The Place of Science in a Liberal Education

by Ralph A. Raimi

                Part 1, The Problem

 The Old Days

        Until the end of the Second World War, the general public -- even the highly educated part of the general public -- didn't much worry about science education. The atom bomb that ended the war in 1945 made all the difference. That is when it was first generally realized how much science had changed all our lives, and was changing the world before our eyes.

        Earlier, of course, science was respected. When Einstein came to America to join the new Institute for Advanced Studies at Princeton in 1933 he was greeted with reverence, and his face in the newspapers became as well known as Clark Gable's. In America technology had been even more than respected, and earlier; the public schools never tired of celebrating the great advances of Henry Ford and Thomas Edison. These men were not exactly scientists, but they used the results of the science of preceding centuries. In the mind of the public they were scientists, and the public wasn't far off. They were certainly more like scientists than like poets or historians. Henry Ford was no historian, after all, and was widely quoted as having said, "History is bunk."

        While there was a difference between the world of science and the rest of the world, everyone in education accepted this calmly. If a child seemed to have scientific talent, well, let him study science. If poetry or history, let him do that. Nobody who majored in history in college, in the days before the atom bomb, felt left out of things, or inferior, if he didn't take any courses in science -- no more than he would have felt guilty or inferior for not having learned to play the violin. People in high school could stop their mathematics with the 10th grade and still be in a program called "college preparatory." Many a high school had a teacher for Latin but none for trigonometry. After 1945, however, the public became avid for news of science.  The political controversy over how to use and control atomic energy made everyone aware that it would be a good thing if the citizen and his legislator in Congress knew something about mass, energy, and the speed of light. There was much talk at the time of the necessity of having journalists who could explain these things in the newspapers, so that people in a democracy could make intelligent choices. Also, America needed more professional scientists, everyone agreed, so that we could maintain our defenses and our prosperity. Other sciences besides physics came into the newspapers, and politics, too, the chemistry and biology, for example, that gave us penicillin, pesticides, and new varieties of rice and corn.

        But nobody went around saying that we should therefore all learn something about biochemistry or nuclear physics. A good thing if we did, surely, but plainly impractical, for there are also geology, astronomy, psychology, mathematics and the medical sciences. We would hope to have a good cadre of professionals to do the work in each of these fields, and another of journalists to teach what we as citizens need to know; but the rest of us, we thought, would go about our other work and enjoy the fruits of science as before, as we might enjoy the work of Heifetz and Shostakovitch without ourselves becoming musicians. Nor did the public attitude really change at first, even as the scientific and technological shocks began to hit us with increasing ferocity.

        The Russians got the A-bomb, we built an H-bomb, the Russians followed close on our heels. But then they overtook us with rockets: The famous Soviet Sputnik, the first earth satellite, was launched in 1957, causing President Eisenhower to appoint a new White House officer, a Presidential Science Advisor, and causing in the educational establishments a flurry of activity designed to increase and improve science education in the public schools. The so-called New Math turned up at this time, and similar efforts in physics and other sciences. Even so, the idea was not to make the general public into scientists, or even very knowledgeable about science, but rather to increase the supply and quality of those people who would then become scientists.  Better physics in high school was mainly intended to make sure every potential engineer would get off to a good start; if the potential housewife or banker also learned a little physics, O.K., no harm done; but nobody believed these other people needed it. Good science journalism in the papers and good science testimony before Congressional committees would be sufficient for the rest of us. 

The Two Cultures

        Then, in 1959, things changed again. That was the year C. P. Snow published his famous essay, The Two Cultures and the Scientific Revolution. Snow, an Englishman, grew up as a physicist; during the Second World War he rose to a position of considerable eminence in directing the war activities of British scientists. When the war ended in 1945, Snow was about 40 years old and at the top of his profession, yet he turned around and became, of all things, a novelist. A successful one, too, well known even in America for such novels as The Affair and The Masters, stories of Cambridge University academics, of government ministries, of scientific fraud. For ten years or so, Snow found himself living in two worlds, for he did not at first entirely give up his scientific or academic connections, continuing to serve, in particular, as an advisor to the government on scientific policy. As he explained it in The Two Cultures, he sometimes spent the day with scientists (and of course some of his best friends were physicists), and then in the evening he would find himself with literary people. The two groups spoke two languages, he found, and lived by different assumptions. It wasn't that the scientists were talking physics while the literati were talking esthetics; Snow was referring to "off-duty" conversation, on interests they supposedly held in common with all educated people.

        In talking politics, or love, or death -- things everyone talks about -- the two groups' attitudes were different; they would respond differently to a given situation. It wasn't that their opinions or conclusions differed, though often they did, but that their processes of reasoning, of assessing evidence, had so little in common that put together in a room they could have no way to say to one another "what was on their minds." In other words, Snow said, they were of two cultures as surely as were the Pilgrims and Indians at Plymouth Rock.

        Now Snow tried to sound even-handed in giving this description, and in the early part of his essay he regretted that the British educational system made scientists as narrow about the humanities as it made the literary folk about science; but in truth Snow was directing his words mainly to the literary culture. He knew how ignorant they were about science, and that having in school begun ignorant they stayed that way.  A humanist cannot just pick up on science in bedtime reading, whereas a scientist who begins adult life ignorant of everything but his science often does mature, bit by bit, into what is called a cultured person. When, as a sort of test, Snow asked some of his literary friends if they could state or explain the Second Law of Thermodynamics, they looked at him as if he were some kind of worm. Thermodynamics?  What has that to do with love and death and the things that truly trouble a civilized mind? Snow observed that if, as a parallel sort of test, he had asked a physicist to tell him who Othello was, and got a similar response, then everyone, scientist and humanist alike, would have been shocked at the ignorance. Yet the Second Law of Thermodynamics is as central to the scientific world view as Shakespeare is to the world of letters. And it is central not just as a framework for technical thermodynamic calculations, but as furniture for the mind in all its thoughts and judgments. In Snow's view, the humanists were a couple of centuries behind the times in their casual attitude towards this awful ignorance of theirs. We will return to the question of what is awful about this ignorance, and what is so important about thermodynamics and its second law; these are not to the present point.    

        Where Snow's essay broke new ground in the postwar debate about scientific education was not in its lamentation of this or that particular deficiency, nor was it a plea for more scientists or technicians, or for better scientific advice for the military or the statesmen, as had been the post-Sputnik reaction. It was new in claiming that some scientific education is a necessity for everyone who would be counted truly educated. The division of the world -- the educated world, that is -- into two camps, scientists and humanists (so- called), who did not communicate with each other was, he argued, a bad thing for the people involved, and especially bad for the literary culture itself, which because of this ignorance read history with a distorted eye, misunderstood the present-day problems of mankind, and were in consequence often politically irresponsible and even dangerous. Here in particular he had in mind exactly the government and commercial establishment of the England of his time, which was almost entirely led by "old boys" of Eton and the like, the sort who went on to win "firsts" in classics or modern lit at Cambridge.

        Snow was not the first to make these observations about the importance of science. The American historian Henry Adams had fifty years earlier written a species of autobiography, The Education of Henry Adams, deploring his own miseducation along these lines. He argued, if somewhat mystically, that only through an understanding of science could the great current of history be properly assessed.  Too, many great scientists of the nineteenth century, Michael Faraday and T. H. Huxley in England, for example, had become popularizers, urging the science of their time on lecture-circuit audiences, for they understood as well as C. P. Snow what their theories meant for the common understanding. In more recent times the novelist H. G. Wells optimistically contemplated the improvement of mankind by the teaching of science, by the clearing away of superstition in favor of rationality, thereby providing a lesson which could than be transmuted into rational thought on social questions too.  (Wells was himself no scientist, of course, and could not in middle age turn as competently from literature to science, even though he wanted to (as evidenced in his The Science of Life), as Snow had done in the other direction. There is a lesson in this asymmetry that Snow might well have mentioned in passing.)    

Responses to C. P. Snow

        But somehow all these urgings did not penetrate the intellectual enclave that Snow called the literary culture of England until Snow himself, a representative of both cultures, offered his ringing challenge.  It was Snow who set the terms of the debate, still unsettled: the question of how to get the lessons of science into the consciousness of those who are not going to make science a profession. Indeed, the debate went deeper than that, for not everyone agreed with Snow, even on the proposition that knowledge of science was in principle desirable for non-scientists. Snow ran into the sternest opposition right in his own home base, in Cambridge University, where the famous literary critic F. R. Leavis, teacher of generations of the British elite, in his retirement year of 1962 loosed upon C. P. Snow a valedictory blast that shook the common rooms on both sides of the Atlantic. 

