George F Carrier on Applied Mathematics

On 4 November 1961, at the meeting of the Society for Industrial and Applied Mathematics held in Washington D.C., a Symposium was held on "Applied Mathematics: what is needed in research and education." It was chaired by H J Greenberg, Thomas J Watson Research Center, and the speakers were: George F Carrier, Department of Mechanical Engineering, Harvard University; Richard Courant, New York University and the AEC Computing Facility, New York; Paul Rosenbloom, Institute of Technology, University of Minnesota; and Chen Ning (Frank) Yang, Institute for Advanced Study, Princeton.

Below we give a version of George Carrier's contribution to the Symposium. Let us make one brief note first, however. The use of "he" when referring to students, teachers and professors looks so wrong that we considered changing it to gender-neutral language. In the end, however, we decided to leave the "he" on the grounds of historical accuracy.

Applied Mathematics: what is needed in research and education.

Dr Greenberg. Our second speaker will be George Carrier, Professor of Mechanical Engineering at Harvard University. Professor Carrier is one of an extremely rare breed. A mathematician who knows engineering and physical science and who, with exceptional skill, formulates and solves problems of interest to engineers, scientists, and mathematicians. With pleasure I introduce Professor Carrier.

Professor George F Carrier. I have been told that Joe Keller once initiated some remarks concerning Applied Mathematics with the definition, "Applied mathematics is that science of which pure mathematics is a branch." As a statement, I can agree heartily with this, but as a definition I feel it is far too conservative. It seems to me that applied mathematics is that science of which not only is pure mathematics a branch, but of which all other sciences become branches as soon as they become sufficiently quantitative. The discussion of this broad a field, of course, would contain nothing but puerile generalities, so I shall confine my remarks to some educational questions associated with what I regard as the core of applied mathematics. I confine myself to educational questions because I feel that when highly qualified men are well educated and prepared for a challenging profession, they will perceive clearly the exciting and worthwhile research needs and will take care of them. I confine myself to the core of applied mathematics because I feel that, in that area, virtually no undergraduate curricula are to be found in the United States.

Perhaps I should first define what I mean by the core of applied mathematics; I shall do so by describing the objectives, abilities and educational needs of the men who populate this core. Their objective is to understand scientific phenomena quantitatively. To do so, such men must be so thoroughly informed in the fundamentals of some broad segment of the sciences (be they physical, biological, economic, or whatnot) that they can pose the question or family of questions they pursue as a mathematical query using, as the occasion demands, either time-honoured and well established scientific laws (as in mechanics) or carefully conjectured models (as in younger sciences). Such an applied mathematician must also have an understanding of mathematics, a knowledge of technique, and such skill that he can use either rigorously founded techniques or heuristically motivated methods to resolve the mathematical problem, and he must do so with a full realisation as to the implications of each with regard to reliability and interpretation of results. In particular, the applied mathematician must be very skilful at finding that question (or family of questions) such that the answer will fill the scientific need while the extraction of the answer and its interpretation are not prohibitively expensive. I must emphasise that such an individual is not a mathematician nor, ordinarily, is he a specialist in a particular branch of science; he is distinguished from these primarily by his attitude and his objectives, but also by the scope of the scientific and mathematical disciplines on which he must draw.

As I said earlier, there are very few undergraduate programs which expose the student early and consistently to these objectives and attitudes. What should such a program be? I firmly believe that this exposure, this program, should begin at the high school level! I think, in particular, that the mathematics curriculum would be enriched and that the student would be better motivated and his perspective improved if, when trigonometry and analytic geometry are taught in the high schools as they should be, the science of statics were presented - so coordinated with the mathematical instruction that the student perceives the need for and utility of mathematical study. Such an exposure is really necessary to balance the new secondary school emphasis on the beauties of precise mathematical reasoning. I think, in fact, that much of the bitter controversy which has arisen over secondary school mathematics might well be resolved if the student encountered not only precise mathematical thought but also well motivated scientific utilisation of that mathematical thought and the techniques it breeds. Furthermore, at this level, such an exposure should include all students who study such mathematics, whatever career they intend to pursue. I find it incredible that a supposedly well educated man should find it unnecessary in our age to understand at least this much of his environment.

This coordinated view of the sciences and mathematics should continue with the college curriculum, necessarily for novices in applied mathematics and, I would hope, for many others as well. When calculus is introduced the student should study dynamics and those rudimentary aspects of other sciences for whose quantitative understanding the calculus suffices. The coordination should be such that in some instances the mathematical topic is suggested by the scientific question and in others the mathematical development precedes its exploitation. In either case, the topic once introduced must be taught as mathematics, remembering that the ideas are far more important than the epsilons!

A more broadly informed person than myself could pursue a description of courses which gives in chronological order the mathematical material and the coordinated scientific material from which an applied mathematics student could profit, but I think that the idea should be clear now without such an outline. In a nutshell, the student must see a consistent pursuit of the under-
standing which the interplay of mathematics and science can give, and he should find it distinguishable from, for example, the mathematician's pursuit of the full implications of fundamental postulates in which he needs and expects no interplay with the physical universe. Or again, he should distinguish applied mathematics from the physicist's pursuit of hitherto unknown fundamental laws and their verification and immediate implications. I draw these lines much too sharply, of course, but I do think such a description clearly distinguishes the main objectives of these various aspects of quantitative science.

