Faculty Forum Weblog

Oxbridge 2005 Faculty Forum Academic Conference

Through its Faculty Forums (2001, 2002, 2003) and Summer Institute – Oxbridge 2002, the C.S. Lewis Foundation has been pleased to host an international Academic Conference of increasing stature. The conference serves as a focal point of the Foundation's mission to advance the renewal of Christian thought and creative expression throughout the world of learning and the culture at large by affording faculty members and independent scholars the opportunity to read and discuss papers addressing the conference theme from the vantage point of all disciplines - the natural, social, and behavioral sciences, the traditional humanities and the professions.

The Academic Conference, featuring the presentation of juried scholarly papers by participating faculty, ran simultaneously with the Institute's afternoon seminars and workshops. View the online handbook at http://www.cslewis.org/programs/oxbridge/2005/academicconference2005.pdf

Review two featured papers below:

C. S. Lewis and Mathematics

David L. Neuhouser
Prof. Emeritus of Mathematics
Taylor University
Upland, Indiana
July 28, 2005

C. S. Lewis has a reputation for hating science and mathematics. This paper will attempt to show that this reputation is undeserved; show what his true opinions about mathematics and science were, and how he got the reputation in the first place. Science is considered along with mathematics because of their close relationship to each other, particularly in Lewis’ thought. This paper will not include Lewis’s use of higher dimensions in mathematics in over a dozen of his books, although his use of this topic shows his understanding and appreciation of mathematics, because I have already written about it in other places.

]]> First of all, Lewis did have great difficulty with mathematics. After receiving a scholarship to University College, Oxford University, he twice failed the mathematics section of the entrance examination. It is probable that he would never have been admitted to Oxford except that the entrance exam was waived for veterans of WWI. He failed this exam because he had trouble with arithmetical and algebraic manipulation. In Surprised by Joy, he said, I could never have gone very far in any science because on the path of every science the lion Mathematics lies in wait for you. Even in Mathematics, whatever could be 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 never could have been a scientist, I had scientific as well as imaginative impulses, and I loved ratiocination. (p.137). A diary entry when he was twenty-three years old reads, “As an intellectual nightcap have been puzzling over our accounts, which come out to a different figure each time.” (All My Road Before Me, p. 18). Although bad at calculation, he appreciated the benefits of mathematics. As he wrote to a young reader, “I wish I was good at Maths.” And to another, “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 a better luck and get over the difficulty! It makes life a lot easier.” (C. S. Lewis: Letters to Children, p. 67 and 75).

Lewis attended Wynyard School, a preparatory school that he referred to as Belson and the cruel headmaster as Oldie. “Except at geometry (which he really liked) it might be said that Oldie did not teach at all.” (Surprised by Joy, p. 27-8). One of Oldie’s idiosyncrasies was to make the students do addition problems for long periods of time and then ask how many they had done. Lewis’s brother Warnie solved the problem by doing the same sums each day. Evidently, Jack was too conscientious or too fearful to follow his brother’s example. It is not surprising that with this kind of a regimen, with the constant threat of beatings, that Lewis did not learn, and even hated, elementary mathematics. He later had a much better teacher, William Kirkpatrick although according to Lewis’ biographer and friend, George Sayer, “Kirkpatrick was not good at teaching math and had no insight into Jack’s problems with math, and, rather than sitting by his pupil’s side and working through his problems, he allowed Jack to do too many other things.” (Sayers, p. 116). So at least two of his teachers were not good at teaching elementary mathematics. All educators know how important the quality of early teaching in mathematics is.

However, this is not the whole story. Lewis said, “I can also say that though he [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.” (Surprised by Joy, p 29). Also, Kirkpatrick taught logic well. This proficiency and love of the use of logic in geometry certainly aided Lewis in his apologetic works.

Mathematics is not just calculation and manipulation. In that part of mathematics Lewis had great difficulty all his life. At a higher level, mathematics is governed by reason and imagination, areas in which Lewis excelled. Thus, Lewis was a better mathematician than he has been given credit for. It is possible to be weak in beginning mathematics and still be good at more abstract mathematics. For example, one of the greatest mathematicians of all time, Evariste Galois, twice failed the mathematics part of the entrance examination to the Ecole Polytechnique. I am not at all suggesting that Lewis could have been a great mathematician, but he definitely had more mathematical ability than has been thought.

There are statements that indicate he had a dislike for mathematics. For example he did say, “I read algebra (devil take it)…” (Surprised By Joy, p. 187). And, “I shudder at the subjects you have to take in High School, and some of them I could not even begin to attempt – Algebra and Calculus for example.” (C. S. Lewis: Letters to Children, p. 106). On the other hand, this diary entry, “After supper I read Anthony and Cleopatra – the most intelligible play in the world – clear through like a theorem – and lovely.” (All My Road Before Me, p. 242) show his appreciation of the beauty of a mathematical theorem or proof. Two more quotations show his appreciation for the applicability of mathematics. ““I said this left out the fact that mathematics did conform to experience wherever the nature of the case allowed them to touch it and that this was in fact the reason why we called them true.” (All My Road Before Me, p. 300). “We have recently been reminded how much mathematics, and how good, went to the building of the [Copernican] Model.” (The Discarded Image, p. 103).

