Étude sur l’espace et le temps. Par Georges Lechalas ingénieur en chef des ponts et chaussées. Paris: Félix Alcan, 1895
This book deals with the mathematical and metaphysical, not with the psychological, aspects of space and time. In the first chapter, on geometrical space, the author discusses the nature of geometrical proof. No postulates are required, since, as rnetageometry shews, all Geometry flows from the mere definition of space, and definitions do not involve the existence of their objects. The justification of a definition lies in the absence of contradiction in its results. Thus general Geometry is apodeictic, but the decision between Euclid and non-Euclid is empirical.
In Mechanics, which is next discussed, we must begin by the choice of a unit-movement, assumed uniform, and chosen from motives of simplicity. We must choose our axes from the same motive; e.g. for axes rotating with the sun, Kepler’s laws would be false. This does not involve absolute motion, but only care in the selection of axes. (The difficulty, however, lies in the fact, overlooked by our author, that the axes have to be fixed by reference, not to particular bodies, but to empty space.) The fundamental notion of Dynamics is not force, but mass; the determination of actual masses is empirical, but apart from this, Dynamics follows apodeictically from Geometry.
After a chapter on the Geometry of our universe, which adds little to Chapter 1, M. Lechalas discusses the problem of similar worlds and the reversibility of the material universe. The former problem is meaningless, since a proportional change of all temporal and spatial magnitudes would be no change. As to the latter, a reversed world would be unstable and improbable. (This answer does not touch the difficulty – apparently insoluble on a purely mechanical level – which lies in the absence of qualitative difference between past and future in mathematical time.)
From a discussion of Kant’s antinomies and Zeno’s arguments against motion, the author is led to declare that motion is discontinuous. The difficulties of space have hitherto proved insoluble; as to time, however, the Transcendental Analytic provides a solution, by identifying temporal succession with causation. The discrete irreducible elements of motion, again, afford a natural unit for time-measurement, and correspond to distinct events in the causal chain.
The book is chiefly useful as a bibliography of recent French works on the philosophy of Mathematics; its own solutions almost always evade the fundamental difficulties they are intended to resolve.
B. A. W. RUSSELL