Philosophy of science is the organized expression of a growing intent among philosophers and scientists to clarify, perhaps unify, the programs, methods and results of the disciplines of philosophy and of science. The examination of fundamental concepts and presuppositions in the light of the positive results of science, systematic doubt of the positive results, and a thorough-going analysis and critique of logic and of language, are typical projects for this joint effort. It is not necessary to be committed to a belief that science and philosophy are or should be one, or else that “never the twain shall meet.” If anything, there is to be expected a wholesome regard for the value of established science in furnishing a foil for philosophy and a check on its old extravagances. This does not mean that even the best-established science may not be subject to most devastating criticism by analysis of its foundations. In fact, despairing of the philosophy of the schools, science has done it largely for itself. The theories of gravitation, atomicity, electro-magnetism, evolution, relativity and quanta have all arisen through drastic revisions of complacent fundamental “truths.”

The disagreement with regard to the validity of the principle of causality, existing to-day among scientists, has its roots in the diversity of definitions of the principle itself rather than in a problematic scientific situation. As far as the formulation of quantum theory is complete its bearing upon philosophical questions can be fixed with precision provided the questions are phrased intelligibly. But a question is intelligible from a scientific point of view only if it satisfies two conditions: (1) the meaning of its terms must be fixed; (2) it must be in accord with the conventions of the science to which the question is put.

The logician recognizes the following propositional forms as necessary and sufficient for the expression of any truth:

The history of Physical Science appears to exhibit, periodically, a race between the acmulation of data and the ability of its codification (usually, a mathematical theory) to find a natural place (in the codifying scheme) for much of the empirical findings. If the codifying scheme is a mathematical theory, capable of interpolation and extrapolation, according to the rules of the particular branch of mathematics employed, the ablest handlers of the theory are frequently confronted with a situation in which mathematical computation alone does not suffice. In such a situation the introduction of symbols, whose interpretation is not direct and simple (from the point of view of the methodology whereby the data were procured) is resorted to. This symbol is usually involved in a mathematical relation by a species of reasoning, which may be somewhat unfamiliar and remote from customary habits of thought. It is generally accepted as physically valid, if the mathematical formulation which involves it can produce results which account for the mass of data uncovered by experimenters. As an example of such a procedure we may mention the case of the Schrödinger Ψ-symbol, a symbol which was involved in Schrödinger's famous matter wave equation. The successes which such a theory enjoyed, particularly, in its initial stages, in formally relating the relative positions and intensities of spectral lines, with the atomistic concepts of gross matter which were inferred from the data of macroscopic measurement, such as the atomic mass, the electronic charge, Planck quantum of action, etc.,—these successes, gave a certain physical dignity to the Ψ-function itself. This dignity was further enhanced by the disclosure, by Schrödinger, of four functions of the Ψ-symbol (and its complex conjugate) which held the same place in the “continuity equation” as the classical charge density and the three components of current density. The successes accompanying this discovery for spectroscopic problems, in particular, are well known to Physicists.

The Bohr formulation—the original one—is now largely a matter of history. The usual career of a theory was its fate. It explained much, but it failed at the first signs of complication, even for the simplest molecules as hydrogen and helium. Some results were either only approximate in a loose or incomplete fashion, e.g. in the application of the correspondence principle to intensities, the only reliable predictions being only for an absence of certain lines, or they quite disagreed with experiment, e.g. in the case of the hydrogen molecule and the hydrogen molecule ion, the normal and excited states of helium, the anomalous Zeeman effect, the Ramsauer effect (the phenomenon of the deflection of electrons at low speeds being less than at higher speeds in atomic collisions), and in the treatment of problems where more than one electron was involved. In the last mentioned cases the perturbation method, so successful in solving “many-body” problems in celestial mechanics, failed completely because systems of several electrons, possessing frequencies of the order of light waves, cannot interact classically. A bit of much forced pragmatism was practiced in treating the many electron atom as a variation of the hydrogen atom. Besides, there were technical oddities, like ½ quantum numbers, and impenetrated anologies between visible and X-ray spectra.

Philosophers have ever declared that their problems lie at a different level from the problems of the empirical sciences. Perhaps one may agree with this assertion; the question is, however, where should one seek this level. The metaphysicians wish to seek their object behind the objects of empirical science; they wish to enquire after the essence, the ultimate cause of things. But the logical analysis of the pretended propositions of metaphysics has shown that they are not propositions at all, but empty word arrays, which on account of notional and emotional connections arouse the false appearance of being propositions. This conception that the “propositions” of metaphysics, including those of ethics, have no theoretical content, is to be sure still disputed. We shall not, however, enter here on its demonstration, but, under its guidance, will limit ourselves to non-metaphysical and non-ethical (non-evaluating) philosophical problems.

