Morris Kline was a slender man, soft-spoken, polite, cultured. For most of his lifetime he was a mathematician, in pursuit of what Alfred North Whitehead called "a divine madness of the human spirit." Yet Kline did not display the madness so often paraded by his fellow mathematicians. He was a champion of common sense, but, as Lord Kelvin put it, "Mathematics is merely the etherealization of common sense." That connection eluded many of Kline's colleagues.
Kline had pursued and taught mathematics since 1930, but he was an outspoken critic of both activities. "Most of the mathematical research done today is a waste of time," he said, sorrowfully about his profession. His indictments of mathematics education stretched from elementary school to graduate school.
Kline's specialty was the mathematics of electricity and magnetism, a high-voltage field that led him into secret research for the US Army during World War II, and then to head a research division in electromagnetics at New York University's Courant Institute of Mathematical Sciences. As a professor emeritus at the Courant Institute, Kline was a Guggenheim fellow and Fulbright lecturer in Germany, and taught at Stanford University and the Technische Hochschule, in Aachen, West Germany. He was, for 11 years, chairman of undergraduate mathematics at New York University and later a Visiting Distinguished Professor at Brooklyn College of the City University of New York.
Despite his stellar career as an applied mathematician and educator, Kline's major contribution to society may well be his writing. He was a graceful, lucid writer with a poetic style and, in only a half-dozen or so books, provided us with what may be the clearest, most accessible window on the nether regions of higher mathematics since Plato.
Perhaps his most widely known book, Why Johnny Can't Add: The Failure of the New Math appeared in 1973, after the so-called new math had been introduced (Kline would say "railroaded") into the nation's elementary schools. For those of you who don't remember, the important thing in the new math, as comedian-mathematician Tom Lehrer put it, "is to understand what you are doing, and not to get the right answer." Kline's outcry was heard, and now new math is only a relic. "There are bits and pieces of it here and there," says Kline, "but the big thing now is 'back to basics.' " (Kline had troubles with the back-to-basics movement, too.) He followed Why Johnny Can't Add with an indictment of university education in the United States, entitled Why the Professor Can't Teach. This book was not so well received, especially in the publish-or-perish groves of academe. Kline believed that a good scholar and a good teacher are not necessarily one and the same thing.
The success of Kline's books, such as Why Johnny Can't Add and Mathematics: The Loss of Certainty, attests to the public's perception of the importance of mathematics. "There are tragedies caused by war, famine, and pestilence," Kline wrote, "but there are also intellectual tragedies.” As he describes, his books “relate the calamities that have befallen man's most effective and unparalleled accomplishment, his profound effort to utilize human reason—mathematics.”
His books generated discussions in such diverse publications as US News & World Report, the New York Review of Books, and New Scientist. Mathematics: The Loss of Certainty, namely, was an intellectual history tracing a complicated line of mathematical thought from the ancient Greeks to the day in 1931 when a paper by Kurt Gödel, the foremost logician of the century, pulled the ethereal mathematicians down from their ivory towers. Gödel's paper not only humbled the mathematicians, but also proved that the ivory towers themselves were foundationless—fictions. Kline's book, though the story it tells ended nearly 100 years before its writing, touched an exposed nerve in the 1980s. It was one more proof of the atomization of our culture, another proof that, as W. B. Yeats put it, "Things fall apart; the center cannot hold." The following interview was conducted by former Omni executive editor Frank Kendig in Professor Kline's office at NYU in 1981.
OMNI: Your new book has been extremely well received, though it is about a complex and difficult subject. Do you think the subtitle, The Loss of Certainty, has anything to do with the book's success?
Morris Kline: Yes, I must credit Oxford University Press for having fixed on that title after I suggested other possibilities. One reason for the interest is that we are living in a new cultural world. We have passed through the Age of Reason, in which people were sure, mathematicians especially, that we could discover the truth about science, about nature, about political and economic systems, all through mathematics. That is no longer the case, and so "the loss of certainty" applies.
Where should we begin trying to understand this loss of certainty?
Well, mathematics begins with the Greeks, although one can find a few predecessors. The early literature is unfortunately scanty. The Greeks believed the universe was mathematically designed. Plato, for example, would say that even if there were no human beings, mathematics would exist. Aristotle differed. He believed that human beings were necessary to abstract the mathematics from physical reality. But both men did say that mathematics itself was truth.
Where does God fit into the picture?
Oh, the Greeks had gods—which the Romans took over, only changing the names—but the gods didn't play a role in mathematics and its relationship to nature. God enters the mathematical picture probably in medieval Europe and carries into the Renaissance and then into the 17th and 18th centuries. The idea was that God designed the universe—a mathematical universe. This belief is very strong in Galileo, in Newton, in Descartes. Not that these men agreed entirely on physical principles, but they agreed that God designed the universe mathematically and that we can discover the mathematical laws if we search hard enough.