        Leavis said: Nonsense! Science doesn't really make any difference to the fundamental concerns of mankind. Knowing a lot of science won't make us stand any straighter, won't contribute to good judgment, good citizenship, or anything really vital in what distinguishes man from beast.  The study of literature, said Leavis, was central in a way that science was not. Sure, said Leavis, we need scientists, just as we need farmers and insurance agents; very good. Let us pay a certain number of people to do our science for us, and then let them get off our backs. The real business of humanity is centered on human relationships: What is love?  What is our duty to our neighbors? What is tragedy, and why?

        These are not scientific questions, said Leavis, and all of the most profound contributions to their understanding have come from people who with good reason cared nothing for the Second Law of Thermodynamics.  Scientists themselves, with their easy optimism, tend to think all problems can be solved, and that tragedy is not the destiny of man. What an illusion, to think their formulas constitute a culture! What arrogance, to call it a culture! There is only one culture, said Leavis, and it is mine.

        Leavis said a good deal more than this, and included an attack on the literary quality of Snow's novels: Snow, he said, didn't begin to understand what literature was about. But this was by the way. The Leavis answer to Snow was really a simple one: There aren't two cultures, and Snow was posing a problem that does not exist.

        Whether or not one follows Leavis, there is still a second argument against Snow's thesis that must be taken seriously, and that is that what Snow was proposing cannot be done. Most people, it seems, simply are unable to learn enough of both cultures to make their union possible. This was not always so. Plato and Aristotle were as much scientists as they were philosophers of ethics and esthetics; and the medieval poets Dante and Chaucer knew more astronomy than today's average citizen of Rochester, New York. But it does not follow that we can expect today's poet or philosopher to be conversant with the science of our own time.

                There is too much of it, for one thing, but more important is that it is too tightly organized. It must be taken step by step, if it is to be learned at all; and the cement of mathematics ties the steps together in such a way that whoever misses out on the first few is, by the time he is twenty years old, forever barred from understanding the rest.  Continued attendance at concerts may, in time, teach the unschooled ear much of what music has to say; and the same can be said about literature, theater, and history, all of which have the power to penetrate the mind by accretion, as it were. Science does not work that way. That science is more than a collection of facts is a truism that doesn't yet explain the difference, for music and literature are not mere collections of sounds or words, either. It is that somehow the accumulation of the "facts" of science does not have the power, of itself, to induce understanding even in a well-disposed mind, as the accumulation of musical experience can. 

        It is well known to all teachers of mathematics and physics, in particular, that certain students, apparently a majority, simply don't get the hang of it, even with good will, with effort, and with enough intelligence to do other intellectual things quite well. By the time today's high school teacher is in a position to present real mathematics or physics to a class, it is already apparently too late to make a difference: The population he faces is already divided into those who can learn, and those whose previous disposition -- perhaps genetic, perhaps educational -- has barred them from understanding. "Good in English, poor in math." Who hasn't heard this assessment? And among this group are found many of the brightest lights of what both Snow and Leavis would call the mainstream of our general, or humanistic, culture. Then what was Snow trying to get his country's educational system to do, the impossible? There is much evidence to argue so, in the history of public and higher education in both England and America, during the more than thirty years that have passed since the publication of The Two Cultures. It is discouraging evidence; there is reason, observational, experimental reason, to be pessimistic about the possibility of accomplishing the union of the two cultures.

Scientific Popularization

        For during these thirty-odd years Snow's influence has made a real difference in the way science is regarded by educated people, but without affecting their actual knowledge of it. All that has happened is that humanists, historians and artists have been made to feel ashamed of their ignorance of physics and mathematics, where in earlier years they wouldn't have cared. To this extent Leavis has lost his argument and Snow has won, but winning an argument is not the same as accomplishing a reform. Books of scientific popularization have earned front page reviews in The New York Review of Books, the New York Times Book Review, and their counterparts in England; and many a professor of linguistics or classics has had a copy of Godel, Escher, Bach at his bedside, and Innumeracy, and A Brief History of Time, and Chaos. But to what end?   Hofstadter's Godel, Escher, Bach (1979) was a noble effort to teach Godel's Theorem to the literate non-mathematician, but the average bookmark at the average humanist's bedside got to page 46 after a couple of weeks and then stopped dead. The book sold well but the message didn't take. Much the same was true of even so modest an effort as Paulos's Innumeracy in 1988. One has only to poll the nearest Department of Foreign and Comparative Literature, asking for a proof of any of the well-known birthday bet: that it is more probable than not that, of twenty-three or more random people, at least two will have the same birthday. As with Snow's asking about the Second Law of Thermodynamics in 1958, one will not, most likely, get a correct argument from anyone but a professional (or student) scientist. 

        It is too late now to test scientific illiteracy by asking around for an explanation of the Second Law of Thermodynamics, for Snow's book had newspaper reporters doing that very thing back in 1959, and by now every humanist has been shamed into the knowledge that it seems to say that heat tends to move from warmer to cooler regions. A second statement of elucidation, an explanation of why so obvious an observation should be called a scientific profundity, will not be forthcoming however, beyond perhaps the widely publicized deduction from all this of "the heat-death of the universe." Widely publicized in 1960, at least; by today it may have been forgotten in most English department common-rooms. Even so, answers of this meagre depth of understanding can be memorized in a minute or two, and are as probing as if, owing to an essay by some humanist counterpart to Snow, every physicist, not having known it previously, now could answer, "Othello? Shakespeare, of course; he also wrote 'To be or not to be'."

        Such an answer from a scientist who could neither recite four lines of Hamlet together nor explain what made Shakespeare great, because he hadn't actually read him, would not indicate a successful effort had been made to induce in him or his fellows an appreciation of poetry. (The average scientist is not this bad off, however.) Perhaps it is too much to ask, that statesmen and professors of literature learn probability theory and thermodynamics; perhaps these thirty-five years since Sputnik and C. P. Snow are only the beginning, and it is to the next generation that the message is addressed and for whom it will prove successful. Maybe it is at least something, that the parents have been made ashamed of not understanding physics, and have gone through the effort, fruitless as it turned out, of reading 45.5 pages of Godel, Escher, Bach. Perhaps now the children will sneak that book out of the bedside table, as their parents took Ulysses from a locked bookcase somewhere, to read by flashlight under the covers. Some will; some always do. But if we look to the public schools for help we had better look again. Consider the so-called "new math."    

The New Math

        Even before C. P. Snow, the 1957 launching of the Soviet Sputnik had sent a shiver through America, and the race was on to "catch up with the Russians." Through various Federal agencies, notably the National Science Foundation, money was funneled to any project that promised to improve American science education, at all levels from kindergarten to graduate school and beyond. The mathematicians of the United States created a project called the School Mathematics Study Group (SMSG) to construct a mathematics curriculum from kindergarten through 12th grade, independent of any textbook publisher, school board, school- teachers' association, or anyone else with a vested interest in the then current schools establishment. Most of the authors were good mathematicians from the universities. Edward Begle, a topologist at Yale University, left his position and took up the directorship of the SMSG project as a full-time occupation for the fourteen years of its life, and was a professor of mathematics education for the rest of his own career. Other mathematicians, some of them quite distinguished, served summers or part-time as writers of textbooks, teachers of experimental classes, and as teachers of schoolteachers already in service, teachers who needed to learn the new approaches and how to convey them to children.

        There was nothing impractical about all this. Everyone involved knew what it means for children to learn, to understand mathematics, not as a catechism or examination-passing technique but as a living framework of ideas expressed in English, logically organized to yield computations when asked for, or proofs of theorems of real meaning. That is, they appreciated what mathematics should be to a child if it is to be useful for structuring real science, for verifying scientific reasoning, for modeling real phenomena.  They not only knew mathematics, these writers and teachers, they learned to know children. (Even the initial group was not entirely made up of research mathematicians, but included a leavening of experienced schoolteachers.) By testing themselves at length in actual classrooms with children of every degree of ability and incentive, they learned what sort of teaching would work and what sort was visionary and would lose the attention of the class. Little by little, over its fourteen years, the SMSG constructed a set of exemplary textbooks and curriculum guides, guides for the instruction of teachers; and they compiled statistics on the progress of the children who studied with them as against those who used traditional materials. All this was well financed and meticulous, and if it had succeeded it could have revolutionized the teaching of mathematics in the United States. It would have created a new generation to whom the statistical truths of thermodynamics would have been as accessible in an early college chemistry course as a novel of Ernest Hemingway might be in an early college English course.

