C.S. Lewis, George MacDonald, and Mathematics

by David L. Neuhouser, Taylor University

The following article first appeared in Wingfold, (Summer 1996, No. 15), a quarterly journal publication celebrating the works of George MacDonald. For Wingfold subscription information write to:  Barbara Amell, ed., 5925 S. E. 40th, Portland, OR 97202.

C.S. Lewis, brilliant scholar, professor at Oxford and later at Cambridge, perhaps the most popular and effective Christian writer of the twentieth century, failed the mathematics portion of the entrance exam at Oxford University, twice! In fact, he never passed it and would never even have been a student at Oxford, if it had not been that World War I intervened. Lewis enlisted in the army and after the war was admitted to Oxford because the entrance exam was waived for veterans. It's a troubling thought that if it hadn't been for a world war, our beloved subject might have kept us from ever enjoying the Space Trilogy, the Narnian Chronicles, and the rest of Lewis' prolific writings.

Lewis often wrote that George MacDonald was his master and one of the greatest influences on his Christian thought. MacDonald was born in Scotland in 1824 and died in 1905 when Lewis was only seven years old. This man, whose books influenced Lewis so much, is quite a contrast to Lewis in many ways. MacDonald was married for 51 years, had eleven children, and suffered from ill health and poverty for much of his life. On the other hand, Lewis was a bachelor for most of his life, was childless, enjoyed robust good health until his last few years, and, although not exactly rich, was always comfortably well off. And, there are other contrasts, but for us, the most interesting contrast is that MacDonald loved mathematics and would never have said, as Lewis did, "I read algebra, devil take it!"

MacDonald loved science as well as mathematics. He taught arithmetic at Aberdeen Central Academy while still a college student and later taught mathematics as well as chemistry at the Ladies College in Manchester. He also taught Euclid to his sons and daughters. One of his majors at Kings College, Aberdeen was chemistry and, on graduation, if he had only had enough money, he would probably have gone to Germany to further study chemistry and medicine. So, the troubling thought in connection with MacDonald is , if it hadn't been for poverty, his love for science and mathematics might have kept us from ever enjoying his delightful fairy tales and insightful novels.

We will return later to consider Lewis' experiences with mathematics and their influence on him and his writing, but first we will take a look at the references to mathematics in the works of MacDonald.

Most of the heroes and heroines in MacDonald's novels study mathematics, especially Euclid. In Warlock O'Glenwarlock, the hero, Cosmo, says, "You see Aggie and I were at school together and we happened to take up with the same kind of thing, particularly algebra and geometry, and can hardly hold our tongues from them whenever we get together." It was this same Aggie who, earlier in their schooldays, asked Cosmo for help with her algebra, saying, "I cannot understand how folks can count with letters and crosses and strokes in place of figures. I have been at it a whole week now - by myself, you know - and I'm no nearer to it yet. I can add and subtract according to the rules given, but that's not understanding." At this point Cosmo understands algebra well enough to help her, but later the teacher says that Aggie understood Euclid and algebra better than any boy, including Cosmo.

Other characters study a proposition of Euclid each night before going to bed or draw geometric diagrams in the sand while studying a proof. Two brothers discuss the "metaphysical necessity for the sine, tangent, and secant of an angle belonging to it's supplement as well" In another novel, the hero is found nearly frozen in a snowstorm and in his delirium is murmuring something about conic sections.

One thirteen year old boy, while on a fishing boat in the middle of a terrible storm, asks the schoolmaster (who just happens to be in the boat also), "Is there a true definition of a straight line, sir? I can't take the one in Euclid." While this suggests that MacDonald may not have understood the necessity of undefined terms, at least he recognized the inadequacy of Euclid's definition of a straight line.

In the Marquis of Lossie, he refers to the "secret of life", actually the Christian Gospel, as "the vital germ of all that is lovely and graceful, harmonious and strong, all without which no poet would sing, no martyr burn, no king rule in righteousness, no geometrician pore over the marvelous must." No matter what he might have meant by all of that, the last phrase shows his love and appreciation for proof in mathematics.

