that ultimate hard and massy particles persist without internal change, that all changes are merely matters of their external “separations and associations,” was the existence of something empty and fixed within which the latter occur. Without such an intervening medium, interaction with one another would be equivalent to internal changes in atoms. Space provided the condition under which changes would be external and indifferent to ultimate physical substances. Since, then, changes have nothing to do directly with the relations of atoms to one another, the temporal order of changes cannot be connected with the atoms themselves. There must be some evenly flowing external change—in reality no change at all—in reference to which they have fixed positions of before and after and of simultaneity. Since velocity and acceleration of observed motions would be disjoined from absolute position and date if they were relative to an observer—to the disruption of the whole physical scheme—motion must also be absolute.

While professing empiricism, Newton thus got the benefit of the rationalistic system of strict deductive necessity. Invariant time, space and motion furnished phenomena those properties to which mathematical reasoning could be attached as a disclosure of inherent properties. The positions of bodies could be treated as an assemblage of geometrical points and the temporal properties of their motions be considered as if they were mere instants. Everything observed had, in its scientific treatment, to conform mathematically to specifications laid down by the mathematics of space and time. Until our own day, until the conception of the determination of simultaneity of occurrence was challenged by Einstein, the system continued to receive at least Pickwickian assent from scientists.

There is no trouble of course about the determination of simultaneity when two events occur within one and the same region of observation. Newton, because of his assumption of absolute time, assumed that the measurement of simultaneity had precise meaning for events not occurring within the same observed field. Einstein saw that this assumption was the Achilles heel of the entire scheme. He demanded an experimental method of determining simultaneity—without which events cannot be dated with respect to one another. He made the demand not on purely general principles, but because of a definite problem with relation to the velocity of light. For the existing state of the doctrine of light presented a discrepancy not to be resolved on the basis of the received scheme. The observed constancy of light with reference to the place from which its direction was observed and its velocity measured, did not agree with a fundamental principle of dynamics; with its postulate concerning frames of reference for coordinate systems having uniform movements of translation. Instead of maintaining the old theory and denying the validity of the observed result of the Michelson-Morley experiment, Einstein asked what change in conceptions was demanded by the experimental result. He saw that the measurement of time relations, centering in the concept of simultaneity, was the crucial point.

So he said, “We require a definition of simultaneity such that this definition supplies us with a method by which in particular cases the physicist can decide by experiment whether or not two events occurred simultaneously.”1 He suggested an arrangement by which two flashes of light, not in themselves capable of inclusion in one region of observation, be reflected to a mirror placed midway between the origin of the two flashes. They are simultaneous if they are then included within one and the same act of observation. To a layman, the suggestion might seem innocuous. But taken in its context, it signified that the temporal relation of events was to be measured by means of the consequences of an operation which constitutes as its outcome a single field of observed phenomena. It signified, in connection with the fact regarding the constancy of velocity of light, that events occurring at different times according to two watches keeping exactly the same time, placed at the points of the origin of the flashes, may be simultaneous. In scientific content, this was equivalent to doing away with Newton’s absolutes; it was the source of the doctrine of restricted relativity. It signified that local or individualized times are not the same as a generic common time of physics: in short, it signified that physical time designates a relation of events, not the inherent property of objects.

What is significant for our purpose is that it marked the end, as far as natural science is concerned, of the attempt to frame scientific conceptions of objects in terms of properties assigned to those objects independently of the observed consequences of an experimental operation. Since the former doctrine about the proper way to form conceptions, to the effect that agreement with antecedent properties determines the value or validity of ideas, was the doctrine common to all philosophic schools—except the pragmatic one of Peirce—the logical and philosophical transformation thus affected may be said to be more far-reaching than even the extraordinary development in the content of natural science which resulted. It is not too much to say that whatever should be future developments in discoveries about light, or that even if the details of the Einstein theory of relativity should be some time discredited, a genuine revolution, and one which will not go backward, has been effected in the theory of the origin, nature and test of scientific ideas.

In respect to the special theme of the nature of mathematico-physical conceptions, the pertinent conclusion is evident. For the conclusion of Einstein, in eliminating absolute space, time and motion as physical existences, does away with the doctrine that statements of space, time and motion as they appear in physics concern inherent properties. For that notion, it compels the substitution of the notion that they designate relations of events. As such relations, they secure, in their generality, the possibility of linking together objects viewed as events in a general system of linkage and translation. They are the means of correlating observations made at different times and places, whether by one observer or by many, so that translations may be effected from one to another. In short, they do the business that all thinking and objects of thought have to effect: they connect, through relevant operations, the discontinuities of individualized observations and experiences into continuity with one another. Their validity is a matter of their efficacy in performance of this function; it is tested by results and not by correspondence with antecedent properties of existence.

