CHAPTER 7 The Undefiled Mathematician.
John von Neumann
 Until about one hundred years ago all mathematicians would have accepted von Neumann?s conception of mathematics. Certainly the three men whom mathematicians nominate as the greatest of all time - Archimedes, Newton, and Gauss - did more scientific than mathematical work and in fact justified their mathematical investigations by mentioning or describing the applications that warranted the mathematical research. Even those whom many cite as pure mathematicians - Carl Jacobi, Karl Weierstrass, and Bernhard Riemann - not only applied mathematics, but pursued mathematics proper to clarify, rigorize, and extend the theory and technique they already knew to be applicable. This is not to deny that some of these men engaged in subjects, such as the theory of numbers, whose attraction is primarily aesthetic value or intellectual challenge; but one need only count the years they devoted to  purely aesthetic subjects as opposed to those bearing on science to determine which research they regarded as more important.
The situation is quite different today. Though the greatest mathematicians of recent times - Hermann Weyl, David Hilbert, Felix Klein, and Henri Poincaré - would have endorsed von Neumann's characterization of mathematics, now only about one out of ten mathematicians devotes himself to problems of the physical and social sciences, and many of these are employed in governmental and industrial laboratories. Among professors, about 5 percent do applied work. The rest are totally ignorant of science and do not undertake any problems bearing on it. The days when mathematicians saw the hand of God in the motion of the planets and the stars are gone. Especially in view of the facts that science and technology have expanded at least as much as mathematics and that social and biological problems are now being tackled mathematically, this reversal calls for an explanation.
In part we have already given it. Professors are expected to publish. The older applied fields - mechanics, elasticity, hydrodynamics, and electromagnetic theory- have been explored for one, two, or three centuries and the outstanding problems are no longer simple. The newer applied fields - quantum mechanics, magnetohydrodynamics, solid state physics, meteorology, physical chemistry, and molecular physics - presuppose an extensive background in physics. As for the social and biological sciences, these are more complicated, and so far successes have defied the best brains. What, then, should professors, especially young people who have yet to earn rank and tenure, publish? The obvious answer is to pick some specialty in pure mathematics and to invent problems that can be solved. Since the editors of the journals come from the same milieu as the  professors, this artificial research is as readily publishable today as are the most profound papers of mathematical science.
What has this alteration in the nature of research to do with pedagogy? The answer is that most mathematics professors no longer teach either the uses of mathematics in science nor how to apply mathematics to scientific problems. Perhaps the best example of the detachment of mathematics from science is furnished by the teaching of calculus. This subject is the crux of applied mathematics, and next to the liberal arts course it is the one that serves the most students. Prospective engineers, physical and social scientists, actuaries, technicians, and medical and dental students take it to learn how to apply the subject. How then do the professors teach calculus?
During the first four decades of this century the calculus course was a series of mathematical techniques taught mechanically and imitated by the students. The students worked but didn't have to think. Of course, this type of course was neither very helpful nor enlightening to the students, but in view of the level of mathematical knowledge in the United States it was about as good as could be expected. As more students flocked to college, and as a college education became a prerequisite for a good job or a professional career, the meaninglessness of the calculus course became more apparent. Though the professors had become more knowledgeable in mathematics proper, they were not prepared to teach a calculus course suited to the interests and needs of the students. The subsequent squirming and twisting reveal the modern professor's evasion of his obligations.
Until about 1945 mathematics students took analytic geometry before calculus. Analytic geometry deals with a new and vital idea, the coordination of curve and equation,  an idea that is used extensively in calculus. To "improve" the calculus course mathematics professors decided to start students with calculus and to teach the requisite analytic geometry as it was needed. Analytic geometry consequently got short shrift. This consolidation also meant asking the student to learn two major techniques simultaneously. Moreover, since the study of analytic geometry obliges students to utilize algebra and trigonometry, when they took analytics before calculus they were better prepared for the necessary uses of these tools in calculus. Subordinating analytic geometry to a topic in the calculus course deprived students of a sorely needed background. Most, therefore, did poorly.
The professors "saw" the remedy for this trouble. They decided to incorporate more algebra and trigonometry, along with the analytic geometry, in the calculus course. This move proved to be still more disastrous, and the professors backtracked. Now the algebra, trigonometry, and analytic geometry are packed into a one- or two-semester course that is called precalculus - a nice semantic device to avoid admission of the original error.
