Review of Psychologie der Axiome. Von Dr. J. Schulz. Göttingen: Vandenhoeck und Ruprecht, 1899. Pp. 232.
The present work has few merits except a lively and amusing style, and a clear statement of the views which it advocates. Since axioms must be defined by a logical criterion, the subject invites to confusions between logic and psychology; and the author does not refuse the invitation. The work should treat either of all that has psychologically held an axiomatic position, and of the causes of past and present opinions on the subject, or of the state of mind involved in correct views on axioms, in which case the most difficult inquiry would be the logical one as to what axioms are. The author, however, confuses the two inquiries, attempts both, and succeeds in neither.
Dr. Schulz’s general position may be described as Kantianism modified by Sigwart. An inquiry into axioms is concerned with the forms of intuition and thought, which are purely subjective, and independent of the stimulus. Axioms are neither analytic nor synthetic; they are, in fact, not propositions, but postulates, innate habits of the soul, at least in man. At bottom, they depend upon the will; a pure intuition is not a describable object, but (!) an activity of the subject. To understand anything is to anthropomorphise it (i.e., we might add, to pretend it is something else). What the world is in itself, we cannot tell, nor even whether Being has any objective meaning (a point, one would have thought, established even by Descartes’ cogito). Not sensation, but association – which is always at bottom association by contiguity – is the beginning of psychical life, and the source of the principles of experience, prior even to the law of identity. This law and that of uniformity both demand comparison, and this consists of two parts: first we feel an experience to be familiar, and then the element with which comparison is to be made can be reproduced. The author inquires at length into the stage of evolution at which animals begin to think in accordance with the various postulates, and tells so many tales of their intelligence as nearly to persuade us of the paradox of Rorarius.
Self-consciousness, we are told, distinguishes men from animals, and is the psychological source of the idea of substance (though the author agrees with Kant that, metaphysically, substance is not applicable to the Ego). Our acts suggest the substitution of causality for mere succession; motions appear not merely to follow volitions, but to follow necessarily from them. Controversies as to the nature of cause have an anthropomorphic origin: if I say I move my limbs, substance is the cause, its own motion is the effect; if I move other bodies, one substance is the cause of motion in another; if my motion moves other bodies, cause and effect are alike events. All three views seem plausible to crude anthropomorphism, and the third is preferred only because it is more convenient in science.
Both substance and cause, says our author, are applied to the material world only by anthropomorphism; nevertheless he insists that they must be applied. Matter is defined by three attributes, extension, motion, and impenetrability. Abstracting from the last two, we get the subject-matter of Geometry. (It is difficult to see how two immovable extensions can fail to be impenetrable, and no explanation is offered.) Abstracting extension too, and leaving only what is common to the psychical and physical, we are left with the thinkable simply, which gives us number. The author proceeds to discuss Arithmetic, Geometry and Mechanics. The account of number agrees in the main with Sigwart. It ignores modern Arithmetic by permitting geometrical arguments on fundamental points, e.g., multiplication and irrationals. The remarks on Geometry are a farrago of logical fallacies, historical blunders, and mathematical errors, culminating in a pretended proof of the axiom of parallels. The discussion of Mechanics is mainly directed against Mach, and insists on the necessity of absolute motion, which the author, like Leibniz, deduces by means of causality. He fails, however, to understand the consequences of absolute motion, since he asserts, contrary to received Dynamics, that two particles alone in the world would have to move in the straight line joining them. Atomism is regarded as a priori necessary, and action at a distance as unavoidable, though the latter, it is confessed, cannot be anthropomorphised. Vortex atoms are dismissed with the single remark that they are the maddest imagination since the time of the Vedas (p. 104), and no arguments are to be found against a plenum. The law of inertia is deduced from the principle that every change is to have a cause; with Heymans, experience is allowed to decide whether the principle is to apply to change of place or change of velocity. It is not realised that both, if the principle be sound, require causes, and that, if one may be uncaused, so may the other.
The book, though it has some good sections, shows, despite violent partisanship, an almost total lack of real argument on controverted points. It also illustrates the fact that philosophers subsequent to Kant, in writing on mathematics, have thought it unnecessary to become acquainted with the subjects they were discussing, and have therefore left to the painful and often crude efforts of mathematicians every genuine advance in mathematical philosophy.