Why Johnny Can't Add - CHAPTER 6: The Illiberal Mathematician

[111] The liberal arts colleges - whether integral parts of universities or independent entities - now cater to a variety of student interests. Adequate instruction calls for courses that, while conforming to the values and objectives of a liberal arts education, serve these interests. Mastering the requisite materials and incorporating them in suitable pedagogical format constitute the major tasks of educators.

But the professors are under pressure to do research, and the graduate students, to obtain a Ph.D. Both are narrowly educated, though in different respects and for different reasons. Thus, both groups of teachers, whether engaged in or training to do research, know some mathematics but are ignorant in science and pedagogy, and both are concerned almost entirely with content and values that only mathematicians prize. Yet both must tackle the courses that are supposed to serve many different student interests. Just [112] what is it that professors and graduate students, taking their cue from professors, teach in some of the most fundamental courses?

The largest single group taking mathematics in the colleges and universities consists of liberal arts students who register for a mathematics course only to meet the requirements for a degree. The typical registrant is indifferent to mathematics or actively dislikes it. Nevertheless, the teaching of these nonspecialists is probably the most important task of the professional mathematician. It is important from the broad sociological point of view and even from the standpoint of mathematics, because many of these students will become leaders in our society and will decide how much support to give to the subject. Also, since the segregation of students into prospective users and nonusers of mathematics is based on interest rather than ability, the nonuser group contains some of the most worthwhile students.

To cater to the nonusers most colleges, whether independent or part of a university, offer what is called a liberal arts mathematics course. Despite the importance of this course, most mathematicians despise teaching it. It is beneath their dignity to bother with nonmathematicians and to waste their precious time and knowledge on mathematical "nonentities." However, many professors are obliged to teach the course in order to fulfill the required number of teaching hours. What, then, do they teach? Because most mathematicians are narcissists, they offer reflections of themselves. They prefer pure mathematics, and even in that area only topics that strike their fancy. Let us examine some of these topics.

Their favorite topic by far is the theory of sets. A set is no more, of course, than a collection of objects. But students are asked to learn specific operations with sets such as union, [113] intersection, and complementation, and are also asked to learn properties of these operations. Set theory also includes infinite sets, such as the set of all even integers. All this material has no, or at best trivial, bearing on elementary mathematics or on real phenomena. In addition, the concept of an infinite set baffled and was rejected by the best mathematicians until the 1870s and is still unacceptable to many today. Since the study of sets, and particularly infinite sets, does not return even infinitesimal riches to the student, he does not see why he should attempt to make his reach exceed his grasp.

Another favorite topic is the theory of numbers, wherein one studies unusual properties of the integers. Some numbers, such as 6, are "perfect" because each is the sum of its divisors (other than the number itself), as 6 is the sum of 1, 2, and 3. "Unfortunately" there are very few known perfect numbers; so the course soon proceeds to study prime numbers. A number is prime if it is divisible only by itself and 1. Thus 7 is a prime, whereas 6 is not. Any number of theorems treat properties of prime and nonprime (composite) numbers. To some mathematicians prime numbers are delightful, intriguing members of the number system. To the students they are hostile strangers. When they learn that there is an infinity of prime numbers they become convinced that the world is full of enemies.

The theory of numbers also includes what is called the theory of congruences, and this topic seems to be a must in liberal arts courses. The theory of congruences concerns an arithmetic that is suggested by how our clocks record time. Six hours after nine o'clock the clock reads three o'clock; that is, 9 + 6 = 3. In other words, twelve and any multiples of twelve are discarded or counted as zero. Certainly this is a peculiar arithmetic. The theory of congruences studies many such varieties. But even ordinary arithmetic is dull, however [114] practical the knowledge. What, then, can students find interesting in clock arithmetic? Moreover, since many are still insecure about the operations of ordinary arithmetic, the new arithmetic shatters what little confidence they had.

A favorite topic in liberal arts courses is axiomatics (see Chapter 3). Because the axiomatic approach to teaching is now the popular one in many courses, this topic warrants special attention here. Every branch of mathematics is founded on axioms, the prototype being Euclidean geometry. There is much to be gained from a study of the axiomatic basis of any branch. Indeed, the most momentous development of the nineteenth century - non-Euclidean geometry - resulted from a change in the axioms of Euclidean geometry. But what features of the study of axiomatics are presented in a liberal arts course? Some professors introduce a system of axioms and merely discuss what properties the system should possess. One of these, for example, is independence; that is, it should not be possible to prove any one of the axioms on the basis of the others, for in that case the provable axiom is more properly a theorem. The study of properties of an axiom system is of great interest to specialists in the foundations of mathematics, but to tyros the axioms are the least significant part of any mathematical development. They are the seeds from which the fruit eventually emerges. Hence, a treatment of the properties of the axioms themselves has little value.