Some professors took another tack. In addition to including algebra, trigonometry, and analytic geometry, they adopted a rigorous approach to calculus. That is, they included the theory as well as the technique. This move, they evidently believed, would make the calculus course understandable. But the theory of the calculus is highly sophisticated. The best mathematicians, from Newton and Leibniz, who worked in the late seventeenth century, to Cauchy, who worked in the early nineteenth century, struggled to understand the logical foundation of the calculus, and Cauchy was the first to make the proper start. A sound foundation was not achieved until Karl Weierstrass, fifty years after Cauchy's breakthrough, cleared up the mess. Certainly, then, the theory is not easy for beginners to grasp. One may be sure that the very same teachers who believe that students beginning calculus can absorb a theoretical foundation would have been swamped, in their own student days, by such a presentation. Nevertheless, having finally grasped the theory after some years of study, they forgot their own experiences and acquired a missionary zeal to spread the light.
The proper pedagogical approach to any new subject should always be intuitive. The strictly logical foundation is an artificial reconstruction of what the mind grasps through pictures, physical evidence, induction from special cases, and sheer trial and error. The theory of the calculus is about as helpful in understanding that subject as the.theory of chemical combustion is in understanding how to drive an automobile. This approach through theory, which had its heyday in the mid-1960s, had to be abandoned.
The prewar teaching of pure technique was not successful; the inclusion of algebra, trigonometry, and analytics was no more so; and the inclusion of rigor was soon found to be a disaster. What measure could the mathematicians adopt? Since the precalculus course, when instituted, took care of the elementary material needed for calculus, the obvious move was to include material beyond calculus. Bits of linear algebra, vector analysis, differential equations, and other topics (some of which have no relevance to calculus) were therefore included in the calculus course, which is now a hodgepodge of topics and a mélange of unrelated techniques. (See also Chapter 10 on texts.)
In all these maneuvers the professors have avoided the one measure that would make the calculus course meaningful and serve the purpose for which it is intended - namely, to make it the introduction to applied mathematics. Mathematics proper and calculus especially are mazes of  symbols and manipulations of symbols. As such, these have no meaning or purpose. They are the shadows of substance and have as much meaning as the notes of a musical score to one who cannot hear the composition they describe. The symbols have no life, but properly interpreted they can tell us about the vital forces that affect almost every aspect of our lives. Only the applications supply meaning and motivation. Since most calculus students will be engineers or physical scientists and intend to use calculus, what better insight could they be given than examples of where calculus achieves results?
Calculus offers excellent opportunities not oniy to apply mathematics but also to show how physical arguments suggest deep mathematical results. For example, we know that a ball thrown into the air rises and then falls. At the highest point in its path the velocity must be zero, else it would continue to rise. Because the velocity is the rate of change of distance with respect to time, what this physical happening suggests is that the rate of change of one variable with respect to another must be zero at the maximum value of the first variable. This is a basic theorem of calculus.
There is, however, an obstacle to the introduction of physical problems that might supply motivation, meaning, and application: Most mathematics professors know no science and will not extend themselves to learn it. Those few who know enough to present the simplest physical applications fear the questions that may ensue.
Many professors realize that calculus proper is dull and meaningless but, not prepared to offer real applications, they put on a pretense of doing so. They assign the following typical problem: Find the velocity of an object that moves with an acceleration of a To find velocity knowing the acceleration is indeed an application of calculus, but what object in this universe moves with the  acceleration stated? Perhaps a drunken driver. Some professors are more realistic. It happens that objects moving near the surface of the earth, if one neglects the resistance of air, are subject to a downward acceleration of thirty-two feet per second each second. Hence, it does make some sense to pose problems of motion involving this acceleration. However, such problems are simple and do not exhibit the power of the calculus. Motion in a vacuum should be followed by more realistic problems involving motion in an atmosphere. The parachutist who could not rely upon the resistance of air would not, after one drop, have to rely upon anything. Perhaps the professors who confine their applications to motion in a vacuum are preparing students for life on the moon, which has no atmosphere, while convinèing them that life on earth is intolerable.
