Creative Intelligence Part 7

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 att.i.tude 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 s.h.i.+ft 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 "things.h.i.+p" is abstraction, and all particular things may be said to partic.i.p.ate in higher, i.e., more abstract, levels of things.h.i.+p. Hence the effort to retain a things.h.i.+p 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 as _actual_ and as _potential_. Of these the former alone changes with the environment, while the latter changes only as the thing ceases to be by pa.s.sing 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 things.h.i.+p, i.e., an abstract element in common, than to think of "_a_ thing" s.h.i.+fting 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 ill.u.s.trating 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 or consciousness (cf. p. 623, (a).) there are in the world as I know it certain colored objects--let the expression be taken navely 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 ent.i.ties 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 part.i.tioned 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 a.s.sumed 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 subst.i.tute an unreal for a real, but selects from reality a genuine characteristic of it which is adequate for a particular purpose. Thus to conceive 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 ent.i.ties of which the real world is taken to be an organized composite aggregate.

The practice of turning qualities into merely conscious ent.i.ties has done much to obscure the status of scientific knowing, for it has left mere quant.i.ty as the only real character of the actual world. But once take a realistic standpoint, and quant.i.ty 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 quant.i.tative 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 quant.i.tative. Quality and quant.i.ty 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 const.i.tuted 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).

V

THE FUNCTION OF THEORY IN SCIENCE

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 ent.i.ties 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 ent.i.ties 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 truth _par excellence_ after the a.n.a.logy 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 to working 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 const.i.tuted of any sorts of ent.i.ties that may suit our whim. And this is because science itself recognizes that its ent.i.ties are not really isolated, but are endowed with all sorts of properties that serve to connect them with other ent.i.ties. 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 a.n.a.lysis 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 pa.s.s 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,"[34] and, 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 formulae 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 its metaphysical nature, or, if you wish, by the a.s.sociation of ideas which the concrete image of the atom carries with it.

"Thus for the plain formulae we have subst.i.tuted, in the chemistry of carbon compounds, structural formulae, 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 cla.s.s, 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 resume 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 ill.u.s.trious 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 furnis.h.i.+ng a model for them, perhaps denote the _positive_ reality 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 manipulations that 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 ent.i.ties 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 Poincare (_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 s.p.a.ce 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. 214 _seq._).

"The three groups of ideas that are connected with the concepts that serve as the basis for the theory of continuum (_a.n.a.lysis 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." Poincare 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 has _adapted_ itself to the conditions of the external world, that it has adopted the geometry _most advantageous_ to the species: or in other words, the _most convenient_."

Now undoubtedly there may be a certain modic.u.m of truth in these statements. As implied by the last quotation from Poincare, 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 ent.i.ties by such abstractions. Russell (_Scientific Method in Philosophy_) is in the curious position of raising arithmetic to a purely logical status, but playing with geometry and sensation after the manner of Poincare, 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 a.n.a.lyzed 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 a.s.sume 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 in question, 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.

VI

MATHEMATICAL INTELLIGENCE

The real nature of intelligence as it appears in the development of mathematics is something quite other than that of sensory a.n.a.lysis.

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 a.s.sumed. 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 a.s.sume 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, or better 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 subst.i.tuted 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 fas.h.i.+on 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 subst.i.tute them for actual situations in such a fas.h.i.+on 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 ent.i.ty 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 and Peano's, where the empty name "ent.i.ty" 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 ent.i.ties const.i.tuted in certain specified ways. And it is worth noting that both factually and logically a collection of ent.i.ties so defined is not a mere aggregate, but possesses a differentiated character of its own which, although the resultant of its const.i.tution, 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 const.i.tuted, just as an atom may be composed of electrons and yet, in valency, possess a property that is not the direct a.n.a.logue of any property possessed by electrons not so organized.

Natural science, however, considers such building up of its fundamental ent.i.ties into new ent.i.ties as a process taking place in time rather than as consequent upon change of form of the whole rendering new a.n.a.lytic 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 distinguis.h.i.+ng of previously unnoted generalizations of relations existent in the realm of facts. But the fact that time has thus pa.s.sed 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 ent.i.ties for which a home must be provided in the part.i.tion 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 cooperation 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 (Levy-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 such fas.h.i.+on 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 att.i.tude 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 intimation that 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."