Another type of "play" with axioms is to show students that the set required to develop a branch of mathematics is not unique. One can change the axioms and still deduce the same body of theorems. Or one can in some branches reduce the number of axioms, though this may involve the need for more complicated proofs of the theorems. The latter activity is relatively simpleminded and usually profitless. One is not surprised to find, then, that students are not exhilarated [115] when they are shown than ten axioms can be replaced by nine or nine and one-half.

One truly liberal arts value to be derived from the study of axiomatics is to help people become conscious of their actual assumptions when they make a decision or adhere to a belief in any sphere. But this "carry-over" of mathematics education to other areas of our culture is never mentioned. The teaching of axiomatics per se reinforces the contention that mathematics does not teach critical thinking. For if it did, professors would certainly ask themselves why they teach axiomatics in a liberal arts course.

Some liberal arts courses and certainly the more advanced courses do indeed derive the theorems that are implied by the axioms, just as the high school course in Eucidean geometry does. This deductive approach to a branch of mathematics is surely the elegant one. Unfortunately, it is almost always a distortion of the more natural thinking that led to the theorems and proofs. The necessity to establish a theorem on the basis of the axioms and previously established theorems obliges the mathematician to recast his original argument to force the theorem into the most suitable place in the logical sequence. This recast proof may be far from the original thoughts that convinced the creator his theorem was correct. Moreover, after at least one successful proof is obtained, its creator or his successors, now able to see how the essential difficulty was overcome, can usually devise a more ingenious or more direct proof. Most theorems have been reproven several times, each successive proof a remodeling of the previous one and often including generalizations or stronger results. Hence, the final theorem and proof are far from the original thoughts. Indeed, they are often shorn entirely of their intuitively grasped form. Some of the logic supplied to shore up the original intuition is entirely artificial and so trumped up and stilted as to [116] preclude understanding. Over one hundred years ago Augustus De Morgan, one of the founders of modern logic, warned,

"The student must not believe that theorems have been invented or perfected by the methods in which it is afterwards most convenient to deduce them. The march of the discoverer is generally anything but on the line on which it is afterward convenient to cut the road."

Because the deductive approach is not the understandable one, and especially because it is a distortion of the natural, intuitive approach, its value to the student is inversely proportional to its elegance. As far as the students can see, the axioms are handed down ex cathedra and then the logical crank is turned to grind out theorem after theorem. The students do not know where the axioms came from, why the particular ones were chosen, and where the seemingly interminable sequence of theorems is going. Even though a student may get to the point of verifying each step in a proof, he usually does not understand the rationale, the basic thought or method behind the proof. Why is this particular chain of steps used rather than some other perhaps more easily understood sequence? The student has a case. Facing such a reworked, more sophisticated, and possibly more complicated result, he cannot grasp it at all. Henri Poincaré, the leading mathematician of the late nineteenth and early twentieth centuries, makes this point:

"Can one ever understand a theory if one builds it up right from the start in the definitive form that rigorous logic imposes, without some indications of the attempts which led to it? No. One does not really understand it; one cannot even retain it or one retains it only by learning it by heart."

Following a proof step by step has been compared to the way a novice at chess observes two masters play. The novice recognizes that each player makes a move that conforms to the rules, but he will not understand why a player makes a [117] particular move rather than any one of a dozen others available to him. Nor will he perceive the overall strategy that suggests a series of moves. Similarly, watching a mathematical demonstration being made step by step, each of which is justified by some axiom or previously established theorem, does not in itself convey the plan of the demonstration, nor the wisdom of that entire method of proof as opposed to some other method. Because he may be called upon to reproduce the proof, the student is reduced to memorizing the prescribed series of steps.

Many professors, having delivered a series of theorems and deductive proofs, walk out of their classrooms very much satisfied with themselves. But the students are not satisfied. They were not involved in the real thinking and derived no stimulation from a presentation they did not understand. The learning of such proofs calls for much activity but little cerebration or celebration. The logical order of mathematical presentations is about as helpful pedagogically as the alphabetic order of words in a literary work. Colleges and universities state in their catalogues that their first objective is to encourage students to think for themselves; yet professors presentations promote not the emancipation but the enslavement of minds.

Mathematicians have a naive idea of pedagogy. They believe that if they state a series of concepts, theorems, and proofs correctly and clearly, and with plenty of symbols, they must necessarily be understood. This is like an American speaking English loudly to a Russian who does not know English, in the belief that his increased volume will ensure understanding.

The deductive presentation of mathematics is psychologically damaging because it leads students to believe that mathematics is created by geniuses who start with axioms and reason directly and flawlessly to the theorems. Given [118] this impression of elevated, far-ranging minds, the student feels humbled and even depressed about his own capacities, especially when the obliging professor presents the material as though he too is genius in action.