It is possible to extend this conclusion to logical forms in general. The fact that there are certain formal conditions of the validity of inference has been used as the ultimate warrant of a realm of invariant Being. But in analogy with the conclusion regarding mathematical conceptions, logical forms are statements of the means by which it is discovered that various inferences may be translated into one another, or made available with respect to one another, in the widest and most secure way. Fundamentally, the needs satisfied by inference are not fully met as long as special instances are isolated from one another.

The difference between the operational conception of conceptions and the traditional orthodox one may be indicated by an illustrative analogy.2 A visitor to a country finds certain articles used for various purposes, rugs, baskets, spears, etc. He may be struck by the beauty, elegance and order of their designs, and, assuming a purely esthetic attitude toward them, conclude that they are put to use only incidentally. He may even suppose that their instrumental use marks a degradation of their inherent nature, a concession to utilitarian needs and conveniences. A “tough-minded” observer may be convinced that they were intended to be put to use, and had been constructed for that purpose. He would, indeed, recognize that there must have been raw materials which were inherently adapted for conversion to such appliances. But he would not on that account believe the things to be original instead of being made articles; still less would he conceive them to be the original “realities” of which crude or raw material were imitations or inadequate phenomenal exemplifications. As he traced the history of these instrumentalities and found them beginning in forms which were nearer to raw materials, gradually being perfected in economy and efficiency, he would conclude that the perfecting had been an accompaniment of use for ends, changes being introduced to remedy deficiencies in prior operations and results. His tender-minded companion might, on the other hand, infer that the progressive development showed that there was some original and transcendental pattern which had been gradually approximated empirically, an archetype laid up in the heavens.

One person might argue that, while the development of designs had been a temporal process, it had been wholly determined by patterns of order, harmony and symmetry that have an independent subsistence, and that the historic movement was simply a piecemeal approximation to eternal patterns. He might elaborate a theory of formal coherence of relations having nothing to do with particular objects except that of being exemplified in them. His tough-minded companion might retort that any object made to serve a purpose must have a definite structure of its own which demands an internal consistency of parts in connection with one another, and that man-made machines are typical examples; that while these cannot be made except by taking advantage of conditions and relations previously existing, machines and tools are adequate to their function in the degree in which they produce rearrangement of antecedent things so that they may work better for the need in question. If speculatively inclined, he might wonder whether our very ideals of internal order and harmony had not themselves been formed under the pressure of constant need of redisposing of things so that they would serve as means for consequences. If not too tough-minded, he would be willing to admit that after a certain amount of internal rearrangement and organization had been effected under the more direct pressure of demand for effective instrumentalities, an enjoyed perception of internal harmony on its own account would result, and that study of formal relations might well give a clew to methods which would result in improvement of internal design for its own sake with no reference whatever to special further use.

Apart from metaphor, the existence of works of fine art, of interest in making them and of enjoyment of them, affords sufficient evidence that objects exist which are wholly “real” and yet are man-made; that making them must observe or pay heed to antecedent conditions, and yet the objects intrinsically be redispositions of prior existence; that things as they casually offer themselves suggest ends and enjoyments they do not adequately realize; that these suggestions become definite in the degree they take the form of ideas, of indications of operations to be performed in order to effect a desired eventual rearrangement. These objects, when once in existence, have their own characters and relations, and as such suggest standards and ends for further production of works of art, with less need for recourse to original “natural” objects; they become as it were a “realm” having its own purposes and regulative principles. At the same time, the objects of this “realm” tend to become over-formal, stereotyped and “academic” if the internal development of an art is too much isolated, so that there is recurrent need for attention to original “natural” objects in order to initiate new significant movements.

The notion that there are no alternatives with respect to mathematical objects save that they form an independent realm of essences; or are relations inherent in some antecedent physical structure—denominated space and time; or else are mere psychological, “mental” things, has no support in fact. The supposition that these alternatives are exhaustive is a survival of the traditional notion that identifies thought and ideas with merely mental acts—that is, those inside mind. Products of intentional operations are objectively real and are valid if they meet the conditions involved in the intent for the sake of which they are constructed. But human interaction is a contributing factor in their production, and they have worth in the human use made of them.

