Applied Mathematics as a Science

We present two examples of Harvey Greenspan's ideas about applied mathematics. The first comes from an interview he gave in 2006. The full interview is available in J Segel (ed.), Recountings: Conversations with MIT Mathematicians (A K Peters/CRC Press, 2009). The second piece below is from a much earlier talk that Harvey Greenspan delivered to the Mathematical Association of America in Washington, D.C., on 27 January 1961. The full talk is available in the article H P Greenspan, 'Applied mathematics as a science', Amer. Math. Monthly 68 (9) (1961), 872-880.

1. Methods of Applied Mathematics

Well, the development of mathematics, for the sciences and for everybody else, does not often come from pure mathematics. It came from the physicists, engineers, and applied mathematicians. The physicists were on to many ideas which couldn't be proved, but which they knew to be right, long before the pure mathematicians sanctified it with their seal of approval. Fourier series, Laplace transforms, and delta functions are a few examples where waiting for a rigorous proof of procedure would have stifled progress for a hundred years. The quest for rigour too often meant rigor mortis. The physicists used delta functions early on, but this wasn't really part of mathematics until the theory of distributions was invoked to make it all rigorous and pure. That was a century later! Scientists and engineers don't wait for that: they develop what they need when they need it. Of necessity, they invent all sorts of approximate, ad hoc methods: perturbation theory, singular perturbation theory, renormalisation, numerical calculations and methods, Fourier analysis, etc. The mathematics that went in to this all came from the applied side, from the scientists who wanted to understand physical phenomena. You most likely couldn't even learn these things in a mathematics department. That's where our curriculum came from. It was called Methods of Applied Mathematics at the graduate level.

So much of mathematics originates from applications and scientific phenomena. But we have nature as the final arbiter. Does a result agree with experiment? If it doesn't agree with experiment, something is wrong. You haven't put the right facts into the model; you haven't analysed the model correctly. So there's formulation of a problem, whether it's about tissue formation, biological growth, whatever, followed by its mathematical statement. And then there has to be a solution of that formal problem.

Well, these are two different tasks. The solution of the problem may involve you inventing mathematics, or doing irregular things to get to the answer. That's what physicists do. But then you've got to say, what does it mean? Is the result or conclusion right? If theory says that such-and-such will happen or be seen, or that the structure of a star is such-and-such, then experimental corroboration or verification is necessary. And so nature is the final arbiter. If it doesn't agree, then the model is not terribly good even if the mathematics is elegant and correct. Something is wrong or at least not quite right or complete. Another tack might be tried that you feel is justified but can't be proved. All right. In applied mathematics, we don't write in a theorem-lemma format. But rather we state the physical problem in mathematical terms; give its approximate solution and describe its relevance to the original problem and the data or experiments. You see? That's science. Scientific study has, as its vindication, that the results are correct when compared to nature and the available data.

2. Applied Mathematics as a Science

The following is an address to the Mathematical Association of America, Washington, D.C., on 27 January 1961.

Pure mathematicians, whatever their specialty, have a common attitude and approach to their work that is easily recognised, and generally agreed upon. Unfortunately, there is as yet no general understanding concerning the nature of study and research in applied mathematics. Perhaps the title itself is at fault but no better name has been suggested. Perhaps the confusion arises from the absence of organised schools or departments of applied mathematics at most of our institutes and universities. Whatever the cause, this vibrant, creative and important subject has been badly (and in part deliberately) neglected and our national position in the scientific world it not as strong as it should and indeed must be.