The logical or deductive approach does not convey understanding. As Galileo put it,

"Logic, it appears to me, teaches us to test the conclusiveness of an argument already discovered and completed, but I do not believe that it teaches us to discover correct arguments and demonstrations."

The logical formulation does dress up an intuitive understanding, but it conceals the flesh and blood. It is like the clothes that make the woman, or make the man want to make the woman, but are not the woman. Logic may be a standard and an obligation of mathematics, but it is not the essence. Nevertheless, the deductive approach to mathematics is almost universally adopted by professors.

The major reason for its popularity is that is is easier to teach. The entire body of material is laid out in a complete, ready-made sequence and all the teacher has to do is to repeat it. On the other hand, to know the intuitive meaning of a concept or proof, to penetrate to the basic idea of a proof, and to know why one proof is preferable to another call for depth of understanding. Even if the professor does acquire this understanding, it is far more difficult to impart it to students. Many teachers complain that students, particularly engineers, wish to be told only how to perform the processes they are asked to learn and then want to hand back the processes. But the teachers who offer only the logical presentation because it avoids the real techniques of teaching - leading students to participate in a constructive process, explaining the reasons for proceeding one way rather than another, and finding convincing arguments - are more reprehensible.

Since the logical approach to mathematics does not [119] convey understanding and even distorts the original thinking, should it be presented at all? The answer is affirmative, and the reasons have already been at least implied. Proof is a check on our intuition. It also refines and sharpens the intuition, much as argumentation with an adversary on, say, a political issue often reveals defects in our thinking.

Whether or not some liberal arts courses do much with axiomatics per se, all do offer as a prime example of "superb" mathematics the logical development of the real number system. Real numbers include the positive and negative whole numbers, fractions, and the irrational numbers such as and the like. Some history here is relevant. These types of numbers were introduced into mathematics on an intuitive and pragmatic basis. Thus, mathematicians learned to add 1/2 and 1/4 to obtain 3/4 because one-half of a pie and one-quarter of a pie amount to three-quarters of a pie. Mathematicians worked successfully with these various numbers for over five thousand years without much, if any, concern about precise definitions or the logical development of their properties. For purely professional reasons mathematicians decided in the late nineteenth century that a logical structure based on a clear, axiomatic foundation should be provided. Of course, the logical structure had to sanction what had already been established on empirical grounds. It proved to be highly artificial, contrived, and complicated. From a logical standpoint, irrational numbers in particular are intellectual monsters, and most who study this apparent aberration wryly appreciate the mathematical term "irrational." Pascal's maxim, "Reason is the slow and tortuous process by which those who do not understand the truth arrive at it," is most appropriately applied to the logical development of the real number system. [120]

Many teachers might retort that the college student has already learned the intuitive facts about the number system and is ready for the appreciation of the deductive version, which exemplifies mathematics. If the student really understands the number system intuitively the logical development will not only not enhance his understanding, it will destroy it. As an example of mathematical structure no poorer choice could be made, because the construction is highly contrived. The development not only stultifies the mind but obscures the real ideas. Yet this topic has become the chief one in college mathematics courses. One may well conjecture that some teachers enjoy presenting the intuitively familiar facts about the number system in the recondite axiomatic approach because they understand the simple, underlying mathematics and yet can appear to be presenting profound material.

To defend teaching the logical development of the real number system many professors extol it as an example of how mathematics builds models for the solution of real problems. This is not the place to discuss applied mathematics (see the next chapter) but it is very clear that the professors who make such a statement haven't the least idea of how mathematics is applied. The example is absurd on many accounts. Let us note two. The real number system had been in use since about 3000 B.C., roughly over five thousand years before the logical "model" for it was constructed. Fortunately, no one waited for the availability of this model to apply real numbers. Nor would anyone use the model today, because the artificial, logically complex construction is as far removed from reality as heaven from earth. No one would ever think of using it even to predict anything about real numbers, let alone for a physical application. The reason for constructing the logical foundation of the real number system had nothing to do with real [121] problems. A few professors may be aware of these facts but perhaps introduce the word "model" in this context because it has another, more pleasing association. But the logical structure in question lacks flesh and blood.

No liberal arts course is considered complete without symbolic logic. This topic, which presents the ordinary principles of reasoning in symbolic form, supposedly teaches reasoning; actually, it is farcical for this purpose. To know what symbolism to use and how to manipulate it, one must already know the common meanings of "and," "or," "not," and "implies." But the students do not have these clearly in mind, and symbolic logic only conceals them under meaningless symbols. How ridiculous to teach symbolic logic to students who still confuse "All A is B" with "All B is A." This topic is of significance only to specialists in the foundations of mathematics. Boolean algebra, which is closely related to symbolic logic, is frequently included in liberal arts courses because it can be applied to the design of switching circuits, and presumably the liberal arts students in question are going to be electronics engineers. To the students the mention of this application may well suggest switching courses.

