It is not the aim of the present essay to defend the principle of empiricism against apriorism or anti-empiricist metaphysics. Taking empirism for granted, we wish to discuss, the question what is meaningful. The word ‘meaning’ will here be taken in its empiricist sense; an expression of language has meaning in this sense if we know how to use it in speaking about empirical facts, either actual or possible ones. Now our problem is what expressions are meaningful in this sense. We may restrict this question to sentences because expressions other than sentences are meaningful if and only if they can occur in a meaningful sentence.

In the fourteenth century William of Occam in the course of his attack on the medieval scholastic philosophy enunciated his famous “razor”: Entia non sunt multiplicanda praeter necessitatem (hypotheses should not be multiplied unnecessarily). This is the classic claim for the description of nature in terms of the minimum possible number of fundamental concepts. It was presumably so recognized by Newton in the third book of his “Principia” in 1687 when he wrote: “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.”

On several occasions I have pointed out that the lesson taught us by recent developments in physics regarding the necessity of a constant extension of the frame of concepts appropriate for the classification of new experiences leads us to a general epistemological attitude which might help us to avoid apparent conceptual difficulties in other fields of science as well. Since, however, the opinion has been expressed from various sides that this attitude would appear to involve a mysticism incompatible with the true spirit of science, I am very glad to use the present opportunity of addressing this assembly of scientists working in quite different fields but united in their striving to find a common ground for our knowledge, to come back to this question, and above all to try to clear up the misunderstandings which have arisen.

Lord Acton, in his letter to the contributors to the Cambridge Modern History, wrote: “By Universal History, I understand that which is distinct from the combined histories of all countries … and is not a burden on the memory but an illumination of the soul.” If we replace “history” by the more general term “knowledge,” we get the statement of an ideal cherished by the great men of every age—those lonely pioneers to whom book-learning is an intellectual gloom more treacherous than the natural gloom of night, since it fools men into thinking that it is a light by which men can see. The pioneers in thought, or indeed in any field, have made a secret treaty with nature. They smile with her at the robed and tassled scholar who, (Goethe's words) “Bewundrung von Kindern und Affen,” can (DesCartes speaks) “lecture plausibly about anything and get himself admired by the less learned.” Those pioneers, those men who—as someone has said—have thought according to no one and have made the human race think according to them, see by an intuitive insight, an illumination of the soul. Einstein, in his prologue to Planck's book Where is Science Going?, writes: “There is only the way of intuition. … The state of mind which furnishes the driving power here resembles that of the devotee or the lover. The long-sustained effort is not inspired by any set plan or purpose. It arises from a hunger of the soul.” We have not quoted a poet, but the greatest of mathematical physicists.

Of recent years there have appeared in the literature several attempts to apply to the subject matter of chemistry mathematical theories which had not thitherto found any use in that field. With the spirit and purpose of such attempts, the author of this paper is thoroughly in sympathy. But some of the investigators who have undertaken this kind of pioneer work in applied mathematics have apparently entered upon their tasks without appreciating the difficulties involved, and have hence produced results more likely to obscure than to clarify their subjects.

When science of the 20th century is spoken of in opposition to that of the 19th century, a particularly characteristic attribute is often cited: namely, that since the time of Galileo and Newton the task of science has been to explain everything mechanistically. By analogy the world was to be conceived as a great machine. But the theories of the 20th century, above all the relativity and quantum theories, caused a revolution in science. It is seen today that nature can be described and understood not ‘mechanistically’ but only through abstract mathematical formulas. The world is no longer a machine but a mathematical formula.

In unseen and unacknowledged specter, an “X,” attends at every economic feast, and stands at the elbow of every modern wiseacre. It confounds the forecaster with seeming malice showing itself only in the event and even then with impish and uncertain effect. It laughs at pretensions to knowledge; it tantalizes all figurers, haunts all generalizers and keeps itself to itself, a shadow, an unknown. But phantom as it is, it is still a force, a potent, active, generative substance, mysterious in genesis, related to a dream, but able to accomplish miracles of reality. For the purpose here it will be called the creative experience.