Conic sections, probability in gambling and life insurance problems, ellipsoids, polyhedra, parallelograms, trapeziums, Arabic numerals, and calculus all appear in his novels. For one young shepherd scholar, who was also in love, "There were moments, doubtless long moments too, in which he forgot Homer and Cicero and differential calculus and chemistry, for 'the bonnie lady-lassie,' - that was what he called her to himself; but it was only, on emerging from the reverie, to attack his work with fresh vigour."

It must be remembered that we are here concentrating on his interest in mathematics. It should not be inferred that he was just a frustrated scientist. His main interest was in poetry and literature in general, but he had a keen understanding of, and appreciation for, a liberal education. His comment on one of his fictional heroes, a college student at the time, in comparing him to some of his fellow students who had a much more narrow range of interests, "And already he saw a glimmer here and there in regions of mathematics from which had never fallen a ray into the corner of an eye of those grinding men. This was because he read books of poetry and philosophy of which they had never heard." In another place, MacDonald quotes Coleridge as saying "that no one but a poet will make any further great discoveries in mathematics."

His belief in the value of mathematics is illustrated by his hero in Lilith who in trying to help a community of good but naive children says, "Knowledge no doubt made bad people worse, but it must make good people better! I was convinced they would learn mathematics."

One of MacDonald's emphases about the Christian life is that only through obedience to what we know will we gain a deeper understanding of God. He illustrates this in several places by pointing out that in learning mathematics we often need to actually obey some rule, performing the operations without understanding it, but that by doing that we will be able to understand.

I think that his best example of this is in Robert Falconer. Robert describes his experience this way to a friend. The schoolmaster told me "if you must always understand before you do as you're told, you'll never understand anything. But if you do the thing I tell you, you'll find yourself understanding it as you're doing it. I just thought I'd try him. It was long division that I boggled at most,. Well, I went on, and I could do the thing well enough, without making one mistake. And I thought the master was wrong, for I never knew the reason of all that beginning at the wrong end, and taking down, and subtracting, and all that. You would hardly believe me, Mr. Ericson; it was this very day, as I was sitting in the church - it was a long psalm they were singing, - long division came into my head again; and first one little glimmering of light came in, and then another, and before the psalm was done I saw through the whole process of it. But you see, if I hadn't done as I was told, and learned all about how it was done beforehand, I would have found out nothing."

MacDonald believed that it was very important to understand, but that sometimes the road to understanding was obedience. His belief in the importance of understanding is shown by his elderly hero David Elginbrod asking a young college graduate, "Do you think, Mr. Sutherland, I could do any thing at my age in mathematics? I understand well enough how to measure land and that kind of thing. I just follow the rule. But the rule itself is a puzzler to me. I don't understand it by half. Now it seems to me that the best of a rule is not to make you able to do a thing, but to lead you to what makes the rule right - to the principle of the thing. It's not that I'm misbelieving the rule, but I want to see the rights of it."

In Lilith, a phantasy for adults, MacDonald uses the idea of higher dimensions to create his phantasy world. However, Lewis uses higher dimensions to a better purpose, so we will return to this concept later.

One of the strange things about MacDonald's love for mathematics and Lewis' distaste for and ineptitude in, at least some parts of, mathematics, is that Lewis emphasizes the rational approach to faith more than MacDonald does. MacDonald is more mystical, while Lewis is more logical. In order to understand this we need to take a look at Lewis' early experiences with mathematics.

Although Lewis was not good at mathematics, I am only aware of one negative reference to it in his books. In Voyage of the Dawn Treader, a government official tries to defend slavery to Prince Caspian by saying "Your Majesty's tender years,... hardly make it possible that you should understand the economic problem involved. I have statistics, I have graphs, I have - "

In a letter to a young fan, Lewis wrote, "I am also bad at Maths and it is a continual nuisance to me - I get muddled over my change in shops. I hope you'll have better luck and get over the difficulty! It makes life a lot easier." And in another letter, he said. "I wish I was good at Maths!"

It is thought by some that Lewis hated science and mathematics. Yet, it seems to me, and I hope to show in this paper, that he really understood the nature of mathematics pretty well and used mathematics effectively in his apologetic works. What he did not like was scientism, (the belief that scientific methods can be applied in all fields of investigation) and, for the most part, he did not like technology.