If the universe is mathematical, and you know the mathematics, then can't you extrapolate in either direction—know the past and predict the future?
Oh, yes. There's a famous statement to that effect. I couldn't quote it verbatim, but basically it says that if you knew all the mathematical laws and the initial conditions—for example, the initial velocities of the objects—you could predict the future.
Doesn't that leave God with very little to do?
Well, there are different views on that. Some philosophers have said that what God has done is done forever. Others believe that God is free to intervene at any time and change the structure of the universe for any reason.
That would be Newton's watchmaker.
That's correct. Newton said that God had to intervene to keep the world functioning according to plan. I think it was Leibniz who said that God can change the design of the world at any time. But the Greek and the Christian mathematicians believed basically the same thing. The Greeks said the universe is mathematical; the Christians said that God made it mathematical.
So when did the loss of certainty begin? Where did we take a wrong turn?
It began around 1800, and it began with geometry. I usually like to quote Mark Twain about this. He said that man is the only animal that has the one true religion—several of them. And that is just what happened with geometry.
The geometry that came from the Greeks is usually called Euclidean geometry, after Euclid. But suddenly at the beginning of the 19th century other geometries were developed—non-Euclidean geometries. Who gets the credit for this is sometimes disputed among historians, but I would say Carl Friedrich Gauss. He was the man who said flatly that we can no longer be sure that Euclidean geometry describes the physical world correctly. The various geometries conflict, although one of them, according to thousands of years of tradition, should describe the truth. You can see the problem.
Can you give me an example of an alternative geometry?
Well, one can cite as an example the theorem of Euclidean geometry that the sum of the angles of a triangle is one hundred eighty degrees. In one of the non-Euclidean geometries, called hyperbolic geometry, the sum is less than one hundred eighty degrees; in another, called double-elliptic non-Euclidean geometry, the sum is always larger than one hundred eighty degrees. Yet all of these geometries are equally accurate insofar as man can measure the sums of angles of triangles.
Are you saying these other geometries work just as well in measuring out a plot of land or constructing a triangle in the living room?
Yes. Gauss would have considered these as small triangles, and, according to the several geometries, the sum of the angles of all triangles approaches one hundred eighty degrees as the triangles get smaller. The departure from one hundred eighty degrees would be there, but it would be too small to measure. Gauss predicted that if we worked with a very large triangle, say the triangle formed by the earth, the sun, and Jupiter, the difference would be quite noticeable. He didn't have the data—nobody did in the 19th century. But he did say we have to allow for the possibility.
What did the mathematicians do when the bottom dropped out of geometry so to speak?
Many mathematicians tried to rescue and maintain as truths the portion of mathematics built on arithmetic, which by 1850 was far more extensive and vital for science than the several geometries. Unfortunately, other shattering events were to follow. Arithmetic and algebra were the next to go by the board.
The best example of this I could give in a semi-popular book was the creation of what are called quaternions, in 1843, by the great mathematical physicist William Rowan Hamilton. Now in the algebra of quaternions, a kind of number known as a hyper-number, multiplication is not commutative. In other words, if I were talking quaternions, I could not say that three times four is the same as four times three. Other strange algebras were created, and it made people start to worry about the laws of ordinary arithmetic. (The one I just stated is known as the commutative law of multiplication). And if we can have perfectly good algebras in which the old familiar laws don't work, then how do we know they work in the case of the real numbers? That's where a mathematician named Hermann von Helmholtz stepped in and told us we don't know it at all. They work in some situations, but not in all.
Are there any elementary examples of these sorts of algebras, where 2 + 2 = 6, or where 5 x 7 = 35, but 7 x 5 is only 34?
I can think of several. Take a quart of water at 40 degrees and mix it with another quart of water at 50 degrees. Do you get two quarts at 90 degrees? You do not. It's more like 45 degrees. So you can't just say I'm going to add 40 and 50 and automatically get 90. It depends on the physical situation.
Consider music, a simple musical tone with a unique frequency and amplitude, say one hundred cycles per second. Now suppose on top of that you impose another note at two hundred cycles per second. Do you get a note at three hundred cycles? Again you do not. It is a note of two hundred cycles, the first harmonic above the one-hundred-cycle note. It is the highest harmonic that determines the pitch—two hundred cycles. This is an important factor in the design of musical instruments.
So algebra and arithmetic went the way of geometry. Did the mathematicians regroup?