The liberal arts courses purport to teach the power of mathematics, and they do this by teaching abstract structures such as groups, rings, and fields. A group, for example, is any collection of objects, such as the positive and negative integers, and an operation that performed on any two members produces a member of the collection. The operation, like the addition process applied to the positive and negative integers, must possess certain additional properties that are the abstract analogues of the familiar properties: 3+ (4+5) = (3+4) +5; there is a zero; and to each integer, 2 say, there is another, -2, whose sum is zero. (See also Chapter 3.) [122]

Abstraction is indeed a valuable feature of mathematics. It reveals properties common to many concrete structures just as knowledge of the structure of mammals teaches us much about hundreds of varieties of mammals. Moreover, one who knows the abstraction can often see at once that it applies to a totally new phenomenon. Abstractions do lay bare the logical structure of several kindred concrete systems, but they are an impoverishment of the concrete as surely as the bone structure of the human body fails to present the whole man. But, some mathematicians rejoin that after presenting the abstract structure they give concrete examples. However, the concrete cases must be thoroughly understood before one introduces the unifying abstraction. To introduce as examples concrete material not yet familiar to the student is of no help in making the abstract concept clearer. In every case learning proceeds from the concrete to the abstract and not vice versa. To see the forest by means of the trees is pedagogically the only sound approach.

Many teachers favor an abstract theme, such as group theory, because they believe it to be an efficient way of imparting much knowledge in one swoop. They are under the impression that if a student is taught group theory he will automatically learn the properties of the rational, real, and complex numbers, matrices, congruences, transformations, and other topics. But a student who learns only the abstract group theory could not on this basis add fractions.

Abstractions do relate and unify many seemingly unrelated developments. However, for young people who possess little background, nothing is unified and illuminated by the abstraction. The abstract structures in question are so remote from their mathematical experiences that the values such structures grant to mathematics are no more evident than the power of philosophy to run a spaceship. To the [123] student the abstractions are shadows that can be perceived only dimly and induce a feeling of mystification and even apprehension. Just as the human body struggles for breath in a rarefied atmosphere, so the mind strains to grasp abstractions. They may pervade the teaching but they evade the student.

What is the major problem facing this nation today? Is it inflation? Unemployment? The absorption of minorities? Women's rights? Retaining the respect of other nations? If one were to judge by the contents of the liberal arts courses, it is the Koenigsberg bridge problem. As we have related (Chapter 1), some two hundred years ago the citizens of the village of Koenigsberg in East Prussia amused themselves by trying to cross seven nearby bridges in succession without recrossirig any one. The problem attracted Leonhard Euler, certainly the greatest eighteenth-century mathematician, and he soon showed that the attempt was impossible. But mathematicians will not let the dead rest in peace, and they revive the problem as though it were the most momentous one facing our civilization.

No worse a collection of dull, remote, useless, or sophisticated topics could have been chosen for a liberal arts course. Many of these topics come from the foundations of mathematics, where only specialized and professional needs justified their creation. With a few exceptions they are late nineteenth- and twentieth-century products that came long after most of the greatest mathematics was created. The best mathematicians of the past - Archimedes, Descartes, Newton, Leibniz, Euler, and Gauss - used almost none of them, for the simple reason that they didn't exist. And even the great mathematicians of the present use most of them only in specialized foundational studies. A liberal arts course that includes such topics must devote a great deal of time to convincing students that they should learn what the entire [124] mathematical world did not miss for thousands of years, and what very few mathematicians need even today. The topics have about as much value as learning to dig for clams has for people who live in a desert. (See also Chapter 10.)

A common alternative to the melange of topics such as set theory, axiomatics, symbolic logic, and a rigorous treatment of the number system is a presentation of technical mathematics that starts about where the high school courses leave off and covers more advanced techniques. This type of course continues an old tradition. The colleges used to require that all students take more algebra and trigonometry, a requirement reminiscent of the treatment doctors of the Middle Ages prescribed for all illnesses. They "cured" every illness by bleeding the patient. Just as the cure relieved the suffering of those who bled to death, the modern technique course drains students of any vestige of respect for mathematics.

Another alternative, very popular today, is known as finite mathematics. Just what is finite about it, except perhaps the students' attention to it, is not clear. It does not include any calculus, but it does use real numbers, complex numbers, and algebraic processes and theorems that involve infinity in several ways. The content, like that of the typical liberal arts course, is a conglomeration of topics having little relationship to each other and little significance for the students to whom it is addressed. This hodgepodge of topics, set theory, symbolic logic, probability, matrices, linear programming, and game theory (we shall not here undertake to explore the nature of these topics) is one of the fads that constantly sweep through mathematics education.

Some such course, if the topics are properly chosen, might be useful to social science students. Presumably then it would contain applications to the social sciences. On examination one finds a mathematical system that describes the marriage rules of a primitive Polynesian society. To [125] select the proper topics the organizers of the course would have to take time to find out what is really useful to the social scientists, but mathematics professors do not do this. Finite mathematics is not just a fad; it is a fraud. In any case it is not a liberal arts course.