Calculus texts offer other "real" applications. A man six feet tall is walking away from a street light at the rate of five feet per second; the problem asks how fast the man's shadow is lengthening when he is ten feet away from the light. The problem deals only with the shadow of reality.
Professors also introduce problems in which physical terms such as "center of gravity" and "moment of inertia" are used. But the physical meanings of these concepts and their uses are not taught. The consequence is that the gravity of these problems produces moments and even hours of inertia in the students.
Of course, curiosity might induce some people to solve any problem. But curiosity not only kills cats; it kills interest in mathematics courses that pose pointless problems. To teach calculus without real applications is to ask people to sit down at a table set for a dinner where no food is served, or to teach grammar but never to mention literature.
Why are professors content to teach artificial, dull and pointless applications? Such problems have been in calculus  texts for fifty or more years. The professors learned how to solve them when they were students. Why bother to dig up new and more significant ones and incur much more work in learning to present them if there is no pressure to improve the course? Surely it is boring to repeat the same deadly material year after year; but then all of teaching is a chore to be disposed of as quickly as possible.
The deficiencies in the calculus course are exemplary of a glaring deficiency in the entire mathematics program, graduate and undergraduate. Though the major reason that students take mathematics is to use it, only a few undergraduate and graduate schools offer applied mathematics. There are courses and texts that are titled "Applied Mathematics," but these are pitiful. They offer mathematics that can be applied, for example differential equations, but at best they mention where the topics are applied. They omit the problem of analyzing physical phenomena to determine which factors or features can be neglected and which must be incorporated in the mathematical formulation; and they fail to teach at all the process of translating physical facts into mathematical language. Since no physical problems are treated, the payoff - what one learns through the mathematics about physical phenomena - is missing. Moreover, most of the texts contain tidbits from various mathematical areas used in applications but no one topic is pursued in depth. If these texts make any impression on the students it is only to bewilder them. They are shifted from topic to topic so quickly that nothing sticks. One is reminded of the whirlwind tours of Europe which cover ten countries in ten days. The tourists return home uncertain as to whether the Eiffel tower is in Paris or Prague.
Mathematicians are, of course, aware of the existence and importance of applied mathematics, and they are sensitive to the charge that they are neglecting it. They justify their purely mathematical offerings on the ground that they are teaching students how to build models for the solution of real problems. As new physical problems are tackled, presumably all the applied mathematician or scientist will have to do is run through his files and select the model that fits his problem. But this type of model-building is a waste of time, and advocacy of purely arbitrary mathematical creations reveals ignorance of what applied mathematics involves.
God may have designed the world mathematically but evidently He did not intend to make that design readily accessible. Mathematics is not emblazoned on the face of nature. Several crucial and difficult steps are necessary to mathematize genuine physical problems. Any real situation contains dozens of elements whose relevance must be considered. If one is studying the motion of a ball, the color can surely be neglected, but the shape and size may not be negligible. On the other hand, if one is studying the reflection of light from some surface or the transmission of light through some translucent material, the color of the surface or the material may be critical. In the study of the motion of a planet around the sun, both the planet and the sun may be regarded as point-masses, that is, the mass of each can be regarded as concentrated at one point. The reason is simply that their sizes are small compared to the distance between them. On the other hand, if one is studying the motion of the moon around the earth, the size and shape of the earth must be taken into account. Should the attraction of both bodies by the sun also be taken into account? That depends upon the problem to be solved. To predict the tides of the oceans on the earth, the sun's attraction does matter. However, to study the precession of the earth's axis, that is, the change in the direction of the imaginary line through the North and South Poles, the sun?s attraction can be ignored. The more exact the answer required, the more care must be  exercised to be sure that the relevant factors are taken into account. Simplification of a problem by discarding the irrelevant factors is a crucial step, and it presupposes an insight that may, of course, be deepened by experience.
After simplifying a problem one must apply physical principles. (In studying the motion of the earth around the sun, the law of gravitation is a fundamental principle.) Such principles are usually supplied by physicists, but the translation of the physical principles and other relevant information about the particular problem into the language and concepts of mathematics must be done by mathematicians. The concepts may not be available and may have to be created. In fact, precisely the need to treat problems of motion motivated the creation of the calculus. New concepts are added constantly as new problems are tackled.