That Hideous Strength is often given as an example of his antagonism toward science. In it the villains are the officers of the N.I.C.E. (National Institute for Coordinating Experiments), ostensibly a scientific institute. However, they are really mostly bureaucrats, not scientists. The one physicist they try to recruit, unsuccessfully, is really an admirable character. (So good, in fact, that the villains have to murder him.) Lewis himself says that the book is anti-bureaucracy, not anti-science and the point is not that scientists will try to take over the world but that if anyone does try, it will be in the guise of science.

In Surprised by Joy, Lewis describes his experiences in a boarding school which he hated. Its cruel headmaster, called Oldie by the students, "decided that <the students> could, with less trouble to himself be made to do arithmetic. Accordingly, when you entered school at nine o'clock you took up your slate and began doing sums. Presently you were called up to 'say a lesson.' When that was finished you went back to your place and did more sums - and so on forever. All the other arts and sciences thus appeared as islands... - the deep being a shoreless ocean of arithmetic. At the end of the morning you had to say how many sums you had done and it was not quite safe to lie." It is not surprising that with this kind of a regimen, with the constant threat of beatings, that Lewis did not learn, and, in fact, hated elementary mathematics.

The other part of the picture is given as Lewis reports, "I can also say that though <Oldie> taught geometry cruelly, he taught it well. He forced us to reason, and I have been the better for those geometry lessons all my life." Later, in Surprised by Joy, Lewis wrote, "I could never have gone far in any science because on the path of every science the lion Mathematics lies in wait for you. Even in Mathematics, whatever could have been done by mere reasoning (as in simple geometry) I did with delight; but the moment calculation came in I was helpless. I grasped the principles but my answers were always wrong. Yet though I could never have been a scientist, I had scientific as well as imaginative impulses, and I loved ratiocination." Notice that Lewis contrasted science and imagination whereas MacDonald combined mathematics and imagination.

A revealing statement on his view of mathematics and of technology is given in The Four Loves. "Egyptian and Babylonian Mathematics were practical and social, pursued in the service of Agriculture and Magic. But the Free Greek Mathematics, pursued by Friends as a leisure occupation have mattered to us more." I believe that his view of mathematics was Platonic and that axioms were self-evident truths. He once wrote, "Human intellect is incurably abstract. Pure mathematics is the type of successful thought."

In Mere Christianity, we find his views on the relationship between mathematics and science. "Many of you no doubt have read Jeans or Eddington. What they do when they want to explain the atom, or something of that sort, is to give you a description out of which you can make a mental picture. But then they warn you that this picture is not what the scientists actually believe. What the scientists believe is a mathematical formula. The pictures are there only to help you to understand the formula. They are not really true in the way the formula is; they do not give you the real thing but only something more or less like it. They are only meant to help, and if they do not help you can drop them. The thing itself cannot be pictured, it can only be expressed mathematically." He used this as an analogy to understand the differences between theological theories and Christianity itself.

An illustration of his grasp of mathematical principle to aid a theological argument is given in an essay on "Modern Theology and Biblical Criticism." To show the weakness of theologians' arguments based on more than one hypothesis, he writes, "Now let us suppose - what I am far from granting - that the first hypothesis has a probability of 90 per cent. Let us assume that the second hypothesis also has a probability of 90 per cent. But the two together don't still have 90 per cent; for the second comes only on the assumption of the first. You have not A plus B; you have a complex AB and the mathematicians tell me that AB has only 81 percent probability. I'm not good enough at arithmetic to work it out, but you see that if, in a complex reconstruction, you go on thus superinducing hypothesis on hypothesis, you will in the end get a complex in which, though each hypotheses by itself has a high probability, the whole has almost none."

Lewis read and enjoyed Flatland by Edwin A. Abbott, and he used the idea in his fiction and non-fiction. In Perelandra he suggests that the appearance of the Oyarsa (angels) was merely the three dimensional cross sections of four dimensional beings.

In Mere Christianity, he uses two and three dimensions to aid in our understanding of the trinity. "On the human level one person is one being, and any two persons are two separate beings - just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we who do not live on that level, cannot imagine. In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course, we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint of something super-personal - something more than a person."

Elsewhere, Lewis writes, "At this point we must remind ourselves that Christian theology does not believe God to be a person. It believes Him to be such that in Him a trinity of persons is consistent with a unity of Deity. In that sense it believes Him to be something very different from a person, just as a cube, in which six squares are consistent with unity of the body, is different from a square. (Flatlanders, attempting to imagine a cube, would either imagine the six squares coinciding, and thus destroy their distinctness, or else imagine them set out side by side and thus destroy the unity. Our difficulties about the Trinity are much of the same kind.)"