The liberal arts courses described above give a low return on the investment of the hard work called for. Students learn brick-laying instead of architecture and color-mixing instead of painting. If these courses exhibit the liberal arts values of mathematics, then certainly student disregard and contempt for mathematics are justified.

Since all this material and its purported values fail to win over students, some professors concentrate on those who, for whatever reason, accept and pursue the material that is taught. With unconscious immodesty the professors label these students keen or bright. The "less intelligent" ones, quantitatively about 98 percent of the total, do want to know why they should learn seemingly useless material. But these professors do not recognize that they have an obligation to teach all the students.

No matter what the choice of topics, a fundamental objection to the usual liberal arts course in mathematics is that all the topics are devoted to mathematics proper. And mathematics proper is remote, unworldly, and even otherworldly. This thought was expressed by one of the greatest mathematicians of recent times, Hermann Weyl:

One may say that mathematics talks about things which are of no concern at all to man. Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold. But it seems an irony of creation that man’s mind knows how to handle things the better the farther removed they are from the center of his existence. Thus we are cleverest where knowledge matters least: in mathematics, especially in number theory.

Moritz Pasch, a leading mathematician of the late [126] nineteenth century, even contended that mathematical thought runs counter to human nature.

The unnaturalness of mathematics is attested to by history. Dozens of civilizations that have existed, some celebrated for their literature, religion, art, and music, did create practical rules of arithmetic and a mixture of correct and incorrect rules for areas and volumes of common figures; but only one—the ancient Greeks—envisioned and created mathematics as the science that establishes its conclusions by deductive reasoning. Even the Greeks regarded mathematics as only a means to an end: the understanding of the physical world. Only one other civilization, the modern European civilization, has surpassed the Greeks in depth and volume of new results, and western Europe learned mathematics at the feet of the Greeks.

The subject matter of mathematics proper cannot be very attractive to most people. It deals with abstractions and this is one of the severest limitations. A discourse on the nature of man can hardly be as rich, satisfying, and life-fulfilling as living with actual people, even though one may learn a great deal about people from the discourse. Beyond the fact that the subject matter is abstract, it is in itself hardly relevant to life. All of mathematics centers on number, geometrical figures, and generalizations thereof. But number and geometric description are insignificant properties of real objects. The rectangle may indeed be the shape of a piece of land or the frame of a painting, but who would accept the rectangle for the land or the painting?

Moreover, the likelihood of interesting students in the material we have described is poor, especially in view of their prior education. The student who is about to begin a college course in mathematics has met the subject to some extent in his earlier education. Unfortunately, the average student leaves these earlier experiences with limited ideas as [127] to what mathematics is and what it has accomplished in our civilization. As far as he knows the subject matter is a series of techniques for solving problems; certainly the techniques are isolated from any easily conceivable use. The failure to see values in mathematics has generally caused the student to do poorly in it, deprecate it as worthless, and shrink from further involvement.

Many professors would argue that the content of the liberal arts course is almost irrelevant. The course teaches students precise reasoning; this usually means deductive reasoning. A course in mathematics proper may teach sharper reasoning, but the students have already had three or more years of mathematics in high school, and it would seem that whatever mental training mathematics can supply would have been supplied already.

Actually, the vaunted value of deductive reasoning is grossly exaggerated. In daily life, business, and most professions deductive reasoning is practically useless. There are no solidly grounded axioms from which one can deduce what career to pursue, whom to marry, or even whether to go to the movies. On the other hand, the distinctions that must be made in analyzing character, personality, values, and good and bad behavior are far more subtle and call for a more highly perceptive and critical faculty than anything mathematics will ever develop. Deductive reasoning is not the paradigm for the life of reason. In fact, whatever faculties equip a man to understand and judge wisely about human problems are not more widely found among mathematicians nor does the study of mathematics contribute to the acquisition of such faculties. Newton was certainly not the critical thinker when he wrote about the prophecies of Daniel. In the contention that mathematics serves to train the mind, even the students smell an aroma of professional humbug.

Moreover, deductive reasoning can be learned more [128] readily in many other contexts. The student who is asked to recognize that because the base angles of an isosceles triangle are equal it does not follow that equality of the base angles makes the triangle isosceles, must first learn the meaning of the terms involved. He can learn the same point from the example that good cars are expensive does not imply that all expensive cars are good.

Actually, most mathematics courses do not teach reasoning of any kind. Students are so baffled by the material that they are obliged to memorize in order to pass examinations. Perhaps the best evidence for this assertion is supplied by the professors themselves. When asked why they do not allow students to use books in examinations, they usually reply, "What, then, can I ask in the examination?"