Yes. In the 19th century mathematicians finally realized that mathematics, whatever it may say in and for itself, is not necessarily the truth about the physical world. But they still believed that mathematics was a correct, sound, logical development in itself. What followed has been called the axiomatization of mathematics. Errors were discovered in past proofs, and those errors were rectified. By 1900, mathematicians believed they could say that they had a wonderful, perfectly logical development. What it has to say about the real world, well, that is up in the air.
But then the mathematicians—you can date it from about 1900—discovered contradictions within mathematics itself. In other words, what they thought was a perfect, logical structure led to contradictions within any one branch of mathematics. Now that is intolerable. If mathematics is not a perfect body of reasoning, if there are contradictions within any particular branch, then you can prove almost anything. If you don't want to use one side of the contradiction, you can use the other. Bertrand Russell was instrumental in pointing out these contradictions.
So, did the mathematicians regroup again?
Yes, they did, this time into four distinct schools. Each sought to rebuild the foundations of mathematics so that these contradictions would be removed.
Is that the problem of consistency? Can you explain the difference between consistency and completeness? I believe those are the important terms.
Consistent means that there are no contradictions within any particular branch of mathematics and that none can ever arise. That brings us to Gödel's 1931 paper. If Gödel's proof is correct, and it seems to be, then we can never establish the consistency of any significant branch of mathematics. We can never prove that there will not be contradictions.
Now completeness is not a matter of contradictions. If a branch of mathematics is complete, you can prove or disprove any meaningful statement belonging to that branch of mathematics. The axioms of that branch contain enough information to deduce any significant assertion from those axioms. Gödel proved that there will always be meaningful statements within a branch of mathematics that cannot be either proved or disproved. He called them undecidable statements.
So this was the prime reason for the loss of certainty. Gödel's proof was the final debacle.
If mathematics has no underlying truth—if it is filled with contradictions and uncertainties, why does it work?
There is no definitive answer to that. It just works. The only test we have that mathematics is reliable—not certain, but reliable—is that one can apply its laws to physical problems and make predictions. If the predictions come through, then we can say that mathematics has some substantial basis, but not certainty. I think people can't help being impressed by what mathematics achieves. Consider the problem of sending a spaceship to the moon and bringing it back. It is entirely mathematical. Of course, there is a tremendous amount of engineering involved in the production of the ship, but the entire plan for it is mathematical. We have a theory about the sun, the planets, and more distant heavenly bodies. We say that what makes them behave as they do is the force of gravity. But nobody knows whether there is such a thing as gravity. We have no physical understanding of it. The theory is mathematical— gravity is a scientific fiction.
The same could be said about electricity and magnetism, couldn't it?
That's exactly right. Everybody today knows what a radio is, and what a TV is, but nobody knows what a radio wave or a TV wave is. You can't smell one or hear one or taste one. But we do have a wonderful mathematical theory developed in the nineteenth century by the mathematical physicist James Clerk Maxwell. The evidence for this wonderful theory is the performance of our radio and TV sets. So we have to accept the fact that mathematics works, or else abandon our radios and our TV sets.
Are most mathematicians since the loss of certainty now working on these physical problems?
No, they aren't. Most of the mathematics created today—maybe ninety percent of it—is a waste of time. That is an opinion, but one that authorities who are far more creative and far better known share with me.
Can you give us an example of mathematics you consider a waste of time?
Some problems now being considered in the theory of numbers, for example, are a waste of time. Take pairs of primes, called double primes. These are prime numbers in a sequence, eleven and thirteen, for example. No even numbers, of course, are primes. Are there an infinite number of these pairs? Are there triple primes? Endless papers are written about these subjects. Who cares?
Speaking of papers, I made the suggestion in one of my books that every paper published in a respectable journal should have a preface by the author stating why he is publishing the article, and what value he sees in it. I have no hope that this practice will ever be adopted.
So you feel that the publish-or-perish system is responsible for this kind of mathematics?
In part it is. It would be hard today to find a really good, active mathematician who is not associated with a university. That was not always the case. Leibniz never had a university job. Descartes never had a university job. But universities today operate on a publish-or-perish system. Mathematicians, like all college professors, are under tremendous pressure to publish, and it is easier to publish in "pure" mathematics because you don't have to know science. You can limit your investigation to a very small area, and mathematics is divided into hundreds of specialties.
If you confront these people and say they are publishing without regard to value, they retort that mathematics is exciting, beautiful, and challenging. But I doubt whether even ten percent of those who devote themselves to pure mathematics are really concerned with beauty, intellectual challenge, or interesting ideas in themselves.
It makes mathematics sound a lot like playing chess or bridge. Exciting, beautiful, challenging; the same words apply to all three kinds of activity.