        The SMSG saw no reason why children should be able to learn the subtleties of English poetry and yet be immune to the corresponding thing in mathematics. But there was a reason, probably more than one, a reason SMSG did not see but which had to have been there, for the experiment was a failure. "The new math" became a laughingstock, derided by everyone from the newspapers to Tom Lehrer. It became a staple of American culture in the 1960s and 1970s that a child brought up in "the new math" could neither reason nor calculate. In a few years parents were demanding that their children be taught again to add a column of figures -- and the Devil take the distributive law and the Venn diagram. The kind of coverage mathematics education was now getting in the newspapers persuaded the general public that the mathematics profession was staffed by fools who had never seen a real live child, and who didn't care if a bank statement actually added up properly so long as the children "understood what they were doing."

        Why did this happen? For one thing, the phrase "the new math" was not invented by mathematicians, let alone the SMSG. It was a schoolteacher's phrase, designed to advertise something phony that was happening in the public schools under cover of SMSG and other experiments of the same nature.  Apart from the few exemplary classroom experiments actually conducted by SMSG professors and specially selected teachers trained by SMSG, the rest of the country, giving lip-service to the same ideals, destroyed the traditional school mathematics curriculum and replaced it with commercial textbooks designed to resemble the SMSG material without actually coming to grips with it. The fundamental problem was that the existing cadre of schoolteachers feared mathematics as much as their students, who, under such instruction, were inevitably growing to resemble their instructors. For years they had been trained in certain routine algorithms, schemata -- templates -- for multiplying and dividing, figuring out the length of the hypotenuse and the area of the circle. They were comfortable with all this and knew the answers.  But these "answers" are not mathematics.

        SMSG pleaded with the publishing world to plagiarize SMSG books, but the publishers found that real books of that nature were not understood by the school textbook selection committees, were feared by the major part of the educational bureaucracy in the departments of education of the states, and by the professors in their teachers colleges,  and  so were not selling. Most teachers and their supervisors wanted only what they already knew, and the only way they could be comfortable with something new was when it was trivialized.  A trivial simulacrum of set theory, for example, found its way into the textbooks and classrooms, was called "set theory," to be sure, but was in fact nothing at all. And the time devoted to silly exercises in Venn diagrams was taken from the (admittedly dull) exercises in adding columns of figures that the earlier books had contained. So it went, through the curriculum, with even the beautiful structures of Euclidean geometry ("old-fashioned" now, like Plato and Shakespeare) removed from the spiritual equipment of the average educated person, in favor of a patina of modernism.    

Acceleration: A Faster Treadmill

        Begle himself, before his untimely death, conceded the failure of SMSG. But he would never have agreed with the newspaper account of what SMSG was and did. And he conceded too much, in fact. There was one unintended benefit of the bowdlerized "modern" mathematics that the schools were choosing over the previous tradition. The removal of years' worth of dull routines, of algorithms for the extraction of square roots and interpolations in logarithm tables, though replaced with a pabulum for the masses, did give superior students a chance to leap ahead and accomplish "12th grade" mathematics two or three years early. This took place at some cost, compared with what had actually been taught such people a generation earlier, but it yielded some benefit, an acceleration, for those who would then become scientists when they went on to college. These lucky few might some day learn Euclid too, though on their own. This leap into calculus for the few was not really what Begle sought, however; and no part at all of what C. P. Snow was advocating.  Today's humanist has been spared meaningless high school exercises in the use of logarithm tables, to be sure, but he has also been "spared" Euclid. He is usually worse off than his predecessor generation, for he will never again see Euclid or his like, while the scientist does in fact see other axiomatic systems, if not precisely Euclid's, with equivalent spiritual benefit.

        It must not be thought that all high school students who are "accelerated" in mathematics, even those who become engineers or scientists, have actually thereby received a benefit. Every college mathematics teacher experiences every year the plague of high school calculus suffered by students who have been told they know something they do not, and who are puzzled and ultimately angry when their acceleration proves to have been nothing at all. The ritual of mathematical rote is not confined to fourth grade exercises; even calculus can be turned into a catechism, and alas, in most cases it is.  In fact, most high school students who show early promise, and are therefore moved along ahead of the rest, end up with as stultifying an experience under the honorific title of "advanced placement" mathematics as they would have received under any older rubric, "trigonometry," perhaps, or "logarithms."

        It is as if they had learned music by being taught, note for note, over a two year period, how to play Für Elise, without learning anything at all about scales, arpeggios, rhythms or modulations. Painless Beethoven, Advanced Placement Beethoven; but what comes next? The student now wants to play Opus 110, but insists on being told, note by note, where to place his fingers. It is his right; why are his college teachers now "making it hard," talking about chord sequences instead? Why do they refuse to teach him the Beethoven he needs? He soon gives up, and so do most of our eighteen year olds who are crippled by high school calculus. Yes, most of them. Others, a minority of the elite (and an infinitesimal minority of the general population), survive. These are the ones who become our scientists, and would have done so under any system that gave them the chance to read a few books and go to a college whose professors know something. Would better teaching in the schools have brought any of the others into the fold of the scientifically educated, as C. P. Snow would have wanted? Academic mathematicians believe so, and experiment showed it is possible for at least some people who today are lost to scientific understanding; but where is that better teaching to be found? 

        Professor Begle and the SMSG had done their best, with a national will behind them, but were stymied by the bootstrap problem.  First, the teachers must themselves be taught, and by whom? A handful of research mathematicians? Too few. A double handful of "master teachers," themselves to be have been taught by the handful of mathematicians? Wherever the SMSG began, it was not far enough back. Thus, even if learning science and mathematics is possible to everyone, possible in some biological sense, it doesn't follow that it is possible in any foreseeable future, beginning with a civilization which is today essentially unchanged from the one Begle faced in 1957.  And granting even this much, there is no proof that it is even in principle possible to all intelligent people. As music is impossible to the tone-deaf, as painting is impossible to the color-blind, without tone-deafness or color-blindness being a definition of stupidity, so perhaps mathematics and its use in science is impossible to another class of people, not yet named in our language, people to whom Snow's message is futile, but who constitute our majority. If this is so, any future SMSG is doomed not only by the existing culture, but by the very nature of man.  But we do not know this, or not yet.  

Part 2, A Restatement of the Problem    

C. P. Snow Again; and F.R. Leavis  

        There are those who can correctly identify the line, "Wherefore art thou Romeo?" as coming from the play Romeo and Juliet, by William Shakespeare, and who therefore score a point on a multiple-choice "achievement test." Such a person might also know that the words are spoken by Juliet from a balcony outside her bedroom at night; he thereby scores an extra point in the "accelerated literature" examination. But it is possible to know all this and yet believe that "wherefore" means "where," that a comma follows "thou," and that Juliet is calling into the night in longing, saying what in modern English would be, "Where are you, Romeo?" This misunderstanding is in fact common. And those who for this reason miss the import of Juliet’s opening lines are the same as those likely to show ignorance of as many other features of Elizabethan diction (and cultural predispositions) as will render them unable to understand anything said by Mercutio or the nurse as well.

        We hope our own children will be better served by their schools than that. But why? Is it really essential that Shakespearean English be understood by every educated person? There are hundreds of languages in the world, and Shakespearean English is but a dialect of one of them.  There are Russians and Brazilians who will never learn it, and whom we may yet call educated. The Russian may have to read his Shakespeare in translation, and the marvelous cadence of "...a rose / by any other name would smell as sweet" will unfortunately never reach his soul quite as it does ours; but is he therefore crippled intellectually or spiritually? The poetry may be missing, but the essential tragedy of the star-crossed lovers can still be understood in translation; and if he puts into the study of Pushkin the energy the English-speaking world applies to Shakespeare, he will have poetry enough to understand the nature of the art, and insight enough to imagine, from the outside as it were, how Shakespeare must sound to those who know English.

        The lesson that Shakespeare offers can therefore be pieced together by one who does not experience Shakespeare directly, partly from one analogue (Pushkin, a poet), partly from another (Shakespeare himself in translation), and doubtless also from a multitude of other "shadows" of Shakespeare, such as references to him in literature, music and oratory. As one may recover the impression of a three-dimensional object from having seen enough two-dimensional photographs of it, from sufficiently many points of view, so may these views of Shakespeare converge towards the experience itself. They will certainly converge to a better view of Shakespeare than is given to the student, taught by ignorant teachers, who appears to have been taught the real thing, who thinks he is learning, who is praised by his examiners for correct identifications on a multiple-choice examination, but who mentally places a comma after Juliet's "thou."