SCIENTIFIC METHOD AND INDIVIDUAL THINKER

GEORGE H. MEAD

The scientist in the ancient world found his test of reality in the evidence of the presence of the essence of the object. This evidence came by way of observation, even to the Platonist. Plato could treat this evidence as the awaking of memories of the ideal essence of the object seen in a world beyond the heavens during a former stage of the existence of the soul. In the language of Theatetus it was the agreement of fluctuating sensual content with the thought-content imprinted in or viewed by the soul. In Aristotle it is again the agreement of the organized sensuous experience with the vision which the mind gets of the essence of the object through the perceptual experience of a number of instances. That which gives the stamp of reality is the coincidence of the percept with a rational content which must in some sense be in the mind to insure knowledge, as it must be in the cosmos to insure existence, of the object. The relation of this test of reality to an a.n.a.lytical method is evident. Our perceptual world is always more crowded and confused than the ideal contents by which the reality of its meaning is to be tested. The aim of the a.n.a.lysis varies with the character of the science. In the case of Aristotle's theoretical sciences, such as mathematics and metaphysics, where one proceeds by demonstration from the given existences, a.n.a.lysis isolates such elements as numbers, points, lines, surfaces, and solids, essences and essential accidents. Aristotle approaches nature, however, as he approaches the works of human art. Indeed, he speaks of nature as the artificer par excellence. In the study of nature, then, as in the study of the practical and productive arts, it is of the first importance that the observer should have the idea--the final cause--as the means of deciphering the nature of living forms. Here a.n.a.lysis proceeds to isolate characters which are already present in forms whose functions are a.s.sumed to be known. By a.n.a.logy such ident.i.ties as that of fish fins with limbs of other vertebrates are a.s.sumed, and some very striking antic.i.p.ations of modern biological conceptions and discoveries are reached. Aristotle recognizes that the theory of the nature of the form or essence must be supported by observation of the actual individual.

What is lacking is any body of observation which has value apart from some theory. He tests his theory by the observed individual which is already an embodied theory, rather than by what we are wont to call the facts. He refers to other observers to disagree with them. He does not present their observations apart from their theories as material which has existential value, independent for the time being of any hypothesis.

And it is consistent with this att.i.tude that he never presents the observations of others in support of his own doctrine. His a.n.a.lysis within this field of biological observation does not bring him back to what, in modern science, are the data, but to general characters which make up the definition of the form. His induction involves a gathering of individuals rather than of data. Thus a.n.a.lysis in the theoretical, the natural, the practical, and the productive sciences, leads back to universals. This is quite consistent with Aristotle's metaphysical position that since the matter of natural objects has reality through its realization in the form, whatever appears without such meaning can be accounted for only as the expression of the resistance which matter offers to this realization. This is the field of a blind necessity, not that of a constructive science.

Continuous advance in science has been possible only when a.n.a.lysis of the object of knowledge has supplied not elements of meanings as the objects have been conceived but elements abstracted from those meanings.

That is, scientific advance implies a willingness to remain on terms of tolerant acceptance of the reality of what cannot be stated in the accepted doctrine of the time, but what must be stated in the form of contradiction with these accepted doctrines. The domain of what is usually connoted by the term facts or data belongs to the field lying between the old doctrine and the new. This field is not inhabited by the Aristotelian individual, for the individual is but the realization of the form or universal essence. When the new theory has displaced the old, the new individual appears in the place of its predecessor, but during the period within which the old theory is being dislodged and the new is arising, a consciously growing science finds itself occupied with what is on the one hand the debris of the old and on the other the building material of the new. Obviously, this must find its immediate _raison d'etre_ in something other than the meaning that is gone or the meaning that is not yet here. It is true that the barest facts do not lack meaning, though a meaning which has been theirs in the past is lost. The meaning, however, that is still theirs is confessedly inadequate, otherwise there would be no scientific problem to be solved.

Thus, when older theories of the spread of infectious diseases lost their validity because of instances where these explanations could not be applied, the diagnoses and accounts which could still be given of the cases of the sickness themselves were no explanation of the spread of the infection. The facts of the spread of the infection could be brought neither under a doctrine of contagion which was shattered by actual events nor under a doctrine of the germ theory of disease, which was as yet unborn. The logical import of the dependence of these facts upon observation, and hence upon the individual experience of the scientist, I shall have occasion to discuss later; what I am referring to here is that the conscious growth of science is accompanied by the appearance of this sort of material.