The discussion so far does not, however, directly touch the question of “pure” mathematics, mathematical ideas in themselves. Newton’s mathematics was professedly a mathematics of physical although non-material existence: of existential absolute space, time and motion. Mathematicians, however, often regard their distinctive conceptions as non-existential in any sense. The whole tendency of later developments, which it is unnecessary for our purposes to specify (but of which the doctrine of n-dimensional “spaces” is typical), is to identify pure mathematics with pure logic. Some philosophers employ therefore the entities of pure mathematics so as to rehabilitate the Platonic notion of a realm of essence wholly independent of all existence whatever.

Does the doctrine of the operational and experimentally empirical nature of conceptions break down when applied to “pure” mathematical objects? The key to the answer is to be found in a distinction between operations overtly performed (or imagined to be performed) and operations symbolically executed. When we act overtly, consequences ensue; if we do not like them, they are nevertheless there in existence. We are entangled in the outcome of what we do; we have to stand its consequences. We shall put a question that is so elementary that it may seem silly. How can we have an end in view without having an end, an existential result, in fact? With the answer to this question is bound up the whole problem of intentionl regulation of what occurs. For unless we can have ends-in-view without experiencing them in concrete fact, no regulation of action is possible. The question might be put thus: How can we act without acting, without doing something?

If, by a contradiction in terms, it had been possible for men to think of this question before they had found how to answer it, it would have been given up as insoluble. How can man make an anticipatory projection of the outcome of an activity in such a way as to direct the performance of an act which shall secure or avert that outcome? The solution must have been hit upon accidentally as a by-product, and then employed intentionally. It is natural to suppose that it came as a product of social life by way of communication; say, of cries that having once directed activities usefully without intent were afterwards used expressly for that purpose. But whatever the origin, a solution was found when symbols came into existence. By means of symbols, whether gestures, words or more elaborate constructions, we act without acting. That is, we perform experiments by means of symbols which have results which are themselves only symbolized, and which do not therefore commit us to actual or existential consequences. If a man starts a fire or insults a rival, effects follow; the die is cast. But if he rehearses the act in symbols in privacy, he can anticipate and appreciate its result. Then he can act or not act overtly on the basis of what is anticipated and is not there in fact. The invention or discovery of symbols is doubtless by far the single greatest event in the history of man. Without them, no intellectual advance is possible; with them, there is no limit set to intellectual development except inherent stupidity.

For long ages, symbols were doubtless used to regulate activity only ad hoc; they were employed incidentally and for some fairly immediate end. Moreover, the symbols used at first were not examined nor settled upon with respect to the office they performed. They were picked up in a casual manner from what was conveniently at hand. They carried all sorts of irrelevant associations that hampered their efficacy in their own special work. They were neither whittled down to accomplish a single function nor were they of a character to direct acts to meet a variety of situations:—they were neither definite nor comprehensive. Definition and generalization are incompetent without invention of proper symbols. The loose and restricted character of popular thinking has its origin in these facts; its progress is encumbered by the vague and vacillating nature of ordinary words. Thus the second great step forward was made when special symbols were devised that were emancipated from the load of irrelevancy carried by words developed for social rather than for intellectual purposes, their meaning being helped out by their immediate local context. This liberation from accidental accretions changed clumsy and ambiguous instruments of thought into sharp and precise tools. Even more important was the fact that instead of being adapted to local and directly present situations, they were framed in detachment from direct overt use and with respect to one another. One has only to look at mathematical symbols to note that the operations they designate are others of the same kind as themselves, that is, symbolic not actual. The invention of technical symbols marked the possibility of an advance of thinking from the common sense level to the scientific.

The formation of geometry by the Greeks is probably that which historically best illustrates the transition. Before this episode, counting and measuring had been employed for “practical” ends, that is, for uses directly involved in nearby situations. They were restricted to particular purposes. Yet having been invented and having found expression in definite symbols, they formed, as far as they went, a subject-matter capable of independent examination. New operations could be performed upon them. They could, and in no disrespectful sense, be played with; they could be treated from the standpoint of a fine art rather than from that of an immediately useful economic craft. The Greeks with their dominant esthetic interest were the ones who took this step. Of the creation by the Greeks of geometry it has been said that it was stimulated “by the art of designing, guided by an esthetic application of symmetrical figures. The study of such figures, and the experimental construction of tile figures, decorative borders, conventional sculptures, moldings and the like had made the early Greeks acquainted not only with a great variety of regular geometrical forms, but with techniques by which they could be constructed, compounded and divided exactly, in various ways. Unlike their predecessors, the Greeks made an intellectual diversion of all they undertook.” Having discovered by trial and error a large number of interrelated properties of figures, they proceeded to correlate these with one another and with new ones. They effected this work “in ways which gradually eliminated from their thought about them all guesswork, all accidental experiences such as errors of actual drawing and measurement, and all ideas except those which were absolutely essential. Their science thus became a science of ideas exclusively.”3