        A small thing, that. A comma, a misconstrued word. Teach him what "wherefore" means, you say; is that so hard? But that is an answer to an infinitesimal part of a mere illustration in what is only an imperfect analogy to the learning of science. For one who already knows modern English the glossary of Shakespeare does not offer an insoluble problem, true; but whether or not "wherefore" is in actuality widely understood or easily learned is not the point. Let us suppose that the ignorance of such words is symptomatic of a profound misunderstanding of the whole. Suppose further that it were demonstrable by logic, or known by bitter experience, that a more or less competent understanding of Shakespeare, his language and his meaning and his place in the history of our literature, our theater, our total sensibility -- suppose all this were impossible of attainment by the vast majority of our high school students? And that their parents, the newspapers, and F. R. Leavis all want them to have an education in Shakespeare? Which would we prefer to accept for them, which would we urge their parents to accept for them, an education in Shakespeare that leaves its students idiotically mouthing a few of his lines with no idea of their meaning, or an education by analogy, as the Russian might acquire it by translation, explanation, history, reference and the like?

        This education in Shakespeare by means of two-dimensional projections will not of itself show the student all of what it is that makes Shakespeare the genius who shakes us to our roots, but if it contains anything at all that is genuine, it must not be scorned. What is not genuine is the analogue of the finger placement in Für Elise; what is not genuine is teaching a Sunday Supplement "Test Your Knowledge of Shakespeare" in twenty questions: Balcony Scene? (answer [a]: Romeo & Juliet.) Ophelia? (answer [c]: Hamlet's girlfriend.) We are all of us condemned to understand most things in (at best) the two dimensional way, if only because of lack of time in a finite life.  Shakespeare is an extreme example, but is Homer any less so? And which of us reads enough Greek to know Homer? Or Plato? There was a time when every future English Prime Minister was taught Caesar and Cicero at least, whatever he studied in University; how does it stand now? For a full understanding, to the degree mankind does in fact understand it, of the balance of payments, of the effects of proportional representation on the assessment of the public will, of the emission of nitrates from factory chimneys, of the probability of death from a new vaccine -- in other words, for an understanding of what a Prime Minister should understand today -- there is no longer time for gerund and gerundive, the Appollonian theory of conic sections, Milton, Spenser and Shakespeare, Rousseau and Adam Smith, Aristotle, Maimonides and St. Thomas.

        A more modern economics, politics and technology are essential to his mental equipment; these things are not. Every professor regrets this fact, if he recognizes it at all. The professor of Plato tells us that it is impossible to understand Plato in translation. All philosophy of the past two thousand years and more has been but a series of footnotes on Plato, he explains, and yet he sees that today's so-called educated man cannot begin at the source, but must make do with the footnotes. Some of the footnotes. Having spent twenty-five years reading Plato and his circle, and steeping himself in the culture of that time, with excursions as necessary into the literature or history of the neighboring Hebrews and Egyptians, he finally sees, at age forty-five, what Plato really was saying, and it seems to him that a citizen without this knowledge is like a one-armed violinist. A Prime Minister without this knowledge is, as he sees it, the blind leading the blind. That would be true, were it true that the only way to understand the lesson of Plato is to understand his inmost language: what Plato said, what he thought he was saying, and what he intended to say. But this is a lifetime's work, as the Professor of Greek well knows, and can be accomplished by only a handful of people -- if that; for who knows but what that handful is deceiving itself?

        The scornful economist denies that so intimate a knowledge of Plato is needed by anyone, Prime Ministers included. He regrets that his Prime Minister is ignorant of calculus and linear algebra, and therefore of the correct interpretation of the simple graphs that decorate his recent book on international trade. No amount of understanding of Plato will reverse or compensate for this blindness, as may be seen from the idiocy of the subsidy bill the Prime Minister has just put before Parliament, thinking it will accomplish what in fact it cannot. So it goes, and there is something in what they are saying. Could God grant us a Prime Minister -- and Parliament, too -- who all, having lived a hundred lifetimes each before taking office, have studied Greek, Latin, Sanskrit, economics, chemistry, Shakespeare, psychiatry and all the rest, as much as the professional scholar in each of these separate studies has done, all with intelligence and good will such as few of those scholars are blessed with by nature, then we would be ruled by Solomons; but who would build our airplanes, cook our food, enforce our contracts and perform our plays? If we accept the view that anything less than a professional scholar's understanding of Plato, chemistry, or international trade cripples our ability to deal with the world around us, or robs us of the joy we could otherwise take in understanding the nature of that world, then the problem of education is hopeless. None of us has the necessary hundred lifetimes, the strength or the goodwill, not even those of us who would be Prime Minister. The members of what C. P. Snow called the literary world have already devoted their lives to literature, with doubtless some history or other humanistic study along the way, in some cases Plato and in others Chaucer; while yet others, less learned, take the time to write novels or poetry themselves.  Snow did not even ask that they learn economics or political philosophy; instead, he concentrated on physics, a study notoriously mathematical and difficult, and as remote from literature as it is possible to get. Why? Was he merely being parochial, like the earlier-mentioned hypothetical professor of Plato, narrowly construing his own particular study as the key to the universe while all other studies are not? In particular, was Snow merely the mirror-image of F. R. Leavis, shouting from his own side of the Combination Room, "There is only one culture – agreed -- but it is mine"?

     The Second Law of Thermodynamics

        No, C. P. Snow had another point. He was not disputing Leavis's notion that love and death, responsibility, honor and the like were the things that truly engage a civilized man. His own novels might not have come up to Leavis's standards of literature, but they were in fact not novels about molecules and nuclear interactions. They were, actually, about honor, truth, and love. He wanted civilized men to be equipped to understand his novels, among other things, of course, but even for this purpose he believed that an education of the sort Leavis would approve, of the sort that Leavis himself had, was not sufficient. He didn't say this in his The Two Cultures, where he spoke more of politics and making the right decisions of state than he spoke of the qualities of character Leavis took as the first requisite of civilization; but it came to the same thing. He simply believed that an education without science was insufficient, not only to make decisions involving scientific judgment, but even to make decisions, or behave honorably, in contexts more strictly humane. In particular, he called the literary-culture intellectuals "natural Luddites," and he emphasized the practical, even the humane, failings of their narrow faith. To Snow, the Second Law of Thermodynamics was a lesson needed by everyone, the historian and the novelist no less than the physicist. 

        Now honor, duty and death have already been studied by Homer, and by him described and dissected to a degree that still commands the admiration of the world; and in Homer's time there was no Second Law of Thermodynamics for him to learn. F. R. Leavis would say that Homer was supreme, not deficient, in his understanding of the nature of man and his world, and would deduce that the Second Law is irrelevant to such understanding, along with all the works of C. P. Snow as well.  Snow disagreed. Snow, optimistic as most scientists have always been, believed there to be more in the modern world than there had been in the ancient, and he believed that the equipment of the modern intellect does have to be more ample than Homer's, genius though Homer was, to match that added complexity. Snow believed that grave and dangerous error lies in wait for one who lives only by -- even the highest -- wisdom of another age.

        One looks in vain at Snow's essay, however, to find out just what it is about the Second Law of Thermodynamics that puts us a step ahead of Homer, other things being equal, in our understanding of what Leavis and Snow both agreed "really matter." He was vague. He said that the physicists he associated with during the day, "had the future in their bones," that they saw things hidden to the literary folk he saw at cocktail time, and that these things were essential to civilized life. That's about all. Leavis more than once, and with heavy scorn, quoted the line about "the future in their bones" as evidence of Snow's poverty of poetic imagination, and the silliness of his argument. (It is a weak line, to be sure.)  But if Snow was but an indifferent poet it doesn't follow that he did not understand the nature of literature and the liberal arts, not to mention twentieth century physics, well enough to avoid a biased judgment. In this he was better off than Leavis, who understood nothing at all of physics. Snow must have meant something, something more than the mere observation that scientists and humanists had little to say to one another, something more than the observation that playwrights seldom fathomed statistical mechanics, and wasn't that a pity, statistical mechanics having so beautiful a structure?  It might be worthwhile here to say a few words about this famous Law.

        The Second Law of Thermodynamics was formulated in the nineteenth century in various guises, not all of them initially recognized as equivalent. All of its statements may be regarded as statements of the limitations nature places on the ability of heat to do work. The First Law of Thermodynamics is the statement of the conservation of energy, whose earliest forms came from mechanics, but which in thermodynamics said (among other things) that there was a fixed equivalence between the quantity of work done against friction and the amount of heat generated in the course of that activity. If heat and work are equivalent, it would appear that a reverse process is possible, that a gas (a steam-filled container, for example) could be made to give up its heat in a suitable engine and thereby perform the equivalent amount of work. Indeed, the steam engine does trade heat for work again, but the steam cannot be made to give up all its heat in favor of an equivalent amount of work; and this is where the Second Law appears: It gives a precise statement of the limitation, in mathematical terms.

        Later in the century was elucidated the theory of statistical mechanics, which equated heat with the kinetic energy of random molecular motion within the material containing the heat. From this arose other formulations of the Second Law, expressed in terms of statistical distributions of velocities among these extremely hypothetical, necessarily invisible, molecules. All these forms of the Second Law recognize the experimental fact that heat tends to flow from a warmer body to the cooler, when the two bodies are juxtaposed, and that the entire observed system has a tendency to approach an equilibrium temperature, after which the flow of heat between the visible parts of that system ceases and no more work is available from it. 