When the defense of mathematics as training in reasoning is deflated, the professors fall back on the aesthetic satisfactions mathematics offers. Though portions of mathematics are beautiful and could be presented in a liberal arts course, there are two limitations. The first is that mathematics lacks the emotional appeal of painting, sculpture, and music. The second is that students who are repelled by the mathematics they are compelled to learn in elementary and high school will not feel moved to pursue the subject to reach the few beautiful themes. The attempt to sell the beauty of mathematics to liberal arts students is doomed to fail.

Many professors proclaim that the goal of a liberal arts course should be to teach what mathematics is and what mathematicians do. No more effective means of driving students away from mathematics has ever been devised. What is mathematics? It is a collection of abstractions remote from life. What do mathematicians do? They strive for personal success, and to arrive at it some even cheat in all sorts of ways, including neglecting the interests of the very students they say they want to attract. But mathematicians create. Do these courses then teach the fumbling, the guessing, the blundering, the mental struggles, the testing of hypotheses, the frustrations, the false proofs, the insights, and other acts of the creative process? No. They teach precise definition, theorem, and proof as though God inspired the elect to proceed directly to the finished product.

Intellectual challenge and thrill of accomplishment are other values cited for mathematics. Mathematicians respond to intellectual challenge much as businessmen do to the excitement of making money. They enjoy the fascination of the quest, the sense of adventure, the thrill of discovery, the satisfaction of mastering difficulties, the pride and glory of achievement -or, if one wishes, the exaltation of the ego and the intoxication of success. Such values are present in mathematics more than in any other subject because it offers sharp, clear problems. But to obtain these values one must be interested in the subject and have already acquired some facility in it. An occasional, exceptional student persevering on his own or fortunate enough to have had one or two fine teachers may come to enjoy these values of mathematics. But such students are exceptions and are almost certainly not to be found among the students taking the liberal arts course. Moreover, people’s intellects make their own claims about what they find challenging. Many find more significant challenge in law and economics.

What should a college course in mathematics for liberal arts students offer? The answer is contained in the question. The liberal arts values of mathematics are to be found primarily in what mathematics contributes to other branches of our culture. Mathematics is the key to our understanding of the physical world; it has given man the conviction that he can continue to fathom the secrets of nature; and it has given him power over nature. We now understand, for example, [130] the motions of the planets and of electrons in atoms, the structure of matter, and the behavior of electricity, light, radio waves, and sound. And we can use this knowledge in man's behalf. Some uses of this knowledge are familiar to all of us: the telephone, the phonograph, radio and television are achievements of mathematics. Mathematics, especially through statistics and probability, is becoming increasingly valuable in the social sciences and in biological and medical research. The search for truth in philosophy or the social sciences cannot be discussed without involving the role that mathematics has played in that quest. Painting and music have been influenced by mathematics. Much of our literature is permeated with themes treating the implications of mathematical achievements in science and technology. Indeed, it is impossible to understand some writers and poets unless one is familiar with mathematical influences to which they are reacting. Religious doctrines and beliefs have been dramatically altered in the light of what mathematics has revealed about our universe. In fact, the entire intellectual atmosphere, the Zeitgeist, has been determined by mathematical achievements. These are the liberal arts values of mathematics and should constitute the essence of a liberal arts course.

Though this is not the place even to sketch the contents of such a course, a few elaborations may help to make clear what it can offer. The average person thinks of science rather than mathematics as providing the explanation of natural phenomena. Yet mathematics is the essence of science. Let us consider an example.

The force of gravity is involved in all phenomena of motion. The action of gravity presumably explains why the planets and their moons keep to their periodic paths and why space ships can be sent to the moon. In all motions on earth - as we walk along a level road or up or down a hill, ride in an [131] automobile or airplane, rise from a sitting position or sit down; in the whirring of machinery; and even in the flow of blood in our bodies - the action of gravity is involved. Presumably an understanding of the force of gravity would clarify all these motions. One might argue that the mathematical law that describes quantitatively the action of gravity is useful, but that gravity is a physical phenomenon. However, to emphasize the physical force of gravity and to regard the mathematical law as an aid in analyzing and predicting the physical action is to miss the main point.

How does the sun's gravitational attraction keep the planets in their appointed paths? Is there a steel cable stretching from the sun to the earth that keeps the earth from flying off into space and confines it to its elliptical path? We have no idea of how gravity acts physically. In fact, there is no force of gravity. As Russell Baker once remarked, "You cant buy it any place and store it away for a gravityless day." It is a fiction introduced to supply some intuitive understanding of the various motions we perceive and undertake. How, then, has science been able to treat gravity, to make such precise predictions of eclipses of the sun and moon, and even to send men to the moon? The answer is that the mathematical law of gravitation is all that we know about this force. By means of the mathematical law and deductions from it, we can describe and predict the behavior of thousands of objects. In fact, one of Newton' s great accomplishments was to show that this very law applies to both terrestrial and celestial motions.