The importance of the intellectual transition from concrete to abstract is generally recognized. But it is often misconceived. It is not infrequently regarded as if it signified simply the selection by discriminative attention of some one quality or relation from a total object already sensibly present or present in memory. In fact it marks a change in dimensions. Things are concrete to us in the degree in which they are either means directly used or are ends directly appropriated and enjoyed. Mathematical ideas were “concrete” when they were employed exclusively for building bins for grain or measuring land, selling goods, or aiding a pilot in guiding his ship. They became abstract when they were freed from connection with any particular existential application and use. This happened when operations made possible by symbols were performed exclusively with reference to facilitating and directing other operations also symbolic in nature. It is one kind of thing, a concrete one, to measure the area of a triangle so as to measure a piece of land, and another kind—an abstract one—to measure it simply as a means of measuring other areas symbolically designated. The latter type of operation makes possible a system of conceptions related together as conceptions; it thus prepares the way for formal logic.

Abstraction from use in special and direct situations was coincident with the formation of a science of ideas, of meanings, whose relations to one another rather than to things was the goal of thought. It is a process, however, which is subject to interpretation by a fallacy. Independence from any specified application is readily taken to be equivalent to independence from application as such; it is as if specialists, engaged in perfecting tools and having no concern with their use and so interested in the operation of perfecting that they carry results beyond any existing possibilities of use, were to argue that therefore they are dealing with an independent realm having no connection with tools or utilities. This fallacy is especially easy to fall into on the part of intellectual specialists. It played its part in the generation of a priori rationalism. It is the origin of that idolatrous attitude toward universals so often recurring in the history of thought. Those who handle ideas through symbols as if they were things—for ideas are objects of thought—and trace their mutual relations in all kinds of intricate and unexpected, relationships, are ready victims to thinking of these objects as if they had no sort of reference to things, to existence.

In fact, the distinction is one between operations to be actually performed and possible operations as such, as merely possible. Shift of reflection to development of possible operations in their logical relations to one another opens up opportunities for operations that would never be directly suggested. But its origin and eventual meaning lie in acts that deal with concrete situations. As to origin in overt operations there can be no doubt. Operations of keeping tally and scoring are found in both work and games. No complex development of the latter is possible without such acts and their appropriate symbols. These acts are the originals of number and of all developments of number. There are many arts in which the operations of enumeration characteristic of keeping tally are explicitly used for measuring. Carpentry and masonry for example cannot go far without some device, however rude, for estimating size and bulk. If we generalize what happens in such instances, we see that the indispensable need is that of adjusting things as means, as resources, to other things as ends.

The origin of counting and measuring is in economy and efficiency of such adjustments. Their results are expressed by physical means, at first notches, scratches, tying knots; later by figures and diagrams. It is easy to find at least three types of situations in which this adjustment of means to ends are practical necessities. There is the case of allotment or distribution of materials; of accumulation of stores against anticipated days of need; of exchange of things in which there is a surplus for things in which there is a deficit. The fundamental mathematical conceptions of equivalence, serial order, sum and unitary parts, of correspondence and substitution, are all implicit in the operations that deal with such situations, although they become explicit and generalized only when operations are conducted symbolically in reference to one another.

The failure of empiricism to account for mathematical ideas is due to its failure to connect them with acts performed. In accord with its sensationalistic character, traditional empiricism sought their origin in sensory impressions, or at most in supposed abstraction from properties antecedently characterizing physical things. Experimental empiricism has none of the difficulties of Hume and Mill in explaining the origin of mathematical truths. It recognizes that experience, the actual experience of men, is one of doing acts, performing operations, cutting, marking off, dividing up, extending, piecing together, joining, assembling and mixing, hoarding and dealing out; in general, selecting and adjusting things as means for reaching consequences. Only the peculiar hypnotic effect exercised by exclusive preoccupation with knowledge could have led thinkers to identify experience with reception of sensations, when five minutes’ observation of a child would have disclosed that sensations count only as stimuli and registers of motor activity expended in doing things.