Perhaps I can emphasise the foregoing further by airing a particular complaint which I have entertained for several years. It is concerned with the education of what I shall call digitally oriented scientists. I must preface the complaint with certain remarks. No responsible person doubts the enormous present and future values of the computational sciences and their attendant machinery. When wisely used, we can extract information which was not remotely within reach before the rapid evolution of this discipline. Nevertheless, this discipline, like any other when unwisely used, can be a mixed blessing indeed. I claim that, when treating a quantitative problem, the man who uses a rigorously justified technique does not need to know nearly as much as one who uses a heuristically motivated method. The more heuristic and the less precise the foundation of the method, the more critical and the less routinely formal must be the procedure. The extreme of this in my experience is the use of approximations involving computational science and machinery. The very youth of the science, the difficulty of identifying the consequences of badly drawn approximations, and the inability of the device to see and describe unexpected singularities of almost any kind imply that the responsible digitally oriented applied mathematician must understand extremely thoroughly the mathematics and the scientific fundamentals whose intricacies his digital science is intended to obviate. Computation is not and cannot be a substitute for thought or a replacement for understanding. Despite this, I have seen several educational channels which eject digitally oriented degree holders whose curriculum has not led them to that understanding of intermediate level analysis which enables them, for example, to solve rather simple Laplace equation problems without computational science, or to detect that certain badly and obviously over-specified problems have no solution. I have even been offered digital solutions of what were stated to be appropriate approximations to the latter. I don't claim that such events are typical! I hope and I think they are not. However, they do exist and my plea to those of you who hold responsibility for digitally oriented education is that you insist on a broad and deep program in those sciences without which digital science could become a sterile and isolated discipline appropriate only to routine operations instead of the powerful adjunct to all of science that it can be.

To summarise then, I think a major deficiency in our educational structure is the very small number of undergraduate educational opportunities wherein the student consistently finds instruction which emphasises and is motivated by the close interplay of science and mathematics on a broad scale. I hope that such programs will find their way into many of our schools.

Dr Greenberg. Thank you very much. When I suggested to George that he was going to talk for 20 minutes, knowing the tradition that when he gives an hours talk it lasts about half an hour, I was afraid he would be done in three minutes. But I guess he feels very deeply about this subject because he practically filled up the full twenty minutes.

Dr Greenberg. If the applied mathematician is going to continue and have influence in new areas, it would seem, if one looks at the courses of applied mathematics which have traditionally been offered and which continue to be offered and which have to do principally with branches of mechanics - solid and fluid mechanics, that substantial changes are needed in the curriculum. The question is: If applied mathematics is going to reach into realms of modern physics and even newer fields like biology and medicine, how is it going to be possible to weave these subjects into the training of the mathematicians so that they can have some influence in the development of these fields, and vice versa?

Professor Carrier. My suspicion is that it will be woven in in the same way that most of the topics which now appear as classical have been woven in. The professional applied mathematician, whose interests are broad and who ventures into these new fields, will see the kinds of mathematics that he has to understand in order to make any progress, and at some stage, probably at the graduate levels of work, he'll introduce this material to students; then, as the need seems to grow, the material will be taught at the earliest level at which the student can assimilate it. I don't think you deliberately put things into the curriculum until you've encountered a need for them, if you're working from the applied mathematician's side. Curricula are already crowded enough with things we know we need. In mathematics departments, there may well be offered very strong courses and very much broader courses than the applied mathematics student can afford the time to look at, but it behoves the applied mathematician to be aware of these fields of mathematics and to be educated as to their general character to the point that he can recognise when one of them is something that he is likely to be able to exploit.

Dr Greenberg. There seems to be general agreement that we need to interweave the subject matters of mathematics and science, and that books have to be written to serve as a basis for the curriculum. Now, we know that books are being written and that there is a crusade. ... I think many people are curious as to the success of these efforts ...

Professor Carrier. I must say that my worries go a little further than the writing of books. One thing I'm concerned with is the training of the high school teacher. As you probably all know, the NSF (National Science Foundation) is supporting summer institutes for upgrading the background of high school teachers and I sat in on some of the decisions as to which ones should be supported and which ones shouldn't. In each of the proposed institutes which I encountered, essentially all of the material to be offered to the secondary school teacher was ,designed to provide depth of background and perspective in mathematics; the very discouraging observation I made was that none of the material would provide any background or perspective regarding the use of mathematics.