        (In the twentieth century both the first and second laws have undergone enormous extension, all of them mathematically expressed.  The mathematical expression of the second law as it relates to the statistical interpretation of heat as molecular motion has given rise to the interpretation of yet other phenomena, quite unrelated to transfer of heat, in "thermodynamic" terms, for example in problems concerning the transmission of information. This is an example of an important feature of the mathematization of science, that if a mathematical structure describes one physical situation, and the same mathematical structure can model another, then experience in the one domain, when mathematically expressed, illuminates the behavior of the other.)

        It is possible to say, simply enough, that heat flows from warm to (adjacent) cool places, and to call that the Second Law, because upon that simple principle (plus conservation of energy, the First Law) was built the theory of heat engines and their efficiency and all that follows thereafter. But everyone already knew, long before Carnot or Maxwell, that heat flows from warm to cold. Aristotle knew it, but certainly had no reason to emphasize it in his physics. Nothing profound in Aristotle's philosophy depended on it. What makes the Second Law profound today is the mathematically expressed theory that is built upon it, not the simple statement itself. Almost the same is true of Newton's laws of motion, especially the one that says that momentum changes according to the impressed force. A very few words are needed to make this statement, but of course one must know what momentum and force are, and the regress grows very long, if not infinite, before one is satisfied. Einstein never could be satisfied, in fact, and so was driven to discover, or invent, relativity. In the other direction, one must know what Newton accomplished with his principle of force and momentum, to fully understand its profundity. He proved mathematically, upon this principle coupled with the inverse square law of gravitation, that the planets were constrained to follow exactly the paths they are seen to follow, i.e., that Kepler's observed laws of elliptical planetary motion were no accident, but a necessary, computable, consequence of the law of gravity.

     A Scientific Question

        Now if one prints a "pop quiz" on all this in the Sunday papers -- and the papers never tire of doing this -- today's high school senior may score well by memorizing something like the five or six paragraphs immediately preceding this one. They might contain wonderful words: thermodynamics, entropy, statistical mechanics, conservation. They might contain sparkling names: Aristotle, Newton, Maxwell, Clausius, Kepler. But they don't really contain any scientific information. They summarize a long, profound and difficult development in the history of science, but they teach no science at all. Memorizing the summary does no more for the understanding than knowing that "The Renaissance put an end to the Middle Ages." It is true, more or less, in a sense and up to a point, providing you know what pitfalls lie behind the words; but even a thousand such statements, without the study of real history, will not add up to an iota of understanding. Those people who already understand what it says don't need to be told, and those people who do not understand what it says are often deluded into believing that what it says is knowledge.

        Here is a sample from a quiz printed in the New York Times Magazine of Sunday, 13 January, 1991, p.24: "Question 2. An atom differs from a molecule because: a. Molecules are made of atoms; b. Atoms are made of molecules; c. Gas is made of molecules, but solids are made of atoms; d. Atoms and molecules are two words for the same thing... Question 6. Galaxies, like our Milky Way, are made of: a. Hundreds and hundreds of stars; b. Thousands and thousands of stars;  c. Millions...; d. Billions... Question 8. The most abundant gas in our atmosphere is: a. Oxygen; b. Carbon dioxide; c. Nitrogen; d. Smog." 

        No; it won't do. And yet, alas, it represents not only what the conventions of today's journalism calls science, but it represents what sort of thing the non-scientist usually has learned in school, and has been taught to call science, believing that the difference between his understanding and that of a professional scientist lies in how many such statements they each know, and with how many decimal places of accuracy. To see that this popular impression of the nature of science is mistaken, one need not go very deep; we can replace any pop quiz with a single question, given below as an example, a question that a good scientist can answer (though it might take more than a few sentences) and that today's non-scientist probably can not. Not only will the average man not be able to answer this question; he cannot even outline the form an "answer" should take.  He is not used to questions of this sort, where a scientist is used to little else. Here it is: "How do we know atoms and molecules exist?"

        There are somewhat more difficult questions than this, for example, "What difference does it make, to know that atoms exist?"  (The two questions are equivalent, actually, according to the philosophic stance taken by scientists today, which is that the "existence" of a scientific entity is instrumental only, and not ideal in the sense of Plato; but this is by the way.) This question, "How do we know that atoms exist?" is never asked by newspaper reporters, partly because the answer is necessarily long, even when given in summary, and partly because most journalists - - and schoolteachers, and television evangelists, and statesmen, too -- don't recognize it as a scientific question. Yet it is, and it is more telling an example than the one C. P. Snow asked about thermodynamics.  For the real difference between the Two Cultures inheres not in whether or not one construes Shakespearean English properly or knows "many cheerful facts about the square of the hypotenuse," nor yet in whether one's bones contain the future. The difference is finally to be found in the philosophical furniture of the mind, not in its precise list of bits of information.

        That difference in “the philosophical furniture of the mind” is possibly one that can be bridged, even if it is hopeless to expect the Plato scholar (or F. R. Leavis) to understand the mathematics and experimental evidence of statistical mechanics, or to expect C. P. Snow to write like Joseph Conrad.  It can be bridged by an education that is quite manageable in principle, though probably its implementation would today run into most of the same difficulties as were experienced by the School Mathematics Study Group. That is, it would be a bootstrap operation for the present generation, even in rich and technologically advanced countries like ours.  Just the same, SMSG was an ideal that will not die, and that may some day, little by little, be realized. In science too it may be worthwhile outlining a program of this sort, in hopes that the time will some day be right for it. It may be that this day will come sooner for science than for mathematics.

Part 3, A Solution of a Sort    

Evolution and Creation

        The controversy attached to the name and teachings of Charles Darwin is still alive today, though it has shifted its domain from the community of scientists and theologians to the community of parent- teacher associations. The controversy is not in the first instance scientific, it is a actually a problem in public policy: shall we use this book or shall we use that book, in the fifth grade curriculum in natural science? There are those who see a choice: on the one hand a Godless curriculum that destroys the moral fiber of our children by telling them the universe is random and without plan, and on the other hand a celebration of the wisdom of the Creator, as it is made manifest in the marvels of His handiwork.  The facts, say the Creationists, are not in dispute here. The children would see the same physiological mechanisms through the microscope under the one plan as the other, and learn the names of the plants and animals, and of their organs, and learn their habitats and their behavior and diseases, exactly the way it is. All that would be changed is a theory, which makes no difference to the observed scientific facts we want our children to absorb, and which cannot be proved anyhow. 

        Perhaps the existence of God cannot be "proved" either, they say, since nobody has actually witnessed the six-day Creation; but there are likewise no witnesses to the millions of years evolutionists have postulated between us and the dinosaurs, nor have any of us witnessed monkeys begetting men. Therefore, they say, why is the Darwinian theory privileged in our schools? Should not the alternative theories be taught side by side?  The response from the spokesmen for science has been quite disappointing, at least as the newspapers understand and report it.  Evolution, they say (or are said to say), is as well established as gravitation, while Creationism, the literal story of Genesis, is just plain wrong.  Fossils are dated millions of years apart with radiation methods; species give way to related species in unmistakable sequence in the fossil record; and so on and so on. Science probably cannot deny Genesis in a symbolic or mythical way, as a metaphor for the history of the earth; but to say that God created it all in a literal week, the fossils and the present species alike is, according to the spokesmen for science, nonsense.

        Another sort of response comes from another direction, from the American Civil Liberties Union. They argue separation of Church and State, and they bring lawsuits. It is not at all that Creationism may be wrong, in their view, as that it is illegal -- in the public schools, anyway.  (One suspects, just the same, that they do think it is wrong.) But legalisms are irrelevant to the philosophical question, which could be reformulated as, "What shall I (“I, and not another”) teach my children about evolution and Genesis?" This is the question here, while legal problems are not to the point. Now, what is inadequate in the response from the scientific community is that, albeit for perhaps valid political purposes, it takes as absolutist a view of the nature of scientific truth as do the schoolteachers who will in the end be teaching these things to fifth or tenth graders.  The scientists testifying before a school board or legislature find themselves forced into an attitude they would never take in their own, in-house debates concerning a controversial theory, where all sides are heard again and again as long as there are those ready to offer a proposition that has real consequences and has not yet been disproved.  They take this stance, which in fact violates scientific norms, because in the public schools -- and in the legislatures -- there never is such a debate, nor even a reasonable explanation of the nature of the debating process in science.