Once the mathematical formulation is achieved, the next stage is the solution of a mathematical problem. There are times when the applied mathematician is lucky. The mathematical problem may have been solved in the course of some earlier study. And one may be sure, if this does prove to be the case, that the solution was originally sought in behalf of a real problem. More often, unfortunately, the mathematical problem is a new one that calls for original work.
Other problems and processes, such as approximation adequate for the use to which the solution is to be put, enter into applied mathematics, but the major point is that mathematical models cannot be constructed a priori and then called upon when needed. One cannot prefabricate useful models. The mathematics involved in real problems is far too complex and special to be conjured up by the free play of the imagination. A physical problem comes to the hands of mathematicians as a rock encrusted with sediment and mud. It is up to the mathematician to remove the dross,  chip away the encrustations, polish the rock, and bring forth ultimately a blazing gem of physical truth. There is an art of applying mathematics and an art of teaching that art.
One cannot expect students to solve new applied problems. But they will be required to do it in their professional work; therefore, they should be taught all that is involved in the entire process. To delude students into believing that the study of solely mathematical structures and processes suffices is to falsify the account. The professors? contention that the study of mathematical models prepares students for applying mathematics is, at best, wishful thinking to rationalize their own ignorance and, at worst, conscious deceit.
When challenged that the values of mathemafics proper do not mean much to potential users, many professors retort that students will learn the applications in other courses. But to ask students to take seriously theorems and techniques whose worth will be apparent one, two, or several years later is a grievous pedagogical error. Such an assurance does not stir up incentive and interest and does not supply meaning to subject matter. As Alfred North Whitehead has advised, whatever value attaches to a subject must be evoked here and now.
Still another argument offered by professors against the teaching of applications is that it imposes a heavy burden on the students; they must learn the mathematics and the relevant physics, say, and they must also learn to relate the two. But this argument is specious. Carefully chosen applications do not require much extramathematical background, and the little that is required can readily be included in the mathematics course. Moreover, the teaching of mathematics is expedited by tying it in with applications. These provide motivation, which mathematics proper does not. Equally important, the only meaning the concepts had  for the mathematicians who created them and the only meaning students will find in courses such as calculus derive from physical or, more generally, real situations.
Professors also use the argument that they cannot cover the syllabus if they include real applications. But even if this argument has force, and it does have some, in what sense is the ground covered? The professors cover the topics in the syllabus but the students are buried so deeply under an landslide of ideas and techniques that they no longer see light. The ground is covered over the students.
In view of the fact that the application of mathematics to science and engineering is its most vital and widespread use, the absence of applied mathematical courses is as deplorable as the absence of honesty in our political leaders. But since most mathematicians are no longer capable of offering applications, they shun them as infections in a sound body. The intransigence of mathematics departments in meeting their obligations to students majoring in areas such as science and engineering is notorious. They teach as though mathematics is all we know and all we need to know. Unfortunately, it is in the most prestigious universities that the professors are allowed to take the position that they are authorities unto themselves. They are autonomous and teach what they will. Syllabi for courses and a planned sequence of courses, essential in a cumulative subject such as mathematics, are detested and ignored by the high and mighty. Professors often palm off the teaching of courses addressed to science and engineering students on the younger members of the faculty or on graduate students, who have no choice of the courses they must teach.
What subject matter do professors teach? Except in a course such as calculus, where the content is prescribed, they favor their own specialties. These in turn are determined by their research. It is not surprising to learn that courses in mathematical logic, abstract algebra, topology, the theory of numbers, functional analysis, and axiomatics dominate the undergraduate and graduate curricula. The technical nature of these subjects need not be examined. What matters is that these subjects, very fashionable today, constitute a one-sided account of mathematics. All are pure; that is, devoid of applications. Professors prefer virginity to bedding with science.
Professors are specialists and they tend to view the world of mathematics through the medium of their own specialty. Certainly there should be courses for prospective pure mathematicians. But all mathematicians should be informed about the chief value of mathematics - namely, its interplay with science and the amazing fruitfulness of that interplay. Here man demonstrates the magical power of his mind, and how mathematics bridges the gap between his mind and the real world. Or, as Max M. Schiffer, professor of mathematics at Stanford University, has pointed out:
Mathematics majors may be free later to pursue any branch of pure mathematics, but not to know the chief role of their subject is to be ignorant, no matter how many research papers they may write later. A goodly number of  the courses should deal with applications to science, and some physics should be part of the education of every mathematician. Further, since the prospective mathematicians are most likely to become university teachers, they should be prepared to teach prospective scientists and engineers.
