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MOLECULAR THEORIES AND MATHEMATICS.1

By ÉMILE BOREL,

Professeur à la Faculté des Sciences de l'Université de Paris, Sous-Directeur de l'Ecole Normale Supérieure.

I.

The relations between the mathematical sciences and the physical sciences are as old as the sciences themselves. It is the study of natural phenomena which leads man to set for himself the first problems from which, through abstraction and generalization, has gone forth the superb complexity of the science of numbers and of space. Conversely, through a sort of preestablished harmony, it has often happened that certain mathematical theories, after being developed apparently far from the real, have been found to furnish the key to phenomena concerning which the creators of these theories had no thought at all. The most celebrated instance of this fact is the theory of conic sections, an object of pure speculation among the Greek geometers, but whose researches enabled Kepler, 20 centuries later, to announce with precision the laws of the motions of the planets. In the same way, in the first half of the nineteenth century, it was due to the theory of the imaginary exponentials that the study of vibratory motions was rendered more profound, the importance of which has been revealed on so large a scale in physics and even in industrial art; it is to this study that we owe wireless telegraphy and the transmission of energy by polyphase currents. More recently still, we know what the utility of the abstract theory of groups has been in the study of the ideas so profound and novel whereby one has tried to explain the results of the capital experiments on relativity made by your illustrious compatriot Michelson. But these illustrations, whatever may be their importance, are special and relate to particular theories. How much more striking is the universal usage of the forms imposed on scientific thought by

1 Address delivered at Houston (Tex.) on the occasion of the inauguration of the Rice Institute (Oct. 10 to 12, 1912).

Reprinted by permission of the author, the publishers, and the Rice Institute, from Revue générale des Sciences pures et appliquées, Paris, Nov. 30, 1912.

the genius of Descartes, Newton, Leibnitz. The employment of rectangular coordinates and of the elements of differential and integral calculus has become so familiar to us that we might be tempted at times to forget that these admirable instruments date only from the seventeenth century. And in the same way the theory of partial differential equations dates only from the eighteenth century. In 1747 d'Alembert obtained the general integral of the equation of vibrating cords. It was the study of physical phenomena that suggested the notions of continuity, derivative, integral, differential equation, vector, and the calculus of vectors; and these notions, by a just return, form part of the necessary scientific equipment of every physicist; it is through these that he interprets the results of his experiments. There is evidently nothing mysterious in the fact that mathematical theories constructed on the model of certain phenomena should have been capable of being developed and of furnishing the model for other phenomena. This fact is nevertheless worthy of holding our attention, for it permits an important practical result. If new physical phenomena suggested new mathematical models, mathematicians will be in duty bound to devote themselves to the study of these new models and their generalizations, with the legitimate hope that the new mathematical theories thus erected will be found fruitful in furnishing in their turn to the physicists forms of useful thought. In other words, to the evolution of physics there should correspond an evolution of mathematics which, without abandoning the study of the classical and tested theories, should be developed in taking into account the results of experiment. It is in this order of ideas that I would examine to-day the influence that molecular theories may exercise on the development of mathematics.

II.

At the end of the eighteenth century and in the first half of the nineteenth there was created on the hypothesis of continuity what we may call classical mathematical physics. As types of the theories thus constructed we may take hydrodynamics and elasticity. In hydrodynamics, every liquid was by definition considered to be homogeneous and isotropic. It was not quite the same in the study of the elasticity of solid bodies. The theory of crystalline forms had led one to admit the existence of a periodic network, that is to say, a discontinuous structure; but the period of the network was supposed to be extremely small with reference to the elements of matter physically regarded as the differential elements. The crystalline structure therefore led only to anisotrophy, but not to discontinuity. The partial differential equations of elasticity, as well as those of hydrodynamics, imply continuity of the medium studied