In another essay he writes, "To explain even an atom Schrodinger wants seven dimensions: and give us new senses and we should find a new Nature. There may be Natures piled upon Natures, each supernatural to the one beneath it, before we come to the abyss of pure spirit; and to be in that abyss, at the right hand of the Father, may not mean being absent from any of these Natures - may mean a yet more dynamic presence on all levels." In reply to Dr. Pittenger who criticized him for this use of geometry in theology, Lewis wrote, "I do not understand what is vulgar or offensive, in speaking of the Holy Trinity, to illustrate from plane and solid geometry the conception that what is self-contradictory on one level may be consistent on another. I could have understood the Doctor's being shocked if I had compared God to an unjust judge or Christ to a thief in the night; but mathematical objects seem to me as free from sordid associations as any the mind can entertain."

I believe that Lewis' most extensive and most profound use of higher dimensions occurs in the sermon "Transposition". In it he points out that whenever something in a richer medium is described in a lower medium there can not be a one to one correspondence. For example, a cube represented in a plane must use parallelograms to represent squares while elsewhere parallelograms to represent parallelograms. In a similar way, if we try to explain a spiritual experience we must use terms which a good psychologist would recognize as psychological experiences with meanings in other settings. In fact, after trying to explain a cube to a Flatlander, the Flatlander would probably respond, 'You keep on telling me of this other world and its unimaginable shapes which you call solid. But isn't it very suspicious that all the shapes you offer me as images or reflections of the solid ones turn out on inspection to be simply the old two-dimensional shapes of my own world as I have always known it? Is it not obvious that your vaunted other world, so far from being the archetype, is a dream which borrows all its elements from this one?" Although we can understand the difficulty the Flatlander has. How else could one try to explain a three-dimensional object to a Flatlander except by using concepts from his world? And, as three-dimensional beings we know that his argument does not destroy the reality of three dimensions. In the same way, the psychologist's argument does not destroy the possibility of a spiritual reality.

He also understands something about the teaching and learning of mathematics. In Mere Christianity, in the chapter "Is Christianity Hard or Easy?", he uses the learning of geometry to illustrate his point. "Teachers will tell you that the laziest boy in the class is the one who works the hardest in the end. They mean this. If you give two boys, say, a proposition in geometry to do, the one who is prepared to take trouble will try to understand it. The lazy boy will try to learn it by heart because, for the moment, that needs less effort. But six months later, when they are preparing for an exam, that lazy boy is doing hours and hours of miserable drudgery over things the other boy understands, and positively enjoys, in a few minutes. Laziness means more work in the long run."

In a letter to a school boy, Lewis comments on a teaching problem that all of us who have ever taught freshman calculus can identify with. "Beware of the Maths. master who over-marks the work. Generous marking is nice at the moment, but it can lead to disappointments when, later, one comes up against the real thing. American university teachers have told me that most of their freshman come from schools where the standard was far too low and therefore think themselves far better than they really are. This means that they lose heart (and their tempers too) when told, as they have to be told, their real level."

So, although Lewis and MacDonald had very different experiences with and attitudes toward mathematics they both used ideas and concepts from mathematics to enrich their writings. I would like to conclude with a passage from MacDonald which shows how much he loved mathematics and how his enjoyment of it led to the praise of God. A character in the novel The Elect Lady, wonders how people can be so indifferent to God and then realizes that at times, he himself, is guilty of the same thing. He begins with a series of rhetorical questions to himself, "Do I meet God in my geometry? When I so much enjoy my Euclid, is it always God geometrizing to me? Do I feel like talking with God every time I dwell upon any fact of his world of lines and circles and angles? Is it God with me, every time that the joy of life, of a wind or a sky or a lovely phrase, flashes through me? - 'Oh my God,' he broke out in speechless prayer as he walked... 'Oh my God, thou art all in all, and I have everything! The world is mine because it is thine! I thank thee my God, that thou hast lifted me up to see whence I came, to know who is my Father, and makes me his heir! I am thine, infinitely more than my own; and thou art mine as thou art Christ's!'"

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