In this study we shall analyze the postulate set into its various elements. In order to carry out our analysis, it is advisable to have before us a typical set.

We take the set of five postulates for Boolean Algebras given by H. M. Sheffer in Trans. Am. Math Soc. 14, p. 482 ff.

The analyses thus far undertaken of cost theory and utility theory have viewed these doctrines as they originally were, namely, as the contentions of two conflicting groups of value theorists. And had the two rival groups been allowed to continue to match strength on the plane on which Macvane, for example, contended, the result would probably have been the destruction of them both. Underneath surface conflicts, however, the rival theories had much in common. As the twentieth century got under way, this state of affairs was discerned, with the result that in the end there was hand-shaking and backslapping all around and the inauguration of a live-and-let-live policy. Although at first regarding each other as mortal enemies, these two groups of price-determination theorists later came to see in one another something resembling blood kinship. But are the underlying assumptions which classicism and neoclassicism have in common any more realistic and scientific than the specific assumptions already examined? This is a question to which we now turn.

Despite the skepticism of many mathematicians and logicians as to the possibility of any test which will show conclusively the consistency or independence of the members of a postulate set, several methods have nevertheless been devised and employed, e.g., the empirical methods of Russell and Huntington, the internal method of Hilbert, and the reflective method of Royce. However, with the possible exception of Hilbert's method (which is still available only in the form of fragments and suggestions), these techniques require us to forsake the purely formal or abstract mode of analysis, and instead to rely in some sense on “concrete representations” or “interpretations” or “modes of action” which are said to “satisfy” the postulates in question. In short, they are all interpretational methods—and here is the crux of the problem. “Is it possible,” queries Weiss, “that the only way we can determine whether a set is consistent is by seeing all the postulates actually exemplified in some object? If so, we must arbitrarily assume that the object is self-consistent, so that the proof of consistency must ultimately rest on a dogma. As independence rests on consistency there are therefore no satisfactory proofs as yet of either independence or consistency.” The question “is thus answered,” according to Young, “only by reference to a concrete representation of the abstract ideas involved, and it is such concrete representations that we wished especially to avoid. At the present time, however, no absolute test for consistency is known.” Moreover, “suppose,” says the late Dr. Eaton, “that the system has no interpretation: how can the consistency (and) independence … of the postulates be shown? There should be some analytic way—purely in the realm of the abstract, without interpretation—of establishing these properties of a set of postulates. This is an important problem that awaits solution.”

So much has been written recently about the marvelous developments in mathematical physics, that we have almost forgotten that the amazing progress of science in the twentieth century was made possible as much by the skill and machinery of the instrument maker as by the genius of the theoretical physicist. The two greatest achievements of modern physics, the relativity theory and the quantum theory, are magnificient examples of the power of experimental instruments to disclose new possibilities and new realms of theoretical study.

Previous claims that psychology is propaedeutic to the other sciences have been met with enthusiastic indifference. Contributing to this indifference has been the fact that psychology, a young and unproved discipline which habitually borrowed the methods of the older sciences, has too frequently revised its notion as to its own nature and subject-matter. More important, however, has been the faith of the physical sciences in the absolute character of their own basic concepts: in the reality of Absolute Space and Absolute Time and in the universality of deterministic principles, for, as long as physics presumed to deal with entities which are what they are regardless of the nature of man, nothing other than these entities appeared to demand prior consideration. The problem of the nature of the observer in the physical experiment became important to physics only as a consequence of the recent ‘subjective’ revolution. It is this revolution which has revitalized the problem of the propaedeutic science and has accentuated the distinction which we must make between the behavioral phenomenon of individuals creating formal arrays of words or symbols which are without operational meaning and the discriminatory responses, or operations, by which the empirical meaning of such arrays is determined.

As a concrete extension of analyses of scientific method and social study already undertaken in the pages of this journal and elsewhere, the present article will endeavor to clarify certain long-standing preconceptions in classical and neo-classical economic doctrine which still persist and which, in the light of 20th century methodology, can be given no other label than that of pseudo-scientific. These preconceptions revolve primarily about the ideas of “value,” “cost,” “utility,” and “price,” which ideas have suffered little change to the present day and which even early writers recognized as being vague and ambiguous.

It was Kronecker who sought to avoid the use in mathematics of all numbers (negatives, fractions, irrationals) other than the positive integers, and he outlined the means for carrying through this program. In the introductory sections of his memoir he briefly indicates the personal philosophy which made such a project appear desirable.

Ten years ago the civilized world was shocked out of an unwarranted confidence in the victory of science over superstition by the news that Tennessee had outlawed the teaching of man's evolutionary origin. Within the next three years both Mississippi and Arkansas joined Tennessee in defense of Biblical truth against the scepticism of science. The furor raised by the dramatic clashes between William Jennings Bryan and Clarence Darrow at the Scopes Trial in Dayton, Tennessee, has now become a faded memory. The struggle between Fundamentalists and Modernists has died down, though the embers still smoulder; and few ever think of the issues then raised, now that war and fascism threaten civilization.

The historic task of Leibniz was to furnish a philosophy of personality, and at the same time, and in harmony with it, a general interpretation of the physical world. He conceives therefore of a plurality of Real Beings which in their most developed form he proposes to call individuals, defining individuality in terms of unique experience. Further, he finds the monads, or so-called metaphysical points, to be centres of life, held together by their own inner or intensive force and therefore impenetrable. Consequently, when we enquire whether the Real Beings are parts of the universe, we find that since the universe is the aggregate of monadic forces, all Real Being is finite, and so in order to deal with the problem of Theism we have to reconcile the concept of divine personality with what we may define as personality in the case of human beings.

A striking characteristic of contemporary science is its emphasis upon probability. This is especially notable in quantum mechanics.

There is a respect in which probability is the same for all scientific theories. Verifiability requires that any theory predict certain numbers which can be compared with the numbers gained by actual operations of measuring. In actual practice these numbers, which we shall term theoretical measurables and operative measurables respectively, never correspond. It becomes necessary, therefore, for the scientist to specify when the deviation between them is such that verification occurs. These specifications are defined by the theory of errors in which the concept of probability has an essential place and a specific meaning. But the theory of errors is the same for all applications; hence, probability as defined by the theory of errors is the same for all scientific theories.

In this section we shall publish brief communications commenting on published papers or bringing up special points of interest which can be discussed by correspondence.