Mathematics, then, is not only a key to our understanding of motion, it is the only knowledge we have. The same can be said of light, radio waves, television waves, X rays, and in fact all of the waves of what is called the electromagnetic spectrum. The student who enjoys radio reception of music, from Beethoven to the Beatles, should bless mathematics. [132] Physical science has reached the curious state in which the firm essence of its best theories is entirely mathematical, whereas the physical content is vague, incomplete, and in some cases, self-contradictory. This science has become a collection of mathematical theories adorned or cluttered with a few physical observables. To use Alexander Pope's words, the mighty maze is not without a plan, and the plan is mathematical.

In fact, it is not hard to maintain that our knowledge of the entire physical world must reduce to mathematics. As Sir James Jeans has put it, "All the pictures which science draws of Nature, and which alone seem capable of according with observational facts, are mathematical pictures." More than that, the pictures are made by man. There is no known physical, objective universe. We, not God, are the lawgivers of the universe.

These examples of the liberal arts values of mathematics have admittedly been drawn from the physical rather than the social sciences. The theory and predictions that mathematics supplies in the former field do describe what takes place. In the latter we have models of what could happen but doesn't; however, the social sciences are young.

Liberal arts values are so numerous and so monumental that only another sample or two can be presented here.*

* For a fuller exposition see the author’s Mathematics in Western Culture.

In the sixteenth and seventeenth centuries, thanks primarily to the work of Copernicus and Kepler, astronomical theory was converted from geocentric to heliocentric, primarily because the mathematics of the latter was simpler. Under the older view the earth was the center of the universe, and since man was obviously the most important creature on earth, man's life, goals, and activities were the most important concerns. If there is a God and prior to the [133]seventeenth century no one professed to doubt it - He certainly would be concerned about humans and had evidently designed the world to favor and further man's interests. But the heliocentric theory shattered all such beliefs. The earth became just one of many planets, all revolving about the sun, and man just one of many insignificant creatures on earth. How, then, at least on the basis of astronomical theory, could one believe in a God concerned with a mite, a speck of dust in a vast universe? As Matthew Arnold put it:

The Sea of Faith
Was once, too, at the full, and round earth's shore
Lay like the folds of a bright girdle furled,
But now I only hear
Its melancholy, long withdrawing roar...
For the world, which seems
To lie before us like a land of dreams,
So various, so beautiful, so new,
Hath really neither joy, nor love, nor light,
Nor certitude, nor peace, nor help from pain...

An even more momentous theme, which certainly belongs in any liberal arts course, is the pursuit of truth. From prehistoric times onward man has sought truths, whether through religion, philosophy, science, or mathematics. Beginning with Greek times, the one universally accepted body of truths was mathematics. The significance of this fact extended far beyond mathematics. The acquisition of some truths gave man the evidence that he could acquire them; and it gave him the courage and confidence to seek them in political science, economics, ethics, and the arts. But the creation of non-Euclidean geometry shattered centuries of confidence in man's intellectual potential. Mathematics was revealed to be not a body of truths but a man-made, approximate account of natural phenomena, subject to [134] change and having only pragmatic sanction.

Though mathematics is a product of cultures, it in turn fashions cultures, notably our own. Just as the meaning of good literature lies beyond the collection of words on paper, so the true significance of mathematics consists in what it accomplishes for our society, civilization, and culture. In particular, mathematics is man's strongest bridge between himself and the external world. It is the garment in which we clothe the unknown so that we may recognize some of its aspects, and it is the means by which, to use Descartes’ words, we have become the possessors and masters of nature. Mathematics proper may be a monument to human inventiveness and ingenuity, but it is not in itself an insight into reality. Only insofar as it aids in understanding reality is it important. And this is what we must teach.

Thus the prime goal of a true liberal arts course should not be mastery of purely mathematical concepts or techniques, but an appreciation of the role of mathematics in influencing and even determining Western culture. Appreciation, as well as skill, has long been recognized as an objective in literature, art, and music. It is equally justifiable as an objective in mathematics. The cultural aspects of mathematics, rather than the narrow viewpoint of the specialist, should be stressed in the liberal arts course to achieve an intimate communion with the main currents of thought in other fields.

Is it surprising that mathematics and the other major branches of our culture are inextricably involved with each other? Knowledge is a whole and mathematics is part of that whole. However, the whole is not the sum of its parts. The present procedure in the liberal arts course is to teach mathematics as a subject unto itself and somehow expect the student who takes only one college course in the subject to see its significance for the general realm of knowledge. This is like giving him an incomplete set of pieces of a jig-saw [135] puzzle and expecting him to put the puzzle together. Liberal arts mathematics must be taught in the context of human knowledge and culture.