        Scientists are asked, as "experts" are asked in courtrooms, such things as, "Which theory is true?", "Could the earth be created in six days?", and "Was Darwin right or wrong?" These are not scientific questions, and to demand that scientists answer them in this form can only muddy the waters. Indeed, it does. Public school teachers and their students are typically assigned an approved textbook that contains the "truth" as it will be asked for on Regents' multiple-choice questions at the end of the year. Such subtleties as the status of a theory, its purpose and its sometimes confused relationship to experience do not get into those examinations, hence not into the textbook, either. Faced with a choice between a creationist absolutism and a Darwinian absolutism, and that is the Hobson's choice our schools afford, scientists will opt for the latter. How much better it would be if the schools had a chance to teach science, instead of the biological, or physical, or chemical, or mathematical, catechism that is called by the name of science in the schools. The problem of "Creationists vs Evolutionists" would simply evaporate, and all students could be encouraged to read not only Darwin and his followers, but the five books of Moses, too (all of it good reading), with no need on anyone's part to lock the bookcase. And what is more, the problem of The Two Cultures would thereby itself be on the way to a solution that might command the approval of C. P. Snow and F. R. Leavis both. To see how these marvels can be accomplished, it is necessary to digress into a consideration of the nature of science. Not the facts of thermodynamics or of any other particular theory, whether of chemistry or biology, but the question of what constitutes scientific understanding itself. The point of view in the following is associated with the name of Karl Popper, a recent philosopher of science, but it has roots even in Hipparchus of ancient Greece and Copernicus of more modern Europe, both of whom put forward a heliocentric model of the planets for predictive purposes, without insistence on its philosophic truth. There is good reason to believe that Popper's view will endure.    

Scientific Statements

        The crucial property of a scientific statement is that it can somehow, at least in principle, be tested against reality. That is to say, it is falsifiable: there is some experience possible that could show it to be mistaken. Thus if one were to say, "Life is like a cup of tea," that would not be a scientific statement, because one cannot imagine an experiment that would satisfy everyone that life fails to resemble a cup of tea. Examples of scientific statements come in all degrees of complexity, and the ones that generate controversy are usually quite intricate, requiring a considerable apprenticeship in the science in question for their mere understanding; but to begin with let us take as trivial an example as is still sufficient to illustrate the point. Suppose we say that light beamed at a reflecting surface leaves that surface at the same angle it arrived with. "The angle of reflection is equal to the angle of incidence" is the usual wording of this venerable law of optics. Of course there are some preliminary hypotheses required, such as the homogeneity of the medium through which the light is travelling (in air, say), and some definitions (of "light," "beam," and the like); but for the present purpose these refinements are inessential. Now the reflection law is a scientific statement, not because one can prove it, but because there is a way of trying to disprove it. Mankind has tried many times. It has been tested on a pool of water, a sheet of glass, a sheet of steel. It has been done with red light, green light and white light. It has been done with angles of five degrees, forty degrees and ninety degrees. It has been tried on mountain tops, under water, and in a vacuum. It has been tried by Greeks, by Chinese, by Jews, by honest men and by thieves. Until now it has proved accurate every time it has been done (with materials and light of frequencies possible in 18th Century laboratories), but did that prove the proposition?

        Not at all. X-rays, for example, don’t bounce off ordinary mirrors in this way.  Is the “law” therefore wrong?  Not this, either. Experiment has shown this "law" of optics fails under those particular circumstances, and therefore must be called false in general, but by the same experiment we know a bit more about what part of the law is actually true.  Or appears to be true, so far. In other words, so long as there remains the possibility that some experiment can deny the statement, then that statement can be called scientific. A statement may be called scientific even when it is practically always false, as for example the statement that the angle of reflection equals twice the angle of incidence. This proposition has already been falsified, and so is of little interest; but it remains a scientific statement, which the one about life and the cup of tea is not. On the other hand, while a scientific statement can be proved false, or limited in its domain of apparent truth, by a single contradictory experiment, it can never be proved flat-out “true”. One experiment denying the statement is enough to reject it, but no number of experiments in its favor will make it certain that all other experiments, an infinitude of trials not yet performed, will also favor it. 

        On this point there is sometimes some confusion, in that mathematical statements can in fact be proved; but mathematics is not science, and mathematical statements are not scientific statements. Mathematics is about numbers, spaces and other abstractions which are defined by man and have no necessary contact with the world of experience. That mathematics is so often applied to real science causes many people to believe that one can "prove" the truth of a scientific statement by some mathematical device, but this is a mistake. What mathematical proof does is to show that one set of mathematical words implies another. It can show how one scientific statement can be converted into another scientific statement saying the same thing, though it looks different; but only experiment can falsify either form of that statement. Mathematics can also prove that, of two different scientific statements suitably related as to subject, it is impossible that both be true, i.e. that they are not equivalent. This is a valuable use of mathematics, but it cannot prove a scientific statement, or even disprove it unless something experimental is also known. Thus "2 x 3 = 6" is true, but not a scientific statement. It cannot of itself prove there are six apples on a table, but it can help persuade us that there are six if there are already known to be three in each of two bowls on that table. "Life is like a cup of tea," which may or may not be true (a point few of us would willingly argue), is also not scientific, because it is untestable. What other sorts of non-scientific statements are worth mention?

        The most important examples, or perhaps the ones that confuse popular discourse the most often, are those that sound like science but do not, by scientific standards, say anything at all. A famous example is from Moličre: Morphine will put a man to sleep because it has a "dormitive" virtue. A dormitive virtue! What a beautiful phrase. It seems to mean that morphine contains a property, or essence, of a sleep-making sort. Now this statement is either a tautology ("Morphine puts you to sleep because it has the property of putting you to sleep."), or it is vague, and says only that something inside morphine puts you to sleep. Neither interpretation gives the least clue about how one could construct an experiment to disprove the statement. There is, actually, a small scientific ingredient in the Moličre statement, which is the implication that morphine always puts people to sleep. It is the beginning of science, after all, to find regularity of behavior, and the fact that this bit of science is contained in the morphine statement is evidenced by the falsifiability of that part: If one found a single person not rendered sleepy by morphine one would have to modify that particular statement. But the main text, the "because" part, is not scientific, and from the scientific point of view it says nothing. Moličre and his audience knew this, while the learned doctors at the Sorbonne (in Molieče's parody of their language and behavior) did not; that's what was so funny. Once one learns what constitutes scientific meaning, one can judge other forms of discourse better than if this idea were not plain.  There are many statements that contain no meaning from a scientific point of view but which still might have validity and importance of some other sort. "2 x 3 = 6" is certainly a venerable example, and statements of particular fact, like "J. S. Bach died in 1750," and "The current in this circuit, as measured by this meter, is now 1.78 amperes," are others.  Some poetic conceits are hard to analyze from a scientific point of view (Try "Truth is beauty, beauty truth") but yet have a poetic validity for the impression they make, the flavor they impart to nearby sentences and to the experience of life generally. Others, like the dormitive virtue of morphine, are of course laughable. It is educational to distinguish all these statements, the true, the valuable, the false and the vacuous, from scientific statements, which might or might not be true or valuable, but simply cannot be vacuous and scientific at the same time. And -- more than educational -- it is essential to recognize that a scientific statement always could be wrong. Always: without the experimental possibility of its being wrong it cannot be scientific.    

Science and Hypothesis

        Somehow the popular understanding has got this message turned exactly around, and in the schools children are taught that the valuable things to be learned are those we can be certain of beyond the possibility of disproof: "the facts" so beloved of Mr. Gradgrind and Sergeant Friday.  There is a reason for this attitude, for the facts are not contemptible, and we all begin our education with them. Some, like particular laboratory observations, bear on science, while others, like the rules of prosody, do not. Each of us knows, and must know, millions of such things, but these are not wisdom; these are mere facts. Science, too, like a child's education, must begin with facts and then move on to simple generalization: the density of copper, the phases of the moon.  But as civilization has developed, it has moved from raw data and primitive observation to form hypotheses and theories that connect these facts to one another and, we hope, to predictions of what will also turn out to be facts in the future. These structures, these theories, are the real science, while the facts, though indispensable, are only its raw material.  As words are to poetry, so are facts to science. A book of words may be a dictionary, and valuable too, but it is not a poem. One can wittily say that the dictionary contains Paradise Lost, as a newly quarried rock contains a Piéta; but it is clear that there is a necessary human imposition of structure involved, and that the art is in the structure, not the material.