That's right. I'm glad you suggested it because it makes the point sharper. People enjoy playing chess. Some people even devote their lives to it. But no matter how ingenious a man is at playing chess or bridge, it isn't going to change this world one iota. Now mathematicians may probe deeper problems, but it is the same thing.
Are there still physical problems to be solved?
Oh, yes. I can mention one that has not yet been solved and is not likely to be solved in the near future—the three-body problem. In other words, if you take the earth, the sun, and the moon and try to predict their motions mathematically, you can't do it with precision. You have to write down a system of differential equations that would incorporate in mathematical form the motions of all three bodies, and such a system has not yet yielded to a solution. Some of the best mathematicians have worked on this problem for nearly three centuries.
Another problem is elasticity. Galileo worked on this problem some three hundred years ago, and it is now a branch of mathematics. We need mathematics to determine the strength of beams and columns—when they will snap, when they will collapse. It is amazing to me that engineers have the courage to put up an 80-story building with what little we know of elasticity. Again it is a problem involving differential equations.
I'd like to ask you about mathematics education. Is publish-or-perish the main problem in the universities?
It's certainly one of them. Teachers, or people who should be teachers, are pressed to publish, and this takes enormous time and energy. Some simply ignore the teaching. They will prepare inadequately or not at all, walk into a class, and just talk off the top of their heads, often about things they are working on rather than about the course material. Moreover, the universities have been using graduate students to do most of the undergraduate teaching. Some universities have large lecture classes where a student can see a graduate assistant for tutorial help. This is not teaching.
Take calculus, which is an applied subject. There is almost no beauty in it; it is a series of techniques for solving scientific problems. But mathematics professors get little training in science. Some of them, as a result, are afraid to use a textbook that brings in scientific problems for fear that some bright student might ask a question the professor can't answer. They are afraid of being embarrassed. They will not take the time to learn the science.
What about mathematics education at the elementary- and high-school levels?
I think that mathematics education—the curriculum, especially—has been horrible in this country from the time we first started teaching mathematics in elementary schools. Incidentally, mathematics used to be taught only in the colleges, even arithmetic. There's been a gradual sifting down of the topics, sometimes for the good, sometimes not.
My main criticism is that the curriculum is not meaningful to the student. The teachers are trained only to teach mathematical techniques, and they're not too secure even about that. I think a student has to be convinced that anything beyond computing the price of three pieces of candy if candy sells for five cents apiece is worthwhile. Problem solving ought to play a significant role, but the problem has to be of interest to the student. There is a particular type of problem that one used to find in the textbooks and that is now returning. One man can dig a ditch in six days; another man can dig the same ditch in eight days. How long would it take for the two men to dig the ditch together? Now what the heck's the interest in that?
This sense of usefulness, of being worthwhile, is extremely important in high school, where the students are asked to learn algebra, geometry, trigonometry. What for? I challenge these teachers by asking them whether they ever in their lives had to solve a quadratic equation outside the classroom. The answer is always no.
Didn't the new math people change the curriculum?
They did, and they were right in saying we must improve mathematics education. They just did it the wrong way. The leaders of the new math were college professors who had no experience teaching in elementary or secondary schools. There were a few exceptions. They presented mathematics as they understood mathematics, and it was much too sophisticated for the students, and that's why the movement failed. That's why I opposed it right from the start. Now it is back to basics, which is really the old curriculum with a little new math thrown in here and there. But it is still not meaningful.
Does it boil down to the fact that mathematics is simply a difficult subject?
It is perhaps the most difficult subject. One of my favorite quotations comes from the great mathematician Hermann Weyl. He died in 1955. Weyl said that mathematics is not a natural concern of men. It has the inhuman quality of starlight—brilliant and sharp, but cold. I think he's right. Sure, some people, perhaps influenced by a very good teacher, take to mathematics without asking why they have to learn it. They are what we call good students, but are they, if they accept unquestioningly what is not meaningful to them? But most students don't behave that way. They react against mathematics, or suffer through it, glad to be rid of it once they have finished their required courses. If the values and relevance of mathematics were presented along with the mathematics, however, I believe that almost all students would take to it very quickly and maybe even enjoy it.
In the above interview, Morris Kline lays out his perception of the flaws in the mathematical system and how the subject can be used to learn more about the world's natural phenomenons. His collection of world-renowned books, including Calculus: An Intuitive and Physical Approach,explore these concepts further.
The subject of calculus is related to science in an introduction that focuses on the application of the subject. Explore the in-depth derivatives with clear-cut examples and many drills. The illustrative equations will polar coordinates, the differentiation and integration of the powers of x, and many more concepts are described in Calculus: An Intuitive and Physical Approach.