All that was required for the development of mathematics as a science and for the growth of a logic of ideas, that is, of implications of operations with respect one to another, was that some men should appear upon the scene who were interested in the operations on their own account, as operations, and not as means to specified particular uses. When symbols were devised for operations cut off from concrete application, as happened under the influence of the esthetic interest of the Greeks, the rest followed naturally. Physical means, the straight edge, the compass and the marker remained, and so did physical diagrams. But the latter were only “figures,” images in the Platonic sense. Intellectual force was carried by the operations they symbolized, ruler and compass were only means for linking up with one another a series of operations represented by symbols. Diagrams, etc., were particular and variable, but the operations were uniform and general in their intellectual force:—that is, in their relation to other operations.

When once the way was opened to thinking in terms of possible operations irrespective of actual performance, there was no limit to development save human ingenuity. In general, it proceeded along two lines. On the one hand, for the execution of tasks of physical inquiry, special intellectual instrumentalities were needed, and this need led to the invention of new operations and symbolic systems. The Cartesian analytics and the calculuses of Leibniz and Newton are cases in point. Such developments have created a definite body of subject-matter that, historically, is as empirical as is the historic sequence of, say, spinning-machines. Such a body of material arouses need for examination on its own account. It is subjected to careful inspection with reference to the relations found within its own content. Indications of superfluous operations are eliminated; ambiguities are detected and analyzed; massed operations are broken up into definite constituents; gaps and unexplained jumps are made good by insertion of connecting operations. In short, certain canons of rigorous interrelation of operations are developed and the old material is correspondingly revised and extended.

Nor is the work merely one of analytic revision. The detection, for example, of the logical looseness of the Euclidean postulate regarding parallels suggested operations previously unthought of, and opened up new fields—those of the hyper-geometries. Moreover, the possibility of combining various existing branches of geometry as special cases of more comprehensive operations (illustrated by the same instance) led to creation of mathematics of a higher order of generality.

I am not interested in tracing the history of mathematics. What is wanted is to indicate that once the idea of possible operations, indicated by symbols and performed only by means of symbols, is discovered, the road is opened to operations of ever increasing definiteness and comprehensiveness. Any group of symbolic operations suggests further operations that may be performed. Technical symbols are framed with precisely this end in view. They have three traits that distinguish them from casual terms and ideas. They are selected with a view to designating unambiguously one mode of interaction and one only. They are linked up with symbols of other operations forming a system such that transition is possible with the utmost economy of energy from one to another. And the aim is that these transitions may occur as far as possible in any direction. 1 “Water” for example suggests an indefinite number of acts; seeing, tasting, drinking, washing without specification of one in preference to another. It also marks off water from other colorless liquids only in a vague way. 2. At the same time, it is restricted; it does not connect the liquid with solid and gaseous forms, and still less does it indicate operations which link the production of water to other things into which its constituents, oxygen and hydrogen, enter. It is isolated instead of being a transitive concept. 3. The chemical conception, symbolized by H₂0, not only meets these two requirements which “water” fails to meet, but oxygen and hydrogen are in turn connected with the whole system of chemical elements and specified combinations among them in a systematic way. Starting from the elements and the relation defined in H₂0 one can, so to speak, travel through all the whole scope and range of complex and varied phenomena. Thus the scientific conception carries thought and action away from qualities which are finalities as they are found in direct perception and use, to the mode of production of these qualities, and it performs this task in a way which links this mode of generation to a multitude of other “efficient” causal conditions in the most economical and effective manner.

Mathematical conceptions, by means of symbols of operations that are irrespective of actual performance, carry abstraction much further; one has only to contrast “2” as attached physically to H, to “2” as pure number. The latter designates an operative relation applicable to anything whatsoever, though not actually applied to any specified object. And, of course, it stands in defined relations to all other numbers, and by a system of correspondences with continuous quantities as well. That numbers disregard all qualitative distinctions is a familiar fact. This disregard is the consequence of construction of symbols dealing with possible operations in abstraction from the actuality of performance. If time and knowledge permitted, it could be shown that the difficulties and paradoxes which have been found to attend the logic of number disappear when instead of their being treated as either essences or as properties of things in existence, they are viewed as designations of potential operations. Mathematical space is not a kind of space distinct from so-called physical and empirical space, but is a name given to operations ideally or formally possible with respect to things having spacious qualities: it is not a mode of Being, but a way of thinking things so that connections among them are liberated from fixity in experience and implication from one to another is made possible.