        One failing of science education in the schools, however, is that theories too are taught as if they were raw facts. Having learned that copper has density 8.92, something easily understood and in fact directly (if not so easily) measurable, the child next learns that it is made up of atoms, each of which has a nucleus surrounded by 29 electrons. Wow!  There is a vast difference in the status of these two "facts" about copper, and a vast tragedy when they are taught as if their status were the same, and asked about on Regents' Examinations as if they were understandable in the same way.  Yet they often are taught as if they had the same standing. A front page feature article in the Rochester Democrat & Chronicle for November 2, 1991 is headlined, "Franklin teacher uses creative touch to help chemistry sink in," and explains,

        Dorway [the teacher] conducts a question-and-answer session with students about the notes they've just read. 'Pepper, how do electrons arrange themselves around atoms?' Dorway asks Pepper-Marie Russell, a 16-year old junior. 'Shells,' Pepper says, answering correctly.

        By the newspaper's lights, this was an example of superior science education, much more modern, much deeper, than learning about the density of copper. Both are science, to be sure, and the falsifiability of the proposed 8.92 density of copper is patent. Let the student try; it will do him good. He might learn more than at first appears, for he will have to satisfy himself at the outset, among other things, that what he has before him is indeed copper; this alone is no small task. But the falsifiability of the 29 electron hypothesis is of another order entirely. We more or less know what copper is, though persuading someone of the purity of a sample raises a good number of non-trivial problems, among them the definition of purity. But electrons? They not only have not been seen, they cannot be seen; even the atoms that carry them are postulated entities, invisible, imponderable, and much more problematic than the notion of "copper." One cannot begin to test the 29 electron assertion until one understands what it says; and what it says depends on some pretty subtle hypotheses on the nature of matter to begin with.

        One such hypothesis is the existence of atoms as the smallest units of copper (for example) that can reasonable be called copper. An atomic hypothesis of a sort was given by Lucretius two thousand years ago, but though some of his formulations have a startlingly modern sound they turn out to have no real scientific content. Today's atoms are more complicated and more valuable. They are testable. Here are a few of the postulated properties of atoms: The atoms of a given substance all have the same weight, and indeed are identical in every way, but the atoms of different substances have different weights. Atoms of a given element are each fitted with a fixed number of hooks ("valences"), by which they combine with atoms of other elements in only a certain few combinations, hook against hook, forming "molecules" which are the minimal amounts of the resulting combined substance. Two gases of different kinds, but of the same volume and at the same pressure and temperature contain the same number of molecules. A molecule of water is composed of two hydrogen atoms combined with one of oxygen. (These are sketchy simplifications of some parts of the atomic theory of Nineteenth Century chemistry, for illustrative purpose only, and each statement has since been substantially modified, but even as quoted here they are scientific statements.)

        With a picture of elements divisible into atoms of this sort, all of it quite invisible and purely conceptual, one can compile a multitude of observations consistent with the picture and no observations (not yet!) that conflict with it. The same is true of the Lucretius model, too, the difference being that it is inconceivable that there could be an observation inconsistent with Lucretius's picture. One might ask, then, whether Lucretius's picture isn't therefore superior to that of Dalton and the others whose researches culminated in the modern notion of atom.  Lucretius had the better poetry, according to those versed in Latin, and it can't be faulted by embarrassing observations; what more can one ask? The value of the scientific picture is not in its poetry, which it may not have, nor in its certainty, which it cannot have, but in its consequences, which it must have if it is to be called scientific. To know that it can in principle be falsified, but to believe that in the event it will not, is exactly to say that certain other things in nature, some of them not yet seen, must take place. Their failure to take place would be the falsification we believe will not occur. The Dalton atomic model, if accepted as true, limits the other possibilities in nature, excluding those which would falsify it; and this limitation is the same as prediction of what must follow from his hypotheses. A "theory" that is consistent with every result, that puts no limitations on what God can do, can tell us nothing about the probable future, about what to expect under certain experimental conditions.

        Having understood atoms as they were developed in the 19th Century, one can go on to neighboring phenomena, both chemical and physical, for a further understanding of atoms. The chemistry of that era, and since, did in fact uncover subtle inconsistencies with the simple picture of atoms (of a given element) all having a single weight and a single number of hooks; and the atomic model did indeed have to be modified again and again. One would never see why electrons had to be invented to complicate the earlier picture, for example, without knowing about those other theory-denying or theory-eluding observations, one of the most important of which was the positions of the spectral lines in light from a burning material, on earth and in the sun, and another of which was the discovery of radioactivity and its curious properties. Until he understands a good part of all this it is perfectly meaningless to tell a high school student that an atom of copper has 29 electrons. It is meaningless because there is no experiment within his power to imagine that would test the statement. To put it another way, he is made to believe, by being told that copper has 29 electrons, that he knows something Lavoisier (and Lucretius) did not; but can we imagine his then going to a reincarnated Lavoisier avid for news of discoveries that had taken place after his death, and telling him such nonsense?  What could Lavoisier learn from such a statement that would clear up a single problem he was pondering in the shadow of the Guillotine?

        The statement that copper has 29 electrons depends for its very meaning on the answer to the accompanying question, "How do we know copper has 29 electrons?" In this question, "know" must be taken with a grain of salt; it serves as abbreviation, without which the question must be put this way: What are the experiments that are consistent with the hypothesis of 29 electrons and inconsistent with any other number? Even this is too brief, for the very notions of atom and electron must, for their meaning, be shown consistent with a large number of experiments, all that anyone has yet been able to imagine, and for which no better notions have as yet been offered as models.    

Science Education in the Public Schools

        Unfortunately, this point of view is very little understood by the teachers of science -- or anything else -- in the schools, and the resulting ignorance is carried down to succeeding generations. Science is treated as a list of facts, the theories themselves having the same standing as raw measurement, and the entire list apparently known to be true because some authority has said so. (History is treated much the same, by the way.) There are teachers who know better, of course, but in the face of the ambient spirit, reinforced by newspapers and Congressmen, they can make little difference. Part of the reason for the treatment of science as a list of truths, of facts to be learned in accumulation, is said to be the complexity of modern science. It is thought necessary for a future engineer (say) to learn at an early age an enormous number of such things, mixtures of facts and theories, many of them related in ways that cannot be understood without some years of college mathematics. Why wait; he needs it now.

        But this is an illusion. Misunderstanding is never needed, and especially not now. There is, actually, a primitive level of understanding that can be useful in routine applications and is still useless as intellectual furniture for the mind. As one may be able to drive a car without understanding anything about the nature of the engine, gears, brakes and the rest, so one can make things out of copper without understanding the atomic hypothesis, let alone the quantum theory that might predict its conductivity in the solid state.  Indeed, most of what is taught as "science" in the schools is philosophically not exacting, nor should it be. The names of the parts of animals, and the species and genera are simple facts, but they are made easier to remember, easier to organize, when simple genetic and evolutionary "causes" are assigned to some of this organization. Electric circuits are "explained" by naming such things as resistance and inductance and mathematically analyzing a diagram using a few handy rules.  There is a level at which the optical behavior of lenses and thin films is best understood by saying flat out that light is composed of waves of such and such a frequency, which can interfere with one another the way water waves do, and yet travel in straight lines too, as does a wave front in water.  For mnemonic purposes, and for calculating things about batteries and cameras, this sort of information is fine, and is a valuable consequence of the work of science, but it is not in itself scientific education. Or not yet.

        When one begins to learn how resistance and inductance are measured, and how two different means of measurement prove equivalent when mediated by a theory, then science has really begun. There is nothing obvious about the statement that light is a wave phenomenon, and there never will be, but the value of the wave model as a hypothesis is not too hard to demonstrate. The future electrician can today learn many cheerful facts about the 29 electrons of copper under the impression that he is learning physics or chemistry, and yet show no sign in later life of having been injured by the mindlessness of the exercise; because all he ever ends up doing with copper is designing wiring with due attention to its conductivity as printed in a handbook. The school thinks that in drilling him about electrons it taught him science. He thinks, at least in his student days, that because he can fill pages with relevant exam-passing symbols, that what he has learned is science. Neither of them knows better because his later work with wiring never tests the result.

        In truth, the that Rochester high school teacher drilled his class on the 29 electrons in shells only because it was in the book. Suppose we changed the book.  Suppose we never even mentioned electrons, but had our future scientist -- and the future non-scientist too -- spend a year in a class called "chemistry" repeating only what the eighteenth century philosophers were doing when they first isolated copper, devising tests to distinguish between certain elements and the compounds that were combinations of them, tests to show that the very notions of "element" and "compound" make sense in a way that other hypotheses on the nature of matter did not. Which way would our high school students learn more chemistry? The answer is: by omitting the electron shells for the time being, and  paying close attention to the experiments of Lavoisier and Priestly.