The rapid development of physics, the result of observations made and ideas introduced within the last few decades, has brought about a change in the whole system of physical concepts. This fact is common knowledge, and has already attracted the attention of philosophers. It is less well known that geometry too has had its crises, and undergone a reconstruction. For centuries, so-called “geometrical intuition” was used as a method of proof. In geometrical demonstrations, certain steps were allowed because they were “self-evident,” because the correctness of the conclusion was “shown by the adjoined diagram,” etc. A crisis occurred in geometry because such intuition proved to be untrustworthy. Many of the propositions regarded as self-evident or based upon the consideration of diagrams turned out to be false. And so Euclidean geometry was reconstructed by methods free from all intuitive elements and strictly logical in nature. Moreover, for more than a century, various other geometries have been devised purely as logical constructions. Since they start with assumptions different from those of Euclid, and lead to conclusions partly in contradiction with his theorems, they are called “non-Euclidean.” Nevertheless, each one of these geometries is a closed system of propositions exempt from contradiction. Recently, some of them have even found application in physics.

Columbus and Galileo are usually considered the prime revolutionaries whose discoveries in the physical world brought on the spiritual revolution in modern life, but during the first full century of the modern world another discoverer was so regarded by Sir Thomas Browne. In the “experiments, grounds, and causes,” of the compass needle, he said, Dr. William Gilbert “discovered more in it than Columbus or Americus ever did by it.” Like Columbus, Gilbert made his discovery unwittingly. The navigator had been in search of Asia when he found a new continent; the doctor in seeking a reason for magnetic action led men's attention to reasoning on a new theory of Matter. Through his magnetical researches, the old logical conception of a continuous, soul-animated matter held by the conservative doctors and alchemists was discredited. The discoveries of all three, Columbus, Galileo, and Gilbert, were in many ways interrelated, and of course no single mind released the curiosities of the modern world unaided. What the scientists had to find out in their own laboratories and their own reasoning was persuasive evidence that matter is divorced from spirit, and for a time in the seventeenth century some of the fundamental hypotheses which indicated the divorce derived from magnetism.

At a certain stage of advance in any science it may be well to re-examine and perhaps to rearrange its fundamentals. One seeks an ideal, logical order of development, which may or may not be the best pedagogical order. Many a high school beginner in physics has become acquainted with “force” as something that gets in between two bodies of matter and pulls them together or pushes them apart, like himself between two carts or like “magnetism” between two magnets. Whether or not this might ever be a proper view, the mature student of today will consider that for him, a knowledge of accelerations and strains, inter alia, is prerequisite to a knowledge of force.

Recent discussions in the physical literature, designed to clarify the logical position of modern physical theory, have brought to light an amazing divergence of fundamental attitudes which may well bewilder the careful student of physics as well as philosophy. Quantum mechanics, representing an abstract formalism, should be capable of having its logical structure analyzed with great precision like any other mathematical discipline. Its consequences in all problems to which its method can be applied are so unambiguous, consistent, and successful in predicting physical experience as to disperse immediately all thoughts of possible discrepancies in its fundamental texture. Yet it must be said that even the founders of quantum theory are not in harmony in their various expositions of the bases of that theory. However, while this situation seems disquieting on the face of it, there is no cause for serious brow raising, for it is a fact that there exists agreement with regard to the central axioms of the theory, and that the ambiguities affect only their philosophical interpretation, a field in which differences of opinion may at present be honestly entertained.

It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draughtsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the “point-planet” of astronomy, the “perfect gas” of thermodynamics, or the “pure species” of genetics are equally remote from exact realization. Indeed the unintelligibility at the atomic or sub-atomic level of the notion of a rigidly demarcated boundary shows that such objects not merely are not but could not be encountered. While the mathematician constructs a theory in terms of “perfect” objects, the experimental scientist observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true. As Duhem remarks, mathematical deduction is not useful to the physicist if interpreted rigorously. It is necessary to know that its validity is unaltered when the premise and conclusion are only “approximately true.” But the indeterminacy thus introduced, it is necessary to add in criticism, will invalidate the deduction unless the permissible limits of variation are specified. To do so, however, replaces the original mathematical deduction by a more complicated mathematical theory in respect of whose interpretation the same problem arises, and whose exact nature is in any case unknown.