In recent years, several institutions, among them Brown, New York University, Harvard, and Massachusetts Institute of Technology have recognised the need and have developed comprehensive programs based on a single common philosophy. Other universities will follow suit. It is this approach that I wish to discuss today. My objectives are simply to describe and illustrate the dynamic nature of applied mathematics and to dispel some of the current serious and even dangerous misconceptions about it. The most common of these is that applied mathematics is more or less just a technical service, as for example, the programming of detailed engineering and technological problems for computing machines. On the basis of this erroneous conception there has arisen a widely held belief that the Russian emphasis on applied instead of pure mathematics is a grave if not fatal error on their part. I would like to quote a few statements emanating from this conference which appeared in the New York Times, Tuesday, January 24, under the by-line of John W Finney. "Russia may be losing its traditional leadership in mathematics because of overemphasis on applied research." "It was suggested here today that the effect could be to weaken Soviet mathematics, since it is on the pure research that future practical uses will be based." "The consensus was that the United States was unequalled in the breadth of its mathematical research - both theoretical and applied." Evidently all these sentiments contribute to a certain smug self-satisfaction among some of our pure mathematicians. If the premise were true, and the Russians did view applied mathematics as a purely sterile service to technology, I would agree emphatically that this would indeed be a serious blunder. However, this is not and that bears repeating, this is not their interpretation nor is it mine. In fact the Russian views on this subject are very similar in spirit to the programs in applied mathematics at the few institutions I previously mentioned. I believe that increased Russian emphasis in this area is really cause for some serious concern on our part. I contend that pure mathematics is much overemphasised in the United States and that a readjustment is long overdue. It seems to me that the vocal pure mathematicians are interpreting the Russian action in the light of their own prejudices and misconceptions; indeed, it is they who are making a serious error. By holding, fostering and acting upon such attitudes and opinions, they retard and in some cases prevent the growth of a vital area within the framework of American mathematics. These are serious matters for they involve an evaluation of the relative importance of pure versus applied mathematics and this in turn relates to our entire mathematics curriculum. The present system does not produce adequate numbers of applied mathematicians and remedial steps must soon be taken. The great majority of the senior faculty in applied mathematics are products of a European education. In fact at a recent symposium in applied mathematics, the guest lecturers, introduced as "America's first team," included not a single native-trained applied mathematician. I hasten to add that we do indeed have excellent and outstanding people exemplified by Professor Keller but they are not the products of any general organised plan. They result, rather, from separate efforts at a few isolated schools. This rather sad commentary becomes alarming when it is noted that the foreign sources of talent are disappearing. Can we replace men like Friedrichs, Courant, Goldstein and Lin? At present I fear not, although it is imperative that we do so. The Russians are faced with no such shortage of trained people of exceptional calibre. As a consequence, the same can be said about the quality of their research in applied mathematics, which in turn has contributed directly and in no small way to their not so surprising scientific successes. I hope to return to these matters if time permits but first I must develop my principal thesis.

Applied mathematics is a branch of science which seeks knowledge and understanding of the external physical universe through the use of mathematical methods and scientific inference. The ultimate goal of the efforts of the applied mathematician lies in the creation of ideas, and that bears repeating, in the creation of ideas, concepts, and methods that are of basic and general applicability to the subject in question, be it elasticity, magneto-hydrodynamics, geophysics, biochemistry, information theory or even economics. The ideal applied mathematician is a truly versatile scientist - a specialist in mathematics - with broad and active interests in many scientific areas. There are three principal facets in his approach to any particular problem, each of equal importance. Firstly, he must formulate physical problems in mathematical symbolism. Secondly he must solve the mathematical problems and thirdly he must discuss, interpret and evaluate the results of his analyses. In this respect the applied mathematician resembles the theoretical physicist in both attitude and approach. The differences are often a matter of degree.

The theoretical physicist has, as his primary interest, the discovery of new physical laws, while the applied mathematician also takes a deep interest in the description and understanding of physical phenomena in terms of known physical laws, and in the creation and development of new mathematical theories. The applied mathematician is generally the more knowledgeable concerning mathematical theories, methods and techniques and is certainly concerned with a broader spectrum of problems. Any and all things that excite his interest and are amenable to mathematical formulation and analysis are within his province and this ranges from studies of liquid helium, kinetic theory, and meteorology to analysis of the circulation of blood, the swimming of fish and automobile traffic control. Mathematics in combination with scientific inference provide the most powerful tool for the advancement of science yet devised. It is the applied mathematician who is best trained to use this knowledge.