        The future engineer would still have to learn some day how to use his materials. To learn a bit of this at an early age might be useful, whether or not it was combined with his education in the nature of science, but his lack of information about electrons would have no bearing on that. If, then, he turned out to be one of the few who wanted and could use a deep scientific education, he would find that he had already achieved a philosophical stance making the quantum theory a hundred times easier to learn, when it came his turn to learn it. Similarly with any other science, and with mathematics too.  And it is not only as preparation for later technical or scientific study that this sort of beginning is important. It is exactly here that the bridge between Snow's two cultures must be built, not in fairy tales about the 29 electrons and E = mc2, but in the tedious construction of a model for what one can himself observe, and in the understanding of what a model is, what it collates, what it predicts, how it may be falsified and perhaps modified. Most important, everyone should understand the philosophic status of a model, how "model" is really another word for "theory," and how a model and its predictions do not form a list of facts analogous to (say) the provisions of the Treaty of Versailles.  

The Case of Evolution

        To anyone who understands the nature of theory, of models and their consequences, the controversy concerning evolution in the public schools is foolish on both sides, in the terms in which it is argued publicly. Both the Creationists who decry teaching evolution and the scientists who defend it are talking as if evolution were true or false, like a theorem of Euclidean geometry or the 8.92 density of copper. They also talk as if the Creation as described in Genesis were made of assertions of the same nature. If they were, there would be something to quarrel about. But the two stories of the origin of species, Genesis and Dar- win's, each have their own philosophical standing. And each is worth learning, though for different reasons, for they do not each have the same sort of consequences. A student learning the Darwin theory should be given to understand that he is not learning facts. The facts are what the dissecting table tells him, the fossils in the rocks, and what the books describe about the experiments of other scientists, all too numerous for any high school student to repeat for himself but not too subtle for him to imagine performing, given the time and the tools. The theory, then, is what places the facts in orderly arrangement to begin with, and then goes on to accomplish a good deal more.

        The biblical story also makes an orderly arrangement of these facts, or at least does not contradict them. To say that what is out there is out there by virtue of God's will is to deny no fact whatever. To know the biblical story in some depth helps understand much of the history of our civilization, its darker moments and its glories both, the Spanish Inquisition and the Bach B Minor Mass. There are scientists, too, practicing Christians or Jews, for whom the biblical story is an inspiration without which they could not imagine working, even in the laboratory. To deprive students of this literature is to make incomprehensible our own history, which every citizen really must understand.  Even the theories of Galileo and Newton sprang from a Bible-inspired view of the universe. The consequences of the Darwin model are of another sort.  Granting the model, biologists looked for a molecular mechanism that might give rise to the variation (later, "mutation") needed to make the model work. With the rediscovery of the work of Mendel the search became more focused still, and the science of heredity had somehow to be reconciled with the doctrine of change. A careful history of this process is not merely a celebration of the road to DNA, it is a lesson in how an ingenious model need not be declared true or false and may yet guide science in fruitful directions that would not have been dreamed of without it. Any educated person must understand the part the Darwinian hypothesis played, and still plays, in the back of the mind of the man with the electron microscope. This educated person will never understand Darwin if he is constrained to call it all true, as if future observation could never require its replacement by something better. He will never understand it if he is constrained to call it false, either. Worst of all, if his education is such as to make him think that if it can't be called true or false, one or the other, it cannot be worth much, then his education has been a total failure.    

Back to the Schools

        The scientific education given by the public schools is exactly this total failure. It is not the failure to teach the Second Law of Thermodynamics that is the fault, nor the failure to teach enough facts, though these are indeed some of its failures. The failure of the schools may be measured exactly by the observation that "creationist" and "evolutionist" textbook committees think they necessarily have a quarrel. It is only as matters stand that they do, because the books they would have the school boards adopt tend to speak of theories as if they were facts; but this is not the way matters have to stand. Changing the biology books alone will not do the job. The proper attitude is one that must be inculcated over many years of education if it is to make sense to the average child of high school age. As it is, the attitude in question is today not even held by his teachers. As with the School Mathematics Study Group and the "new math," a change in this state of affairs appears to be a bootstrap operation impossible of accomplishment. If that is so, one can only be hopeless about the future of the world, for this implies that religious and ethnic strife will never yield to reason, and that disasters along other fault lines -- religious, racial, linguistic, economic, legal -- will forever keep mankind from happiness. A pessimist too has a model of history, and it is not yet possible to call him wrong. It is possible to put his model to the test, though, and it is our duty to do so. We therefore should bend our efforts to teach science from the earliest grades as a process of discovery (mankind’s discovery, not discoveries by the student except in a limited way, for the history of scientific discovery cannot be reproduced in a single lifetime in a high school class or laboratory.) 

        One must, of course, begin with a budget of facts, some of them facts the child can observe for himself. There is no need to pretend, by the way, as some sentimental school teachers try to do, that what is happening is really discovery; the student should understand that history is being repeated here, though usually in a simplified way, and certainly much faster than the first time around. In every case, the fitting of facts into a problematic pattern must be attempted, even if only a principle of classification.  Whatever it is, the problem must be made clear before a solution is attempted. Little by little, hypotheses should be insinuated, much as mankind has done in its own history, to organize these facts whose concatenation forms the problem, and to serve as a model from which predictions arise. Not every model must be complete, and some should be wrong enough to provide the student with the thrill of himself finding the falsification.

        Modern technology cannot be hidden from these students, of course, but "explanations" of how a computer or television set works should always make explicit what is being left out. And what is being left out -- for the time being – cannot help but include much that the student could not in principle test for himself. To explain electric current as a flow of electrons teaches nothing, until -- much later in the educational process, probably in college -- the student is prepared for such a model.  To explain it as a flow of something, however, provides a model that can be understood and worked with.  To say, as Faraday said, that a "current" is an imaginary fluid, that seems to move when a wire is placed in the vicinity of a moving magnet, does make sense. It guides our thoughts and causes us to ask questions of measurement. What sort of "fluid"? Let's try a few experiments and see. In what measurable way is the wire different when the magnet is in motion, from what it is when the magnet is still? Does it matter what the wire is made of? This can be made to make sense to everyone. Until it does make sense it is futile to go further into the story of electrons.

        Good teaching will make students look forward to the day when they will have the answers to what is now necessarily obscure.  For the best students, such education is a preparation for research of their own; but all students will benefit by knowing how models are formed, and by knowing that answers don't grow in books. Until the language of scientific hypothesis and falsification replaces the language of scientific truth and verification in the schoolbooks and the newspapers there is little hope that any bridge between the two cultures can be formed, and little hope, too, that any but a very few of us will overcome that handicap, to become useful scientists. The "delay" of scientific education, that teachers fear might result from the systematic introduction of the historical method of teaching science, instead of today's drumfire list of scientific truths, is illusory. It might look like a delay to a ten year old deprived of knowledge of the number of electrons in an atom of copper, but that would be empty knowledge. The speed of understanding of really sophisticated theory at a later stage, that results from a proper understanding of the nature of science, albeit based on experiments made centuries ago, would outweigh this delay a hundred to one.

        To the schoolteachers and the schools, here is the prescription:  Slow it down. This is not the same as "dumb it down," which is the ironic description so true of many recent educational innovations. The cramming of information about E=mc2 and the planetary electron shells of copper is of no value until certain more basic ideas, patiently arrived at, have been found to model a visible reality. All this is not to say that we know nothing, that everything we do proceeds according to a model we may yet find false. Students must understand the convenience of the common understanding too, and that it really is true that putting one's hand in the fire is hurtful, whatever a philosopher might say about models and falsifiability. But the common understanding (according to the evolutionary model of the history of mankind) was formed in our species to apply to common experience, which must be carefully distinguished from what we think we know about thermodynamics and electrons.

        The greatest advantage of a rational, historically based, scientific education would accrue to the most ordinary citizen, the one to whom the detailed understanding of the exact meaning (today) of a statement about the electrons of copper will never be of importance. He would know what science is about, and what it can and cannot say and do. He would end up with a scientific picture of the universe to which he can add for the rest of his life, something that the person with either a list of forgettable facts or a bag of garbled theoretical phrases cannot. It is not possible for every architect, baker and statesman to understand algebraic topology and quantum theory; but it is worse than useless to have them mouth some of the vocabulary of these sciences as if they had learned something, and then argue seriously about whether evolution "has been proved, or is only a theory."  It may be that special talent or interest is necessary to make a child into a scientist, and that even with talent the needed education is too arduous and long for more than a few. What C. P. Snow wanted may not ever be possible, and the view of science available to the average literate person might well be only a small set of two-dimensional views; but every literate person should be able to learn enough of the mode and meaning of science to appreciate the idiocy of the phrase, "only a theory." It must be possible to everyone to understand enough of science as a human activity, and as a philosophic stance, to know that a political debate about evolution is nonsense unworthy of our energies.

© Ralph A. Raimi

12 December 1991, revised 17 June 2004