I shall attempt in this paper to give a rounded, if schematic, account of the concept of probability. My central concern will be to clarify the sense in which law-like statements (including ‘statistical’ law-like statements) are made probable by observational data which, in a sense equally demanding analysis, ‘accord’ with them.

The Gibbs’ Paradox is commonly explained by invoking some type of “principle of indistinguishability,” which asserts that the interchange of identical particles is not a real physical event, i.e., is operationally meaningless. However, if this principle is to provide a satisfactory resolution of the Paradox, it must be operationally possible to determine whether, in fact, two given systems are identical or not. That is, the assertion that the Gibbs’ Paradox is resolvable by an indistinguishability principle actually is an assertion that we can in principle possess a complete set of effective procedures for determining the identity or non-identity of arbitrary physical systems. We show that, in rather general situations, an assertion of this type is not well founded. It is further pointed out that a failure to recognize an incomplete set of “sameness criteria” can lead to serious blunders in physics and in biology.

One style of conducting scientific research involves the application of theoretical propositions to an empirical domain under investigation. The investigator plans his research in terms of the theoretical propositions and states his findings in these terms as well. We wish to examine not the merits of this style of research but a difficulty inherent in it. The difficulty concerns what is to be done when disconfirmative results are produced by some discrete study or experiment in the course of such research.

Newtonian theory has usually been accepted as a paradigm example of an explanation. There are two widely known analyses of what makes it so. According to one analysis, the deductive and predictive nature of the theory is what counts. The second analysis emphasizes the ability of the theory to connect widely different events and laws. The present paper proposes a third analysis stressing three characteristics. (1) The explanation includes a description which is in part of something unobserved. (2) The description is true in the sense of corresponding to the facts. (3) Through the description, the explanation confers “naturalness” upon the thing explained.

In this paper I shall argue that to a very significant extent mathematics is concept analysis, and that though the analysis of mathematical concepts is in a number of ways different from the analysis of philosophic concepts, the similarities between these two types of concept analyses are as important and far reaching as the differences. I shall argue that because mathematics and philosophy are each concerned with the analysis of concepts, they are much more like one another epistemologically than is often recognized. In insisting upon fundamental similarities between mathematics and philosophy, I shall be agreeing with the classical rationalists, but on a very different conception of both philosophy and mathematics from that held by the rationalists. The rationalists wished to assimilate philosophy to mathematics as understood in their time; viz. as a body of necessary propositions, which followed from self-evident axioms and postulates, revealed to the natural light of reason. As against this rationalistic position, I wish to make a comparison in the reverse direction, in which I shall presuppose a certain conception of philosophy as something given, and then insist that mathematics is in many important respects similar to philosophy as so understood. In particular, I wish to insist that there is a significant comparison between mathematics on the one hand, and philosophy as understood by probably a majority of philosophers today on the other—viz., philosophy understood as concept analysis—, where it is conceded that the analysis of philosophic concepts is inherently a tentative matter, wherein it is impossible—at least in the usual case—to offer any one analysis of a given philosophic concept as absolutely certain and beyond all revision. I shall argue that by virtue of the fact that mathematics, like philosophy, is concerned with the analysis of concepts, many at least of the propositions advanced within it are inherently revisable, and do not possess the kind of certainty the rationalists ascribed to them.

A recent proposal by Dr. W. H. McCrae, cosmologist and mathematician, to the effect that decisions between such cosmogonies as those of Hoyle and of Gamow are experimentally impossible by virtue of a general cosmological indeterminacy principle, is here examined and elaborated upon. Some comments on the “antinomies” in Kant's “Critique of Pure Reason” are made in reference to this principle as well as to the Heisenberg indeterminacy principle. If McCrae's principle is accepted, we will have moved a long way toward Kant's conclusion that a logically consistent understanding of an underlying “ultimate reality” is outside the confines of science—whatever one may think of Kant's own line of reasoning.

The possibility of a sub-quantum mechanical realm has been investigated in recent years by DeBroglie, Bohm, Vigier, and others. It is felt that what is needed in this investigation is some simple and direct resolution of the problem as to whether sub-quantum entities exist or not. By restricting attention to quanta of light energy there is presented a theoretical expression for a sub-quantum or micro-photon which has proven to be testable. It is possible by this development to bridge the particle-wave duality wherein these two conceptions become arithmetically commensurate with one another. The theoretical expression derived for the micro-photon also leads to a different interpretative approach to Planck's constant as well as the appearance of a new unit or element of mass.

During the last decade, in 1958 to be precise, an organizational confrontation took place: the essential issue before its membership was this: should the American Sociological Society (ASS) be rechristened the American Sociological Association (ASA). The results of the vote conclusively demonstrated that for a majority of the participants, it was preferable to be known as a group of ASA's rather than a bunch of ASS's.

As anyone who has ever seriously attempted to analyze the semantico-epistemological status of scientific theories has soon discovered, it is not easy to reconcile the belief that theoretical terms (i.e., terms which cannot be explicitly defined in the observation language) have genuine cognitive properties with the empiricist tenet that all knowledge derives from experience. Even if it be granted that knowledge can originate in experience without being about experience, it still remains to develop a coherent metalinguistic account of the truth-conditions of theoretical propositions and the designata of denotative expressions in the theory language. In a recent article, “On the use of Hilbert's e-operator in scientific theories” [1], Carnap has broadened his analysis of theories to include provision for the referential properties of individual theoretical terms. But while his account focuses more closely than ever upon the fundamental semantical problems of a theory language, his specific proposals amount to a repudiation of the very logical empiricism which he has so strongly championed these many years. In what follows, I use Carnap's notation and terminology throughout (including a pejorative sense of the term “metaphysics” borrowed from Carnap's earlier writings).

In 1958 I set down some thoughts that arose from an attempt to consider epistemological problems on the assumptions that (1) The biology of the human nervous system is relevant to epistemology and (2) The human nervous system, like every other object of biological investigation, is a product of evolution by natural selection. These thoughts lay more or less neglected until they were brought stunningly to mind by Professor George Gaylord Simpson's [1] recent paper on “Biology and the Nature of Science”. In that paper Professor Simpson writes:

It will be recalled that Mr. Mattick stated that Marxism does not derive its social theory from physical theory, and that any attempt to do so is an aberration from marxism. It is maintained that Marx is not a determinist or indeterminist in the ususal sense of these terms. Furthermore, it was argued that ideologies are no longer decisive weapons and that Marxists put little trust in the dialectical course of history.

In his recent paper Gellner singles me out for special comment and some reply is called for. He attributes to me several propositions which he says I made in my note on ‘Physical and social kinship’ in this journal, and he then refutes them. Reading his paper I cannot avoid thinking that he exaggerates the differences between us, thereby apparently strengthening his argument. Some substantial differences there are, but others are fictional. A line-by-line analysis of what he says about me in his paper might be amusing but would be of only ephemeral interest. In this reply I confine myself to a few important issues.