That the typical mathematics professor was not required to learn any science can be charged to the graduate school professors, who (with few exceptions) are a collection of specialists in various areas of pure mathematics. Thdse professors, who face the task of educating scientists, engineers, and future teachers of such students and refuse to do so, are irresponsible. Were they in an industrial or commercial organization, they would be fired. That such dismissals do not take place in the universities is due only to the fact that mathematicians as a group are nearly all in the same position, and not even the chairman, who is one of the group, would wish to take action. Many professors claim that they take pride in their work and do fulfill their obligations as teachers. But their actions belie their words.
What is wrong, then, is not that professors do not know what they are teaching but that they do not know how to relate their subject to the rest of knowledge and to life. Education is evaded within academic walls as well as without, and it is more often the professors rather than the students who do so. Professors may not be consciously dishonest or aware of their pedagogical ineptness. However, the lack of clear standards of teaching, the professors? ignorance of pedagogy, and the obligation, which many take too literally, to cover ground prescribed in syllabi produce the same effect as incompetence and dishonesty. The practices of the past are not re-examined, and the challenges that true education might pose are ignored.
The student who goes to college to prepare for a career is  certainly justified in doing so, and the colleges have been implicitly accepting such a goal while explicitly talking about liberal education. Hence, courses for business-oriented students, statisticians, actuaries, and scientists must be given the attention other courses get. True, vocational or professional interests clash to some extent with purely academic interests, but the former should not be submerged or ignored. Nevertheless, myopic professors impose their own interests on the students, with the result that their courses are largely useless to most students and to society. Students are accused of resistance to intellectuality, but their resistance is to arrogant and indifferent professors who, in the name of academic freedom, serve themselves.
Mathematicians have abandoned science in an age whose major achievements are scientific and most of whose principal problems will be solved largely by resort to science. They live in a self-imposed exile from the real world. Blinded by a century of ever purer mathematics, the professors have lost the will and the skill to read the book of nature. Like the mathematicians Gulliver met in his voyage to Laputa, they live on an island suspended in the air and leave to others the problems of earthly society.
The extent to which they have abandoned science is perhaps best indicated by the words of Marshall Stone, who served over the last few decades as a professor at Yale, Harvard, and Chicago:
Stone's words have not gone unchallenged. Professor Richard Courant, formerly head of the pre-Hitlerian world center for mathematics at the University of Gottingen and more recently the founder and head of what is now called the Courant Institute of Mathematical Sciences of New York University, has denounced this abrogation of the essence of mathematics:
The various sins of pedagogy were attacked again by Courant:
The abandonment of applied mathematics by most mathematicians is a blow not merely to pedagogy. It is a threat to the very existence of mathematics itself. Problems which stem from the real world are the lifeblood of mathematics. It must remain a vital strand in the broad stream of science or it will become a brook that disappears in the sand. The entire history of mathematics shows that physical science has supplied the inspiration, vitality, and fruitfulness of this subject.
Nor should one overlook the value of applied mathematics to technology and thereby to humanity. That men and women now work thirty-five or forty hours a week instead of eighty, that their homes are better built and more comfortable, that they enjoy quality phonograph records and television, that they can receive medical treatments which cure diseases or at the very least prolong their lives, these and a multitude of other benefits are due in large measure to mathematics.
To speak of an applied mathematics program as though it were one of many programs or as though there were two kinds of mathematics, pure and applied (or monastic and secular, as some would put it), is a concession to the current practice. But it is a misrepresentation of mathematics and mathematics education. There is just one subject: mathematics. The chief function of that subject, and its chief claim to support by society and to an important role in education, reside in what it does to help man understand the worlds about him—physical, social, biological, and psychological. So far the successes have been mainly in the physical sciences, but judged by the time scale of civilizations, mathematics is young.