Let me examine the general attitude and approach in somewhat greater detail. To formulate a mathematical model usually requires a detailed knowledge of the observational and experimental facts related to the particular phenomenon under consideration as well as penetrating insight and mature judgment. The aim is to simplify the description without losing the physical core of the phenomenon. Experience is essential to develop the flair for making just the right approximation in the always complicated physical situation. This talent is not easily come by. Thinking in physical terms must be started early in the training of the applied mathematician.

Once this important and difficult phase is completed, the mathematical solution should be found. The applied mathematician must, of course, be extremely proficient and often creative in the use of mathematical methods and techniques. Much of modern mathematics arose from the need to solve physical problems. I need only mention the names of Riemann, Hadamard, Gauss, to illustrate the point. In fact, most of the great mathematicians of the last century were both pure and applied, attesting to the stimulation of one upon the other. Physics has probably given more to pure mathematics in the way of stimulation than it has received, although one may document either case with numerous examples. The fact that much of modern mathematics is now so far removed from physical reality leads me to question any statement which suggests that "it is in pure mathematical research that future practical uses will be based."

There are fundamental differences between pure and applied mathematicians even in the strictly analytical aspects of solving a well-set mathematical problem. As one wit put it, the applied mathematician can find the solution to any difficulty, whereas the pure mathematician can find the difficulty to any solution. Since the primary aim of the applied mathematician is directed to the understanding of physical phenomena, he must often use approximate methods or invoke physical reasoning to achieve a simplification, often without being able to estimate the errors involved. In this way, an analysis might be made more amenable to solution, simplifications introduced or arguments made plausible. All of these means are employed because most current research involves highly nonlinear problems about which very little is as yet known. Although his treatment is at all times responsible and disciplined, the applied mathematician is not a deductive logician interested solely in the beauty of form and the power of abstraction. In other words he is not a pure mathematician. He must, however, have enough background to distinguish between clear demonstration, plausible arguments, and hopeful speculation-all of which are used and used often. Rigour is of course always desirable, but not at the price of sterility. Professor Goldstein puts it this way: "Applied mathematics cannot and should not wait until mathematical knowledge has improved sufficiently for him to turn out a polished piece of work. His real business is primarily with the understanding of physical phenomena and if he can contribute to this end he should proceed, doing always the best he can with the mathematics and struggling always to improve it."

I would like to comment upon ... evaluating the results of an analysis.

All results and conclusions must be checked against the existing body of knowledge. Any new inferences or predictions are subject to verification by further experimentation and observation since their truth cannot be determined by purely logical means. The results are tested by usual scientific means - simplicity, generality, consistency and agreement with observation. The nature of the universe is such that it provides no deductive guarantee of truth.

In the course of this talk I have mentioned many areas under development by applied mathematicians. One important discipline, magnetohydrodynamics including plasma dynamics, though still in its infancy, presents enormous possibilities. The unity of both gas dynamics and electromagnetic theory challenges the inquiring and curious mind. These studies will develop understanding and knowledge of numerous subjects including the structure of the galaxies, the origin of cosmic rays, sunspots, radiation belts, the control of nuclear fusion, plasma rockets for interplanetary travel to name a few. Gradually substantial progress is being made. Almost all the other fields I mentioned are laden with possibilities and abound in important and intriguing problems-astrophysics, meteorology, oceanography, information theory, economics and eventually, I am sure, even the social sciences. Our best scientific tools go unused because they require a mathematical sophistication not always possessed by even the foremost specialists in many of these subjects. How much there is to be done, to be understood, appreciated and incorporated into the body of knowledge. We need not look too far to find a "New Frontier." It is no wonder that the Russians consider applied mathematics important and emphasise it strongly. It is time we did too.

In conclusion, I maintain that there should and must be room in mathematics proper for both the pure and the applied. The present situation is grossly unbalanced and actually represents a potential source of difficulty. A readjustment and reappraisal is vitally necessary. Mathematics curriculums must begin to account for the fact that not everyone desires to be a pure mathematician nor is such training the only or best possible or even the most desirable. Finally, I ask you to ask yourselves, "Is the domination of American mathematics by the pure mathematicians warranted? deserved? desired? necessary? and is it really in the national interest?"

Last Updated November 2020