The distinction between physical and mathematical conception may be brought out by noting an ambiguity in the term “possible” operations. Its primary meaning is actually, existentially, possible. Any idea as such designates an operation that may be performed, not something in actual existence. The idea of the sweetness of, say, sugar, is an indication of the consequences of a possible operation of tasting as distinct from a directly experienced quality. Mathematical ideas are designations of possible operations in another and secondary sense, previously expressed in speaking of the possibility of symbolic operations with respect to one another. This sense of possibility is compossibility of operations, not possibility of performance with respect to existence. Its test is non-incompatibility. The statement of this test as consistency hardly carries the full meaning. For consistency is readily interpreted to signify the conformity of one meaning with others already had, and is in so far restrictive. “Non-incompatibility” indicates that all developments are welcome as long as they do not conflict with one another, or as long as restatement of an operation prevents actual conflict. It is a canon of liberation rather than of restriction. It may be compared with natural selection, which is a principle of elimination but not one controlling positive development.

Mathematics and formal logic thus mark highly specialized branches of intellectual industry, whose working principles are very similar to those of works of fine art. The trait that strikingly characterize them is combination of freedom with rigor—freedom with respect to development of new operations and ideas; rigor with respect to formal compossibilities. The combination of these qualities, characteristic also of works of great art, gives the subject great fascination for some minds. But the belief that these qualifications remove mathematical objects from all connection with existence expresses a religious mood rather than a scientific discovery.4

The significant difference is that of two types of possibility of operation, material and symbolic. This distinction when frozen into the dogma of two orders of Being, existence and essence, gives rise to the notion that there are two types of logic and two criteria of truth, the formal and the material, of which the formal is higher and more fundamental. In truth, the formal development is a specialized offshoot of material thinking. It is derived ultimately from acts performed, and constitutes an extension of such acts, made possible by symbols, on the basis of congruity with one another. Consequently formal logic represents an analysis of exclusively symbolic operations; it is, in a pregnant and not external sense, symbolic logic. This interpretation of mathematical and (formal) logical ideas is not a disparagement of them except from a mystical point of view. Symbols, as has already been noted, afford the only way of escape from submergence in existence. The liberation afforded by the free symbolism of mathematics is often a means of ulterior return to existential operations that have a scope and penetrating power not otherwise attainable. The history of science is full of illustrations of cases in which mathematical ideas for which no physical application was known suggested in time new existential relations.

The theory which has been advanced of the nature of essences (universals, invariants) may be tested by comparing the conditions which symbolic operations fulfill with the attributes traditionally imputed to the former. These attributes are ideality, universality, immutability, formality, and the subsistence of relations of implication that make deduction possible. There is a one to one correspondence between these characters and those of objects of thought which are defined in terms of operations that are compossible with respect to one another.

The correspondence will be approached by pointing out the traits of a machine which marks its structure in view of the function it fulfills. It is obvious that this structure can be understood not by sense but only by thought of the relations which the parts of the machine sustain to one another, in connection with the work the machine as a whole performs (the consequences it effects). Sensibly, one is merely overwhelmed in the presence of a machine by noises and forms. Clarity and order of perceived objects are introduced when forms are judged in relation to operations, and these in turn in relation to work done. Movements may be seen in isolation, and products, goods turned out, may be perceived in isolation. The machine is known only when these are thought in connection with one another. In this thought, motions and parts are judged as means; they are referred intellectually to something else; to think of anything as means is to apprehend an object in relation. Correlatively, the physical effect is judged as consequence—something related. The relation of means-consequence may thus justifiably be termed ideal in the sense of ideational.

Operations as such, that is, as connective interactions, are uniform. Physically and sensibly, a machine changes through friction, exposure to weather, etc., while products vary in quality. Processes are local and temporal, particular. But the relation of means and consequence which defines an operation remains one and the same in spite of these variations. It is a universal. A machine turns out a succession of steel spheres, like ball-bearings. These closely resemble one another, because they are products of like process. But there is no absolute exactitude among them. Each process is individual and not exactly identical with others. But the function for which the machine is designed does not alter with these changes; an operation, being a relation, is not a process. An operation determines any number of processes and products all differing from one another; but being a telephone or a cutting tool is a self-identical universal, irrespective of the multiplicity of special objects which manifest the function.

The relation is thus invariant. It is eternal, not in the sense of enduring throughout all time, or being everlasting like an Aristotelian species or a Newtonian substance, but in the sense that an operation as a relation which is grasped in thought is independent of the instances in which it is overtly exemplified, although its meaning is found only in the possibility of these actualizations.