Now, while I fully concur that these teachers must acquire the appropriate precision of mathematical thought and a thorough understanding of certain mathematical disciplines, it is equally essential that he acquire some perspective and understanding of the way in which mathematics is used. In fact, he really should acquire some skill at using mathematics to deal with rudimentary quantitative questions in science. After all, the vast majority of students who will be taught by our secondary school mathematics teachers will not become professional mathematicians but will have occasion to use mathematics in connection with science and other professional activities. Thus it seems to me that the secondary school mathematics teacher must study carefully prepared and presented material which prepares him both to lead the student into the habits of precise mathematical thought and to introduce him to the attitudes and skills which underlie and exemplify the challenging variety of uses of mathematics. In particular, these teachers must be led to recognise that the currently popular notion that the beauty and purity of mathematics are somehow contaminated when they are related to questions involving the real world is not a very intelligent view. The use of mathematics in science and elsewhere can be as challenging, as aesthetically pleasing, and as valuable to society, as the "pure" self-contained discipline. Our teachers and our students had better find this out - soon !

Dr Greenberg. Let me ask a question that we have not considered at all; that is, the graduate schools in applied mathematics in the United States - are they doing well or are they doing poorly? One question which someone raised, and it's a nice one, asks us to discuss the role of the Ph.D. thesis problems in regard to the present state of pure and applied mathematics. Are we doing good graduate work in applied mathematics in our institutions as far as the Ph.D. theses which are being assigned? Are they meaningful in furthering the role and status of the applied mathematician in the sense that he is led to do good scientific work? I think Professor Carrier ... may have the answer to this.

Professor Carrier. The obvious answer is that you can't answer that question uniformly. There are some graduate schools that I think are doing exceedingly fine work. They have people on the staff who know what worthwhile research is (by some reasonable criterion); they assign meaningful work to students, and, while guiding these students through their thesis work, they see to it that the student uses his own imagination, that he assimilates the attitude of the applied mathematician, that he learns a great deal, and that he really does contribute something of research value rather than just manipulation. In such schools the research is worthwhile and, more important, the student emerges with an appropriate background on which to build a productive research career. There probably are other schools that fail to provide a good Ph.D. education, frequently because the staff does not do research itself; whether the Ph.D. research is worthwhile usually rests with the research advisor or the research committee of the student; obviously there are able ones and there are less able ones. I like to think that we do a good job (at Harvard), and I'm sure that everybody here feels exactly the same way about the work at his own institution. In any event, I'm sure that good research in applied mathematics is being done in many schools in the United States.

Dr Greenberg. If the source of good applied mathematics of the future is going to come from science and, to some extent, also from technology - where engineering attempts to exploit new discoveries - what about the interaction of the graduate school with the industrial community as a source of problems, and so on?

In other words, does the applied mathematician in his department have access to new problems, unless he is a consultant on the outside and has a lot of contacts? Of course, a lot of people do, and that is a source of fresh, new problems, but is it a necessary source of fresh, new problems?

Professor Carrier. I don't know. I know some people who do very good research who have essentially no industrial contact; they're very fine applied mathematicians and some of them are fine engineers too. On the other hand, many of the problems that I personally find exciting and stimulating are ones that in one way or another I encounter from industrial contact - it may be casual or it may be actually consulting. But I know other people who from their knowledge of the state of the art and their knowledge of the literature and so on, detect and work on very worthwhile scientific questions without such external stimulus; so I repeat that there is no single answer. It depends on the individual who is doing the research and who is providing problems for his students (when that needs to be done) whether an outside stimulus is needed. Different people are stimulated in different ways.

Professor Carrier. I have something to add to avoid misinterpretation as to what I mean by the word "suggested." I said that many of the problems that stimulated me are suggested by contact with the industrial world. I don't mean that the question that they ask one to answer is literally the question that the applied mathematician or any other scientist will necessarily find stimulating (although in many instances it may be). The stimulating line of research may be an abstraction or a generalisation of such questions, so when I say "suggested," I mean it in a very broad sense; I certainly don't mean that a productive research career is likely to be one in which one attempts to answer the questions that literally are put to him by some external concern. On the other hand, one can very profitably concern himself with the fundamental questions which underlie the phenomena which give rise to such "industrial questions."

Dr Greenberg. There are a couple of more questions from the floor that I would like to try to cover. One is directed to George Carrier. In singling out numerical calculations for special remarks, this person feels that you implied it may not be the function of the applied mathematician to actually touch the machine, that they should have other bodies such as programmers and so on. He wants to know whether the applied mathematician should do the whole problem right through the numerical analysis, even if it means doing some programming himself.

Professor Carrier. Ordinarily I think the answer to this is "yes." The most consistently successful use of machines that I have seen has been at Los Alamos, where just about every man follows his research right through any necessary calculations involving machines. But this answer is not uniformly appropriate, of course. There are certain routine problems for which all the programming is really already available except for certain formal substitutions; the inversion of not too large matrices I believe is in this category, although my own ignorance may now be showing. There is no doubt, I think, that routine operations can be done by someone who has the job of doing routine operations, and is not an applied mathematician in the sense that I would like to define it. But when dealing with subtle research problems where, in the first place, the procedure is not necessarily routine, where the structure of the result cannot be accurately anticipated, and where a great deal of understanding is required in order to decide what one wants the machine to do, the applied mathematician should carry out the non-routine aspects of the investigation himself.

Last Updated January 2021