Professors must learn that mathematics proper is not the most important subject for the nonprofessional. Even some of the best professional mathematicians did not grant the subject supreme importance. Newton regarded religion as more vital and said that he could justify much of the drudgery in his scientific work only on the ground that it served to reveal God’s handiwork. But of course Newton was just a lowly physicist. Gauss ranked ethics and religion above mathematics, but Gauss, too, was as much a physicist and astronomer as a mathematician. Weyl's words - it is an irony of creation that man is most successful where knowledge matters least, in mathematics - bear repetition.

Why do mathematics professors teach pointless material to liberal arts students and ignore the truly cultural values of mathematics? The sad fact is that most professors are themselves ignorant of these values. Some may be powerful engines of mathematical creativity but limited to their tracks. They work in their own mental grooves and naively assume that what they value is eminently suitable. If the thought of including, say, applications to science should occur, they would banish it because they know they might be embarrassed by the students' questions. The elitist, narcissistic mathematician who presents his own values, curiosities, and trick problems is totally unfit to be a teacher in any course.

The charge that most mathematics professors are culturally narrow may seem incredible. But one must remember that mathematics is in large part a technical subject in which one can be highly proficient as a cabinetmaker is proficient among carpenters; and one would not necessarily be surprised to learn that a cabinet-maker [136] is neither cultured nor a pedagogue. Competent researchers need not and generally do not know the broader values of mathematics. Certainly most do not care about the art of pedagogy. These professors present to all students just those values that they as professional mathematicians see in their subject - and they do not question their own values. They wish to have students appreciate the abstractions, the rigorous reasoning, the logical structure, the crystalline purity of mathematical concepts, and the presumed beauty of proofs and result. Non-Euclidean geometry, the most dramatic and shocking event in recent intellectual history, is to them just another topic.

The damage done by such professors extends beyond what they inflict on students. Because they are so involved in their own research, they ignore and even disparage anything outside their own specialty. This attitude discourages young professors, many of whom, less indoctrinated, do recognize the need for a truly liberal arts course but refrain from any action for fear that the older men will regard them as trafficking in trivialities. They are made to feel that any talk about music or philosophy is a default on their real obligation to teach mathematics, and in fact to train future mathematicians. Hence, departure from the norm is discouraged.

It is tragic that professors teaching in a liberal arts college, which is purportedly devoted to educating the whole person and to instilling interests and attitudes, are not themselves interested in learning material closely related to their own subject.

C.P. Snow, in his famous lecture, The Two Cultures and the Scientific Revolution, deplored the gulf separating the scientists and humanists. The former, self-impoverished, disdain the humanistic culture and even take pride in their ignorance of it. The latter respond by wishful thinking to the [137] effect that science is not part of culture but rather mechanization of the real world. They wish neither to understand nor to sympathize with the nature and goals of the scientific enterprise, and they pity the scientist who does not recognize a major work of literature. Snow's more severe criticism is directed toward the humanists who pretend that their cultural interests are the whole of culture and that the exploration of nature is of no consequence. Actually, it would have been more just to rail at the scientists, particularly the mathematicians. One cannot expect them to make exposition of their subject their mission in life; but where they have the chance, even the obligation, to offer a liberal arts course to about two million freshmen each year and thereby make the meaning and significance of their subject clear to themselves and the students, they default.

The students in a liberal arts course, many of whom are the most intelligent of our youth, are our best bets for producing broadly educated men and women who would be unhampered by petty parochialism and fully alive to the interrelationships of not just two cultures but of all knowledge. Some might even become members of a now rare breed of politicians who have some idea of what scientists are doing. Surely the mathematics course should aim at such goals.

Mathematicians are digging their own graves. Student protests for relevant education, though not always wisely formulated, are fully justified in the case of mathematics. A course in college mathematics is fast disappearing as a general requirement, and mathematicians will lose not only jobs but also any significant role in the liberal arts college. The mere fact that mathematics has been taught for centuries certainly is no assurance that it will be retained. Both Latin and Greek are the languages of the cultures that have contributed most to the fashioning of Western culture, [138] and Latin was the international language of educated people until about one hundred years ago. But both languages have practically disappeared from modern education. Indeed, even the study of classical Greek and Roman culture has practically disappeared.The danger that mathematics, too, may be dropped from the liberal arts curriculum has already occurred to many mathematicians, and some are growing concerned about the public image of their subject. But apparently they are not sufficiently aroused to utilize the best medium already at their disposal to improve that image.

Socrates was condemned to death for corrupting the morals of the youth of Athens. What punishment should be meted out to professors who degrade mathematics, cause the students to hate the subject or intensify the already existing hate, and in many cases poison the students' minds against all learning? And should the universities be exculpated for entrusting the education of liberal arts students to specialists or graduate students who are the living refutation of liberally educated people?

I am very grateful for the kind permission of Professor Kline's widow, Mrs Helen Kline for this book to be reproduced.