The relation, between things as means and things as consequences, which defines a machine is ideal in another sense. It is the standard by which the value of existential processes are estimated. The deterioration or improvement in use of a concrete machine and the worth of an invention are judged by reference to efficiency in accomplishment of a function. The more adequately the functional relation can be apprehended in the abstract, the better can the engineer detect defects in an existent machine and project improvements in it. Thus the thought of it operates as a model; it has an archetypal character with respect to particular machines.

Thought of an object as an ideal therefore determines a characteristic internal structure or form. This formal structure is only approximated by existing things. One may conceive of a steam engine which has a one hundred per cent efficiency, although no such ideal is even remotely approached in actuality. Or, one may like Helmholtz conceive an ideal optical apparatus in which the defects of the existing human eye are not found. The ideal relationship of means to ends exists as a formal possibility determined by the nature of the case even though it be not thought of, much less realized in fact. It subsists as a possibility, and as a possibility it is in its formal structure necessary. That is to say, the conditions which have to be met and fulfilled in the idea of a machine having an efficiency of one hundred per cent are set by the necessities of the case; they do not alter with defects in our apprehension of them. Hence essences may be regarded as having Being independent of and logically prior to our thought of them. There is, however, in this fact nothing of the mystery or transcendental character which is often associated with it. It signifies that if one is to attain a specified result one must conform to the conditions which are means of securing this result; if one is to get the result with the maximum of efficiency, there are conditions having a necessary relationship to that intent.

This necessity of a structure marked by formal relationships which fulfill the conditions of serving as means for an end, accounts for the relations of implication which make deduction possible. One goes into a factory and finds that the operation of reaching an end, say, making in quantity shoes of a uniform standard, is subdivided into a number of processes, each of which is adapted to the one which precedes, and, until the final one, to that which follows. One does not make a miracle or mystery of the fact that while each machine and each process is physically separate, nevertheless all are adapted to one another. For he knows that they have been designed, through a “rationalization” of the undertaking, to effect this end.

The act of knowing is also highly complex. Experience shows that it also may be best effected by analysis into a number of distinct processes, which bear a serial relation to one another. Terms and propositions which symbolize the possible operations that are to control these processes are designed so that they will lead one to another with the maximum of definiteness, flexibility and fertility. In other words, they are constructed with reference to the function of implication. Deduction or dialectic is the operation of developing these implications, which may be novel and unexpected just as a tool often gives unexpected results when working under new conditions. One is entitled to marvel at the constructive power with which symbols have been devised having far-reaching and fruitful implications. But the wonder is misdirected when it is made the ground for hypostatizing the objects of thought into a realm of transcendent Being.

This phase of the discussion is not complete till it has been explicitly noted that all general conceptions (ideas, theories, thought) are hypothetical. Ability to frame hypotheses is the means by which man is liberated from submergence in the existences that surround him and that play upon him physically and sensibly. It is the positive phase of abstraction. But hypotheses are conditional; they have to be tested by the consequences of the operations they define and direct. The discovery of the value of hypothetical ideas when employed to suggest and direct concrete processes, and the vast extension of this operation in the modern history of science, mark a great emancipation and correspondent increase of intellectual control. But their final value is not determined by their internal elaboration and consistency, but by the consequences they effect in existence as that is perceptibly experienced. Scientific conceptions are not a revelation of prior and independent reality. They are a system of hypotheses, worked out under conditions of definite test, by means of which our intellectual and practical traffic with nature is rendered freer, more secure and more significant.

Our discussion has been one-sided in that it has dealt with the matter of conceptions mainly in reference to the “rationalistic” tradition of interpretation. The reasons for this emphasis are too patent to need exposition. But before leaving the topic, it should be noted that traditional empiricism has also misread the significance of conceptions or general ideas. It has steadily opposed the doctrine of their a priori character; it has connected them with experience of the actual world. But even more obviously than the rationalism it has opposed, empiricism has connected the origin, content and measure of validity of general ideas with antecedent existence. According to it, concepts are formed by comparing particular objects, already perceived, with one another, and then eliminating the elements in which they disagree and retaining that which they have in common. Concepts are thus simply memoranda of identical features in objects already perceived; they are conveniences, bunching together a variety of things scattered about in concrete experience. But they have to be proved by agreement with the material of particular antecedent experiences; their value and function is essentially retrospective. Such ideas are dead, incapable of performing a regulative office in new situations. They are “empirical” in the sense in which the term is opposed to scientific—that is, they are mere summaries of results obtained under more or less accidental circumstances.