The atomic theory, whose tradition goes back to the Greek philosophers, was not abandoned during that period Independent of the confirmation that it found in the properties of gases and in the laws of chemistry, it was by means of that theory that one was obliged to explain certain phenomena, such as the compressibility of liquids or the permeability of solids, in spite of the apparent continuity of these two states of matter. But the atomic theory was placed in juxtaposition with physical theories based on continuity; it did not affect them. The rapid advances in thermodynamics and in the theories of energy contributed to maintain this sort of partition between the physical theories and the hypothesis of the existence of atoms, which became so fruitful in chemistry. To the majority of physicists half a century ago the problem of the reality of atoms was a metaphysical question properly beyond the domain of physics; it mattered little to science whether atoms exist or are simple fictions, and one might even doubt if science were able to affirm or deny their existence.

However, thanks especially to the labors of Maxwell and of Boltzmann, the definite introduction of molecules in the theory of gases and solutions showed itself fruitful. Gibbs created the new study, to which he gave the name "Statistical mechanics." But it is only in the last 20 years that all physicists have been forced, by the study of new radiations on the one hand and the study of the Brownian movement on the other, to consider the molecular hypothesis as one that is necessary to natural philosophy. And, more recently, the thorough study of the laws of radiation has led to the unlooked-for theory of the discontinuity of energy-or of motion. It does not come within my subject to expound the experimental proofs through which these hypotheses are each day becoming more probable. The most striking of such experiments are perhaps those which have made it possible to observe the individual emission of the a particles, so that one actually obtains one of the concrete units with which the physicist constructs the sensible universe, just as the abstract universe of mathematics can be constructed by means of an abstract unit.

For definitely formulating their hypotheses and deducing therefrom results susceptible of experimental verification the theorists of modern physics make use of mathematical symbols. These symbols are those which have been created in starting out with the notion of continuity. It is, therefore, not astonishing that difficulties sometimes appear, the most real of which is the contradiction, apparent at least, between the hypothesis of the quanta and the older hypothesis that phenomena are governed by differential equations. But these difficulties of principle do not prevent the success of what one might call partial theories, by which a certain number of experi

mental results may, in spite of their apparent diversity, be deduced from a small number of formula which are coherent among themselves. Usually the employment of mathematics in these partial theories is quite independent of the ultimate bases of the theory. And so it is that for many of the phenomena of physical optics the formulæ are the same in the mechanical theory of Fresnel and in the electromagnetic theory of Maxwell. In the same way the formulæ used by electrical engineers are independent of the diversity of theories concerning the nature of the current.

If I have been obliged to point out, though beyond my subject, this employment of the mathematical tool as an auxiliary to the partial physical theories, it is in order to prevent all misunderstanding. It appears certain that for a long time to come, as long, perhaps, as human science shall endure, it will be under this relatively modest form that mathematics will render the greatest service to the physicists. There is no reason why we should be disinterested in the general mathematical theories whereof physics has furnished the model, whether we may be concerned with speculations on partial differential equations suggested by the physics of the continuum or with statistical speculations pertaining to the physics of the discontinuum. But it should be well understood that the new mathematical theories which discontinuity of physical phenomena might suggest can not have the pretention of entirely replacing classical mathematics. These are only new aspects, for which it is proper to make room by the side of older views in such a manner as to augment as much as possible the richness of the abstract world in which we seek models suitable for making us better to comprehend and better to conjecture concrete phenomena.

III.

It is frequently a simplification in mathematics to replace a very large finite number by infinity. It is thus that the calculus of definite integrals is frequently more simple than that of summation formulæ, and that the differential calculus is generally more simple than that of finite differences. In the same way, we have been led to replace the simultaneous study of a great number of functions of one variable by the study of a continuous infinitude of functions of one variable; that is to say, by the study of a function of two variables. By a bolder generalization Prof. Vito Volterra has been led to define functions which depend upon other functions-that is to say, in the most simple case, functions of lines-in considering them as the limiting cases of functions which would depend on a great number of variables or, if one prefers, on a very great number of points of the line.

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