During the course of the last century it has grown increasingly clear that not all the issues with which an experimental science can be faced are experimental issues. If there were no other ground for this belief, history itself would force upon us some such conviction. For there are differences of opinion dividing men today that have divided men from the earliest times recorded, and in every one of the ages in between the self-same issue will have involved in dissension, not an occasional disputant here and there, but all the best thinkers in all the known sciences of their day. Yet at no time has either party to such strifes been in possession of facts ignored by the other; for had this been the case, the ignorant party must ere now have been driven from the field by force of the facts it ignored. It is fair to suppose that an issue which has come no nearer to solution as century after century added its contribution to the store of human knowledge, cannot depend for its decision on any information to be looked for from the centuries to come.

The controversy which has raged between those who accept Aristotelian Logic and those who reject it, seems to continue. It is, I believe, advisable to consider the arguments on both sides. Modern Logicians for the most part insist that Aristotelian Logic breaks down, i.e. it is not true for all meanings of the terms. On the other hand there are those who insist on the validity of Aristotelian Logic claiming that those values for which it fails are not permissible values. But there is one who stands almost entirely alone in his position—namely Professor H. B. Smith—who maintains that Aristotelian Logic breaks down because and only because its interpretation into Boolean terms has been accomplished by means of a definition that is not forced upon us and which, since it gives up desirable principles, is to be rejected.

Following the general program, outlined in a previous paper on the Foundations of Mathematical Biophysics, we shall outline some physico-mathematical aspects of the Gestalt-problem. Faithful to the principles laid down before, we shall not attempt to give a theory of such and such psycho-physiological phenomena connected with our problem, but we shall investigate in a preliminary way the abstract conceptual basis, which must underlie the construction of any concrete theory. To make our aim quite clear and to avoid at the same time unnecessary repetition, we refer the reader to our previous paper. Furthermore we shall give here only an outline of the physico-mathematical aspects; and of necessity we must leave a number of details untouched and unanswered.

If you wish to learn from the theoretical physicist anything about the methods which he uses, I would give you the following piece of advice: Don't listen to his words, examine his achievements. For to the discoverer in that field, the constructions of his imagination appear so necessary and so natural that he is apt to treat them not as the creations of his thoughts but as given realities.

In considering this problem a distinction should first be made between its scientific and it philosophical aspects. The scientific problem is that of defining in exact understandable terms those conditions and factors (with their various types of combination and interplay) which make possible the synthesis of the living organism from the simpler elements of the non-living environment, and also its maintenance in the adult state as a fully developed and autonomous organic individual. The problem as thus stated is one to be approached by methods of observation and experiment, leading to verifiable results which are then expressed in systematic and intelligible form. The final statements or representations thus reached, the accepted scientific generalizations, inevitably take on a theoretical form, and as such are necessarily partial and abstract. Many of them, in fact, are to be regarded primarily as simplifications or models devised in the interest of clear representation or understanding. Their relation to concrete reality may range from attempts at exact portraiture to formulæ or diagrams having little more than heuristic significance. Any realistic philosophy of nature must always remember how incomplete and provisional are the theoretical conceptions of any natural science.

The purpose of this paper is to make clear (1) that the widely recognized formulations of the principle of induction do not express the most fundamental rule of induction; (2) that the current view concerning the probability of induction must be revised in terms of a frequency theory of probability; (3) that on this basis the problem of induction in its traditional form is a pseudo-problem; and (4) that the principle of induction must be interpreted as a pragmatic or operational maxim.

The striking synthetic effort made by Bohr to unite radiation facts with atomic structure facts led to the formulation of his “quantum theory of atomic structure,” which with a large group of subsequent developments, is now usually referred to as the “classical quantum theory.” As a conceptual aid it takes over the idea of an “astronomical atom” consisting of a positive nucleus around which electrons revolve in definite orbits, more or less in the fashion of the solar system. More or less. The similarities and differences as a bit of description are interesting and impressive, but when the theory is reduced to its postulates—and it is this which interests us here—it sounds very much like a re-statement (for a purpose) of the Planckian decisions.