Our next chapter will be devoted to explicit consideration of the historic philosophies of empiricism and rationalism about the nature of knowledge. Before passing to this theme, we conclude with a summary statement of the more important results reached in the present phase of discussion. First, the active and productive character of ideas, of thought, is manifest. The motivating desire of idealistic systems of philosophy is justified. But the constructive office of thought is empirical—that is, experimental. “Thought” is not a property of something termed intellect or reason apart from nature. It is a mode of directed overt action. Ideas are anticipatory plans and designs which take effect in concrete reconstructions of antecedent conditions of existence. They are not innate properties of mind corresponding to ultimate prior traits of Being, nor are they a priori categories imposed on sense in a wholesale, once-for-all way, prior to experience so as to make it possible. The active power of ideas is a reality, but ideas and idealisms have an operative force in concrete experienced situations; their worth has to be tested by the specified consequences of their operation. Idealism is something experimental not abstractly rational; it is related to experienced needs and concerned with projection of operations which remake the actual content of experienced objects.

Secondly, ideas and idealisms are in themselves hypotheses not finalities. Being connected with operations to be performed, they are tested by the consequences of these operations, not by what exists prior to them. Prior experience supplies the conditions which evoke ideas and of which thought has to take account, with which it must reckon. It furnishes both obstacles to attainment of what is desired and the resources that must be used to attain it. Conception and systems of conceptions, ends in view and plans, are constantly making and remaking as fast as those already in use reveal their weaknesses, defects and positive values. There is no predestined course they must follow. Human experience consciously guided by ideas evolves its own standards and measures and each new experience constructed by their means is an opportunity for new ideas and ideals.

In the third place, action is at the heart of ideas. The experimental practice of knowing, when taken to supply the pattern of philosophic doctrine of mind and its organs, eliminates the age-old separation of theory and practice. It discloses that knowing is itself a kind of action, the only one which progressively and securely clothes natural existence with realized meanings. For the outcome of experienced objects which are begot by operations which define thinking, take into themselves, as part of their own funded and incorporated meaning, the relation to other things disclosed by thinking. There are no sensory or perceived objects fixed in themselves. In the course of experience, as far as that is an outcome influenced by thinking, objects perceived, used and enjoyed take up into their own meaning the results of thought; they become ever richer and fuller of meanings. This issue constitutes the last significance of the philosophy of experimental idealism. Ideas direct operations; the operations have a result in which ideas are no longer abstract, mere ideas, but where they qualify sensible objects. The road from a perceptible experience which is blind, obscure, fragmentary, meager in meaning, to objects of sense which are also objects which satisfy, reward and feed intelligence is through ideas that are experimental and operative.

Our conclusion depends upon an analysis of what takes place in the experimental inquiry of natural science. It goes without saying that the wider scope of human experience, that which is concerned with distinctively human conditions and ends, does not comport, as it currently exists, with the result that the examination of natural science yields. The genuinely philosophic force, as distinct from a technical one, of the conclusion reached lies in precisely this incongruity. The fact that the most exacting type of experience has attained a marvelous treasury of working ideas that are used in control of objects is an indication of possibilities as yet unattained in less restricted forms of experience. Negatively, the result indicates the need of thoroughgoing revision of ideas of mind and thought and their connection with natural things that were formed before the rise of experimental inquiry; such is the critical task imposed on contemporary thought. Positively, the result achieved in science is a challenge to philosophy to consider the possibility of the extension of the method of operative intelligence to direction of life in other fields.

• 1.

  Einstein, Relativity, New York, 1926, p. 26. Italics not in original.

• 2.

  The phrase “conception of conceptions” is used to suggest that the interpretation is self-applying:—that is, the conception advanced is also a designation of a method to be pursued. One may lead a horse to water but cannot compel him to drink. If one is unable to perform an indicated operation or declines to do so, he will not of course get its meaning.

• 3.

  Barry, The Scientific Habit of Thought, New York, 1927, pp. 212–213.

• 4.

  “The long continued and infrequently interrupted study of absolutely invariant existences exercises a powerful hypnotic influence on the mind …. The world which it separates from the rest of experience and makes into the whole of being is a world of unchanging and apparently eternal order, the only Absolute cold intellect need not reject. A conviction thus establishes itself which finally affects the whole of waking thought: that in this experience one has at last discovered the eternal and ultimate truth.” Barry, Op. cit., pp. 182–183.