The third edition of Mr. Walras's book presents the author's theories in their complete form. Mr. Walras is among those who believe that the domain of mathematical analysis cannot be extended too far, and that the study of the phenomena of life and society is in no way inherently incompatible with purely quantitative methods.
The opponents of this tendency are generally unfamiliar with physico-mathematical theories or those of rational mechanics. They might see there that the application of analysis to phenomena depends above all on the ingenuity deployed by the founders of the exact sciences. Whatever the intrinsic value of Mr. Walras's results may be, we can only approve the direction of his work and highly appreciate the effort he has made, following Cournot, Gossen, and Jevons, to establish a mathematical Economics.
We could not provide here a summary of this work. By its very mathematical nature, it cannot be narrated or summarized. One may say of good mathematical works that the ratio of their form to their substance is constant.
Mr. Walras begins by extracting, from the tangle of social phenomena, those that will enable him to create an "economic space" — that is, a set of variables linked by purely quantitative relations, upon which one need only operate mathematically to discover the properties of their combinations and variations. It is on this indispensable starting point that criticism must concentrate its efforts.
We regret finding in it several paragraphs lacking in rigor and critical scrutiny, and an entire chapter (the second lesson) on the distinction between science, art, and morality, which shows us that Mr. Walras's analytical mind has not often ventured beyond the domain of pure economics.
We also regret not finding at the beginning of the volume — instead of a sketch of the theory of functions that is insufficient for mathematician and non-mathematician alike — a general idea of the physico-mathematical method, and above all a brief exposition of the theory of units. Every explanation is subject to the necessity of expressing an entire order of phenomena through the combinations of a limited and generally small number of carefully determined variable quantities. Thus, rational mechanics has the sole aim of reducing every problem it poses to an equation between time, space, and mass, and this restriction is obviously founded on the philosophical and logical nature of an explanation. Mr. Walras, for having neglected the exposition we speak of, for having passed over in near silence such important ideas as continuity and the generalization of the idea of measurement, was compelled to state his definition of value and price in a rather awkward manner. Even the remarkable equation of number 117:
x₁ + y₁ pᵇ + z₁ pᶜ + w₁ pᵈ = 0
which expresses that the sum of quantities offered and demanded at various prices is zero, is presented somewhat disadvantageously on account of the insufficiency of the surrounding context.
But all of this is merely a quarrel over form. The misfortune is that it appears to us to bear on the important portion of Mr. Walras's work — on the preliminary analysis of facts that must precede mathematical analysis. Once calculation can intervene, one may say that the difficulties are over, and there remains at last only the reading of the results, or their verification — that is, the comparison with reality of the new relations that algebraic operations bring to light.
Let us recall, finally, that speculations such as Mr. Walras's deserve more than any others to be encouraged. Beyond the special attraction they hold and the document they offer to the so fascinating history of the expansion of mathematics, they imply a true courage and a noble disdain for immediate success: let us not forget that mathematical economists are as rare as economical mathematicians, and that no one is kind to the founder of anything.
Paul Valéry