Reprinted from JOURNAL OF THE AERONAUTICAL SCIENCES A Special Issue Commemorating the Seventy-Fifth Birthday of Dr. Theodore von K&rmLn Copyright, 1956, by the Institute of the Aeronautical Sciences and reprinted by permission of the copyright owner M~~,1956 VOLUME 23,No.5 Applied Mathematics An Art and a Science M. A. BIOT* Cornell Aeronautical Laboratory and Shell Development Company WHAT Is APPLIED MATHEMATICS? If mathematicians were to be assembled to agree on a formal definition of the fieldof applied mathematics, T HAS BEEN SAID that France is divided into 40 I believe the result would bear a rather embarrassing I million Frenchmen. To raise the question of similarity to the emasculated and noncommittal definition for applied mathematics is to invite perhaps resolutions that the present generation has resigned it- as many answers as there are applied mathematicians self to accept as the normal product of international or rather people who call themselves by that name. political gatherings. As a definition would surely be There have been numerously quoted remarks on this presumptuous, is it not preferable to recognize that we subject, some by individuals bearing great names in are dealing here not with an established body of science mathematics. I personally recall a lecture by the but rather with a way of life of the scientific mind in late Harry Bateman in which the applied mathema- general, best understood by a description of its activ- tician was characterized as a “mathematician without ities and of its peculiar creative tools and processes? mathematical conscience.” There is no doubt that the It may be truly said that mathematics today is not statement reflects the existence of a deep and insoluble only the very fabric and soul of the physical sciences conflict-the contradiction between the ideal of clarity, but also their universal language, providing the essen- precision, and rigor of mathematics as a science and the tial symbols through which the key concepts are not adventurous and imaginative urge of the mind, whether merely conveyed, but created, molded, and refined. it is striving for the ultimate in its impatient search for truth and artistic expression or laboring under the fierce If there is a conflict, we should recognize that it is not compulsions of the pragmatic and competitive world between people but that it arises in the individual and is of technology. One could understand the feelings of reflected with all the gradations and shades of a spec- the artist who undoubtedly benefits from the scientific trum-between those whose preoccupation is mainly study of colors but who would be constantly reminded of with the creation of a systematic and rigorous body of proceeding with rigorous conformity to the dictates of science and those whose interest in mathematics is physics and psychology. purely pragmatic. Such disharmonies may be some- An excellent illustration of the conflicting mental times trying, but we should look upon them as a sign attitudes referred to here is given by Sir Geoffrey of inner vitality. For in this dual aspect and con- Taylor in a recent witty essay.t His advice to those flict lies the most fertile source of progress, from the who would seek counsel from a mathematician is to use early attempts at land surveying in primitive agri- the approach one would make to a child : “Make your cultural’ societies and their influence on the develop- question as simple as possible and don’t be disappointed ment of abstract geometry to the complexities of pres- if he finds that he cannot answer the exact question ent-day technology. The contrast is reflected on many you put to him but can answer a related but rather planes in the teaching and practice of applied mathe- different one.” matics, as it is only partly a science and mainly a craft Evidently it would be a drastic fallacy to envisage and an art. This is particularly apparent in the teach- this conflict as opposing the pure and the applied ing, for, as experience shows, it is usually much easier sciences. Indeed it has generally been the rule that for a student to acquire the formal trappings of a theory many significant discoveries in pure mathematics were than the ability to use it and adapt it to a particular made by a process of trial and error which did not situation. exactly reflect a spirit of conformity with the conven- In this respect, the achievements of von K&m&r tional requirements of rigor. This is not to say, of as an applied mathematician are perhaps better ap- course, that rigor lies outside the realm of aesthetics, as praised by stressing the imponderable rather than by mathematicians well know, but simply that many referring to specific contributions that are amply dis- times, because of human limitations, it cannot be cussed elsewhere in this issue. He gives innumerable attained except through lengthy and laborious meth- examples of how to tackle difficult and complex prob- ods. lems in widely different ,fields, providing the student with the opportunity of living through the same creative * Research Consultant. experience and inspiring him with an indelible vision t Taylor, G. I., Rheology for Mathematicians, Proceedings of the Second International Congress on Rheology; Academic and an enduring faith in the effectiveness of such Press, Inc., New York, 1954. methods in pure and applied science. 406 THE INTERRELATIONBETWEEN PURE AND science always proceeds in steps, from the collection APPLIED MATHEMATICS of unbiased and precise empirical facts to their general- ization in the form of systematic scientific laws. The We can best characterize the relationship between prevalence of this viewpoint, due perhaps to the per- pure mathematics and the surrounding sciences as a vasion of our elementary and undergraduate teaching continuous process of cross-fertilization. From the by the positivism of Auguste Comte, has had an im- applied fields questions constantly arise-problems of ponderable but pernicious influence on the layman’s precise definition, existence, unicity, convergence, conception of creative scientific thought and on scien- and others that are a challenge in pure logic and belong tific progress itself. There are numerous instances of to the classical realm of pure mathematics. In fact, scientific discoveries where the pattern is actually it is sometimes difficult to say whether some questions of reversed, and where mathematical theories determine this type belong to the pure or applied category. Take, directly the type and scope of physical measurements for example, the problem of unicity of solution of the and observation. One of the earliest and most striking Navier-Stokes equations in fluid dynamics. We may examples is the discovery of radio waves. The exist- consider this as a problem for a specific equation ence of these waves was not suspected and could not concerned with a physical problem, but obviously be derived mathematically from the laws of electro- the question immediately arises of the properties of dynamics as formulated by Faraday and Amp&e. It equations of a similar type, devoid of any physical took Maxwell’s addition of a term known as the dis- application and of the possibility of generalizing par- placement current in the equations to make it possible ticular theorems to more general classes of objects. to derive theoretically the existence of electromagnetic One can cite innumerable occurrences of this process. radiation, later confirmed in most striking fashion by One of the most striking is the recent impressive de- Hertz’s famous experiment. It must be emphasized velopment of theories in the functional and abstract that Maxwell’s brilliant intuition was essentially a spaces which found their origin and inspiration in the modification of the physical laws motivated solely familiar eigenvalue problems of classical and quantum by considerations of mathematical symmetry, con- physics. As another example in a totally different sistency, and aesthetics. Some of the most spectacular branch of physics we could mention the boundary- advances in physics have been consequences of this layer theory, whose progress is closely associated with type of thinking. Its latest triumph is the experi- the name of von Karman, and which has led to a new mental confirmation of the existence of the antiproton. and interesting study of the properties of differential Having considered what we have called the heuristic equations when a high-order derivative is multiplied by value of mathematics in the discovery of physical a vanishingly small coefficient. laws, let us turn our attention to another aspect In turn, such progress in the domain of pure form which can be characterized as the qualitative in con- and concept suggests new viewpoints and novel ways in trast with the quantitative. It is represented by the the formulation of physical problems. The tensor approximate laws and the mathematical models whose calculus, a primary conceptual tool in the creation of object is to give a compact and simple mathematical the theory of Relativity, was itself the result of abstract representation of basic phenomena. developments and generalizations arising from the In this category we have, for example, sources and sinks, the radiation study of geometric surfaces. Here we have a typical of a Hertzian dipole, etc. They may or may not be example of the stimulation and the molding of physical realizable physically, but they furnish a precise formu- thinking by purely mathematical concepts. lation of an intuitive concept, combining the advantage In this sense, the mathematical sciences may be of simplicity while containing the essence of the physics. looked upon as providing not only an indispensable Such models are further useful as building blocks in the and universal language for our description and under- representation of a more actual and complex situation.