In these days of Gestalttheorie it seems not untimely to inquire whether psychology itself does not stand in need of being recognized as a configuration of complementary parts. The suggestion, it is true, tends at the first to be repudiated by the implacability of the schools with their claims and counterclaims resounding on every side. On further analysis, however, such controversy seems to acquire the very different significance of having permitted certain partial aspects of the whole field to become well-articulated, even if in somewhat exaggerated form. In other words, if there is any underlying unity in psychology, then the very overstatements of the school zealots while unwittingly defeating their own purpose may have been doing the science a real service by throwing its basic pattern into bold relief. In broadest terms this seems to describe what has been happening in psychology in the recent past and to define the present stage of its development.

To be scientific is to be objective. This is a sine qua non in the social scientists’ efforts to keep their researches on a par with those of the natural scientists. The extreme differences of subject matter in the two fields, however, make it difficult to maintain an even standard of objectivity. Whether objectivity is defined in the strict Cartesian sense or in a broader sense also makes a difference. The Cartesian meaning threatens to enslave the social sciences to Physics as the Lord of All Sciences, and this possibility is so familiar that it needs but brief comment. A broader interpretation of objectivity meets with complications which are only inadequately recognized as yet, and which may call for reinterpretation of the scope of science or else may open up new and independent avenues of opportunity beyond the limitations of science.

When we consider closely any highly integrated vital process, like embryonic development, or animal behavior of the end-subserving or purposive type, we are inevitably impressed with the importance of those special controlling factors, collectively termed “regulative,” which appear chiefly responsible for the unified and finalistic character of the whole sequence of events. These factors are persistent in their influence although they may act intermittently. Without their presence the sequence would soon lose coördination and “run wild,” just as an automobile runs off the road unless occasional compensating touches are given to the steering wheel. They are the factors which give unity and direction to the total process. Generalizing, we may say that in any complex unified process, especially a synthetic one—where the progress is from less to greater complexity—integration seems always to imply or presuppose the operation of regulative factors having the character which we call directive. These keep the sequence of events to a definite course which in the living organism usually terminates in some outcome having special biological significance, or survival value.

It is the aim of this paper to examine certain aspects of a point of view which has attracted much attention in physical methodology. This is the standpoint known as operationalism. We wish to discuss its significance in the construction and interpretation of physical theories.

The essential meaning of operationalism in physics is that physical concepts should be defined in terms of actual physical operations. On this view there is no meaning to a concept unless it represents an operation which can be performed in the laboratory. Thus the term “pressure of a gas” signifies nothing until an operation is described which constitutes the measurement of pressure. When one thinks of pressure one must think. of some actual apparatus, such as, for example, some glass and rubber tubing and some mercury and certain operations performed using these until finally one gets a pointer reading which one calls the pressure of the gas. The same stand is taken with respect to length, time, velocity, acceleration, temperature, etc.

There is sufficient evidence from more recent experiments in psychology that equal retinal stimulus-elements do not lead to equal experiences and reactions except under certain rather specific conditions. An unsophisticated observer will find himself surprised to be able to cover with his own finger a person entering the door of his living room. When the finger is moved to the right or left, thus doing away with the precise retinal coincidence with the person, the observer will soon become unable to recognize intuitively the actual retinal stimulus equality of the two distant things. This usually holds even for the case when he is making every inner effort towards an antagonistic, analytic perceptual attitude of the type which is used by painters or draughtsmen in order to represent the environmental situation in a similar way as it would project itself on a photographic plate or on a retina.

Mr. A. F. Bentley in his Linguistic Analysis of Mathematics has attacked the problem of the “embedding language” of mathematics; “This essay” we read in the Foreword, “deals with the language of mathematics, including not only the mathematical symbols, but also those immediately surrounding forms of expressions and assertions through which the symbols are developed, communicated and interpreted. The writer seeks to establish a firm construction for this embedding language.” Inevitably, in the first instance this embedding language must be, as he puts it, “every-day language”; but this is sadly inadequate: “The every-day language reeks with philosophies—the absolutisms of pointing. It shatters at every touch of advancing knowledge. At its heart is paradox… Mathematicians know this. Yet they feel ever the compulsion to interpret their mathematics in terms of every-day language. So proceeding, their harvest is super-paradox.” Yet the writer recognizes after all that “every-day language is the basal medium of communication between men,” even when we seek to “extend standards of symbolic consistency into the regions of the embedding language.”