There were two fields of ancient science, those of mathematics and of astronomy, within which very considerable advance was achieved, a fact which would seem therefore to offer exception to the statement just made. The theory of the growth of mathematics is a disputed territory, but whether mathematical discovery and invention take place by steps which can be identified with those which mark the advance in the experimental sciences or not, the individual processes in which the discoveries and inventions have arisen are almost uniformly lost to view in the demonstration which presents the results. It would be improper to state that no new data have arisen in the development of mathematics, in the face of such innovations as the minus quant.i.ty, the irrational, the imaginary, the infinitesimal, or the transfinite number, and yet the innovations appear as the recasting of the mathematical theories rather than as new facts. It is of course true that these advances have depended upon problems such as those which in the researches of Kepler and Galileo led to the early concepts of the infinitesimal procedure, and upon such undertakings as bringing the combined theories of geometry and algebra to bear upon the experiences of continuous change. For a century after the formulation of the infinitesimal method men were occupied in carrying the new tool of a.n.a.lysis into every field where its use promised advance. The conceptions of the method were uncritical. Its applications were the center of attention. The next century undertook to bring order into the concepts, consistency into the doctrine, and rigor into the reasoning. The dominating trend of this movement was logical rather than methodological. The development was in the interest of the foundations of mathematics rather than in the use of mathematics as a method for solving scientific problems. Of course this has in no way interfered with the freedom of application of mathematical technique to the problems of physical science. On the contrary, it was on account of the richness and variety of the contents which the use of mathematical methods in the physical sciences imported into the doctrine that this logical housecleaning became necessary in mathematics. The movement has been not only logical as distinguished from methodological but logical as distinguished from metaphysical as well. It has abandoned a Euclidean s.p.a.ce with its axioms as a metaphysical presupposition, and it has abandoned an Aristotelian subsumptive logic for which definition is a necessary presupposition. It recognizes that everything cannot be proved, but it does not undertake to state what the axiomata shall be; and it also recognizes that not everything can be defined, and does not undertake to determine what shall be defined implicitly and what explicitly. Its constants are logical constants, as the proposition, the cla.s.s and the relation. With these and their like and with relatively few primitive ideas, which are represented by symbols, and used according to certain given postulates, it becomes possible to bring the whole body of mathematics within a single treatment. The development of this pure mathematics, which comes to be a logic of the mathematical sciences, has been made possible by such a generalization of number theory and theories of the elements of s.p.a.ce and time that the rigor of mathematical reasoning is secured, while the physical scientist is left the widest freedom in the choice and construction of concepts and imagery for his hypotheses. The only compulsion is a logical compulsion.

The metaphysical compulsion has disappeared from mathematics and the sciences whose techniques it provides.

It was just this compulsion which confined ancient science. Euclidian geometry defined the limits of mathematics. Even mechanics was cultivated largely as a geometrical field. The metaphysical doctrine according to which physical objects had their own places and their own motions determined the limits within which astronomical speculations could be carried on. Within these limits Greek mathematical genius achieved marvelous results. The achievements of any period will be limited by two variables: the type of problem against which science formulates its methods, and the materials which a.n.a.lysis puts at the scientist's disposal in attacking the problems. The technical problems of the trisection of an angle and the duplication of a cube are ill.u.s.trations of the problems which characterize a geometrical doctrine that was finding its technique. There appears also the method of a.n.a.lysis of the problem into simpler problems, the a.s.sumption of the truth of the conclusion to be proved and the process of arguing from this to a known truth. The more fundamental problem which appears first as the squaring of the circle, which becomes that of the determination of the relation of the circle to its diameter and development of the method of exhaustion, leads up to the sphere, the regular polyhedra, to conic sections and the beginnings of trigonometry. Number was not freed from the relations of geometrical magnitudes, though Archimedes could conceive of a number greater or smaller than any a.s.signable magnitude.

With the method of exhaustion, with the conceptions of number found in writings of Archimedes and others, with the beginnings of spherical geometry and trigonometry, and with the slow growth of algebra finding its highest expression in that last flaring up of Greek mathematical creation, the work of Diophantes; there were present all the conceptions which were necessary for attack upon the problems of velocities and changing velocities, and the development of the method of a.n.a.lysis which has been the revolutionary tool of Europe since the Renaissance. But the problems of a relation between the time and s.p.a.ce of a motion that should change just as a motion, without reference to the essence of the object in motion, were problems which did not, perhaps could not, arise to confront the Greek mind. In any case its mathematics was firmly embedded in a Euclidian s.p.a.ce. Though there are indications of some distrust, even in Greek times, of the parallel axiom, the suggestion that mathematical reasoning could be made rigorous and comprehensive independently of the specific content of axiom and definition was an impossible one for the Greek, because such a suggestion could be made only on the presupposition of a number theory and an algebra capable of stating a continuum in terms which are independent of the sensuous intuition of s.p.a.ce and time and of the motion that takes place within s.p.a.ce and time. In the same fas.h.i.+on mechanics came back to fundamental generalizations of experience with reference to motions which served as axioms of mechanics, both celestial and terrestrial: the a.s.sumptions of the natural motion of earthly substances to their own places in straight lines, and of celestial bodies in circles and uniform velocities, of an equilibrium where equal weights operate at equal distances from the fulcrum.