When I undertook to write an article for mathematical laymen on the mathematical infinite. I did not realize the depths of my own layness, I do now. Having refreshed my memory of the classics of infinity by re-reading among other things the famous papers of Cantor and Zermelo, and having struggled like a boa constrictor to swallow the latest papal bull on the human significance of the infinite, I am completely reduced to what Professor E. W. Hobson aptly and somewhat indignantly calls Mathematical Nihilism.

The mind-body problem is—despite appearances—still the inevitable basic issue of unending discussions in recent philosophy. Various types of epistemologies and metaphysics, European and American, have offered their widely divergent “solutions” of the dreaded Cartesian tangle. Is there any hope of reaching a universally acceptable view?

The extent to which Ernst Haeckel was hated and attacked by reactionary scientists, philosophers, litterateurs and preachers, is, as Lenin has pointed out in his “Materialism and Empirocriticism,” a measure of the success with which he originally expounded the results of natural science and drove them home to their logical conclusions in the interpretation of nature on a materialistic basis. Haeckel, in the days of his greatest mental vigor, made himself the spearhead of the scientific attack upon the then dominant forces of theology and obscurantism, showing fearlessly how revolutionary were the results of modern science, and particularly those concerned with evolution, in altering fundamentally man's outlook upon the world in which he lived and his own nature. It has become the mode among many intellectual circles in Western countries to refer slurringly both to the more technical scientific work and to the more general reasoning of Haeckel, as being crude and out of date; but this derogation has its origin in the despair of the teleologists and is not founded in the facts of modern scientific development. Certainly, so far as genetics is concerned, the most modern results are not merely in harmony with, but brilliantly confirmatory of, those main tenets of Haeckel's general interpretation of nature which involved him in the bitterest controversy. In addition they help to show the general soundness of his more technical zoological contributions. By contrast, Haeckel's deficiencies in the realm of social affairs stand out the more strongly, although his own materialism of natural science is a necessary basis for an adequate social materialism. We will not concern ourselves here, however, with Haeckel's own social theories, but only with those in the field of natural science; it was only in the latter field that he attained his real prominence.

All of Freud's more recent writings extol the virtues of the scientific method. “The Future of an Illusion” ends with the definite suggestion that religious instruction be taken from the list of topics taught every child and that in its place children be taught to interpret reality in terms of natural science. A similar theme runs through “Civilization and its Discontents”; if there be any hope for humanity, it lies in the scientific method. Finally, Freud's “New Introductory Lectures” ends with the following lines—lines which must be very reassuring to many admirers of Freud (and the present writer counts himself one of these) who have been somewhat discomfited by some of the recent highly speculative passages from his pen:

The foundation of the mathematical theory of probabilities is still a controversial subject. There are schools of insufficient reasoning and of cogent reasoning, of a priori determination and of frequency determination, of subjective and of objective probability. Two main difficulties exist. The first is the definition of equally like events. The second difficulty is the relation between the laws of causal natural science (mechanics, electrodynamics, etc.) and the laws of statistical regularity. Is it really necessary to add to the laws of mechanics one or more independent “laws of large numbers” to explain the regularities in dice throwing? In all these cases different answers exist, and the existence of a mathematical theory of probabilities remains a kind of miracle, as esoteric to the further domain of natural science as the resurrection of the dead.

Mathematical methods in biology occupy a somewhat peculiar position, and the attitude of many biologists toward them is similar to that of many practical engineers toward what is called pure scientific research. The modern progressive engineer recognizes the value of pure science, which seeks for truth regardless of any possibility of practical applications; yet he still frequently shows a definite dislike towards such investigations. The whole history of civilization demonstrates that discoveries which, at the time they were made, did not show any signs of the possibility of being turned into practical use, now lie at the basis of many useful engineering devices. As Professor R. A. Millikan has said, we owe our present-day civilization to Galileo. Every educated person, be he pure scientist or engineer, recognizes this fact. And yet the frame of mind of the “practical” man is not too receptive to investigations which do not show at least some remote prospects of practical applications. Unless he can see at least hypothetically some practical results of such research, he will not give it his whole-hearted enthusiasm. The ultimate criterion in the evaluation of any research still remains its practical use. Those pure scientists who have had close contact with what is called “industrial research” know only too well the truth of the above statement.

There is a significant and a too little noted analogy between the histories of the physical and the “mental” sciences—especially psychology—an analogy capable of throwing much light upon the present-day problems of the latter. Some aspects of this analogy have become commonplaces, but others have been almost completely overlooked.