Many mathematicians today and of recent years would dispute this evaluation. Mathematics, they say, is what mathematicians do, and since most mathematicians have no interest in anything but the subject itself the relationship to other fields is irrelevant. No doubt many are sincere. Since they do not know the magnificent and powerful uses of mathematics they can be indifferent to them. But they should heed more of von Neumann's words:
Does the abandonment of science by most mathematicians mean that science will be deprived of mathematics? Not entirely. The Newtons, Laplaces, and Hamiltons of the future will create the mathematics they need, jusf as they did in the past. Formerly such men, though they were honored as mathematicians and served as mathematics professors, were basically physicists. In today's world they are cast out by the mathematicians but they find their place in science departments. Nevertheless, the services of many able people are now lost to science and to the necessary pedagogy.
The failure of mathematics departments to cater to science, engineering, and social science students has had the expected effect. The physical science, social science, and engineering departments of many universities offer their own mathematics courses. In some institutions, statistics and probability courses are given in half a dozen departments. In almost all colleges and universities computer science has broken away from mathematics and is a separate department. Clearly, the users of mathematics have decided that mathematics is too important to be left to the mathematicians. Should this practice expand, it will be the end of mathematics education as such. Curiously, though some mathematics professors resent this competition for students  and the takeover of what they regard as their province, others exhibit a remarkable "broadmindedness." Their reaction is, Well, now we don't have to teach the useful mathematics; we can teach what we please.
Though the teaching of mathematics courses by other departments and schools such as engineering may seem to resolve the problem of giving students the kinds of mathematics courses that would be more useful to them, it is not the realistic solution. A relatively minor objection is that it leads to endless duplication of courses, which the universities can ill afford. More consequential is the fact that mathematics, physics, economics, and the various branches of engineering are now vast domains and a professor is hard put to it to master even a portion of any one of these domains. A physics professor, for example, may know selected portions of mathematics, but he cannot teach mathematics effectively. He does well if he can teach successfully his area of physics. If, on the other hand, these several users of mathematics were to hire mathematicians to teach the mathematics they want taught, they may not require the services of a full-time professor and may settle for a graduate student or adjunct. However, even if a number of full-time professors are employed but are segregated in nonmathematical departments, they will be isolated and will fail to benefit from contact and communication with colleagues who can stimulate and educate each other. The departmental organization of disciplines has drawbacks, but it is the best we have. Mathematics is far too important to be taught in dribs and drabs.
The neglect of applied courses and the poor teaching by mathematicians has had the consequence that more than half of the students who enter college as science majors either drop out or turn to other fields. About 50 percent of the engineering students do not graduate. Other factors enter, of  course, but poor mathematics education is certainly a major cause. The courses are lethal weapons that snuff out the intellectual interests and sometimes the academic lives of the students.
One who peruses the current mathematical journals will find articles advocating the institution of applied mathematics programs in the undergraduate and graduate schools, and may therefore infer that mathematicians are now genuinely concerned with this obligation. But closer investigation will reveal that this sudden interest is caused not by any recognition of student needs or guilt about the poor courses that have been offered but is, rather, selfishly motivated. Academic positions for Ph.D.'s in mathematics are now far fewer than the number of Ph.D.'s being turned out by the graduate schools. However, industry and the federal government do employ applied mathematicians. Hence, graduate students and Ph.D.'s are being urged to prepare for the available jobs. Were they to accept this recommendation, most would have great difficulty in acquiring the proper training.
If one were to judge mathematicians by their pedagogy, one could take either of two positions, depending upon whether he was charitably or critically disposed toward them. Under the former, they would be characterized as so introspective as to be unworldly, deeply in love with their subject, intensely concerned with the progress of mathematics, and so far superior to ordinary mentalities that they cannot appreciate normal problems. Under the second view, their preference for their own values and interests marks them as selfish, overly absorbed in themselves, culturally narrow, and indifferent to society?s interests and to the needs of students. In either case the mistakes made by professors bury their pupils just as surely as the mistakes of doctors bury their patients.
 Nevertheless, the professors who teach pure, abstract mathematics and maintain that they are successful do provide some lift to all of us. We can be quite sure once again that there is a heaven and that there are angels. Certainly the students must be angels to be able to absorb what the professors say they teach them, and the professors evidently derive their sustenance from heaven because they have their heads in the clouds.
I am very grateful for the kind permission of Professor Kline's widow, Mrs Helen Kline for this book to be reproduced.
Copyright © Helen M. Kline & Mark Alder 2000
Version: 22nd March 2001