He also had an appreciation for mathematicians starting with his mother who “had been a promising mathematician in her youth and a B. A. of Queens College, Belfast.” (Surprised By Joy, p. 4). I believe this passage from Mere Christianity shows his appreciation of mathematicians,

Good tennis players “have a certain tone or quality which is there even when he is not playing, just as a mathematician’s mind has a certain habit and outlook which is there even when he is not doing mathematics. In the same way a man who perseveres in doing just actions gets in the end a certain quality of character. Now it is that quality rather than the particular actions which we mean when we talk of a ‘virtue.’” (p. 80).

This quality of mind that the mathematician shares with some scientists is commented on in an essay in God in the Dock:

That the continued application of scientific methods breeds a temper of mind unfavourable to the miraculous, may well be the case, but even here there would seem to be some difference among the sciences. Certainly, if we think, not of the miraculous in particular, but of religion in general there is such a difference. Mathematicians, astronomers and physicists are often religious, even mystical; biologists much less often; economists and psychologists very seldom indeed. It is as their subject matter comes nearer to man himself that their anti-religious bias hardens. (p. 135).

That he understood the mathematical method is shown by the following statement:

Unless you accept these [principles from the Tao] without question as being to the world of action what axioms are to the world of theory, you can have no practical principles whatever. You cannot reach them as conclusions: they are premises. … If nothing is self-evident, nothing can be proved. Similarly if nothing is obligatory for its own sake, nothing is obligatory at all. (The Abolition of Man, p. 40).

His appreciation for pure mathematics is shown in the following statements. "Human intellect is incurably abstract. Pure mathematics is the type of successful thought.” (God in the Dock, p. 65). “Egyptian and Babylonian Mathematics were practical and social, pursued in the service of Agriculture and Magic. But the free Greek Mathematics mattered to us more.” (The Four Loves, p. 69).

Another aspect of mathematics that Lewis appreciated was its permanence. In this regard, he compares mathematics to morality and religion.

Wherever there is real progress in knowledge, there is some knowledge that is not superseded. Indeed the very possibility of progress demands that there should be an unchanging element…. I take it we should all agree to find this sort of unchanging element in the simple rules of mathematics. I would add to these the primary principles of morality…. As regards material reality, we are now being forced to the conclusion that we know nothing about it save its mathematics…. Like mathematics, religion can grow from within, or decay…. But, like mathematics, it remains simply itself, capable of being applied to any new theory of the material universe and outmoded by none.” (God in the Dock, p. 45-7).

Many have accused Lewis of being opposed to science. In reply to J. B. S. Haldane’s charge that That Hideous Strength showed anti science bias, Lewis pointed out that the only good character and the only natural scientist in the evil National Institute for Coordinated Experiments (N.I.C.E.) is a physical chemist.

The good scientist is put in precisely to show that ‘scientists’ as such are not the target. To make the point clearer, he leaves my N.I.C.E. because he finds he was wrong in his original belief that’ it had something to do with science.’ (p. 83). To make it clearer yet, my principal character, the man almost irresistibly attracted by the N.I.C.E. is described (p. 226) as one whose ‘education had been neither scientific nor classical – merely ‘Modern’… Lest even this should not be enough, the hero…is made to say that the sciences are ‘good and innocent in themselves; (p. 248) though evil ‘scientism’ is creeping into them… If anyone ought to feel himself libeled by this book it is not the scientist but the civil servant. (Of Other Worlds, p. 78).

Lewis goes on to say that if any one now attempts to take over the world he would do it under the guise of science. In other words science would be used to disguise their real intent. “Technocracy is the form to which a planned society must tend. Now I dread specialists in power because they are specialists speaking outside their special subjects. Let scientists tell us about sciences.” (God in the Dock, p. 315). Lewis then acknowledges that it is another of his books that could more easily be misunderstood to show anti science bias but he explains what it is really against.

If any of my romances could be plausibly accused of being a libel on scientists it would be Out of the Silent Planet. It certainly is an attack, if not on scientists, yet on something which might be called ‘scientism’ - a certain outlook on the world which is causally connected with the popularization of the sciences, though it is much less common among real scientists than among their readers. It is, in a word, the belief that the supreme moral end is the perpetuation of our own species, and that this is to be pursued even if, in the process of being fitted for survival, our species has to be stripped of all those things for which we value it – of pity, of happiness, and of freedom. (Of Other Worlds, p. 76-77).

Here the charge of an anti technology bias on Lewis’ part could be defended but even here we must be careful to understand his view. He did say, “Their labour-saving devices multiply drudgery; … And as for permanence – consider how quickly all machines are broken and obliterated”. (The Pilgrim’s Regress, p.187). He preferred buttons to zippers because a button could be easily replaced while a zipper could not. He never learned to type, in fact, wrote with a dip pen, several levels behind the then current level of writing technology. However, “’I agree Technology is per se neutral,’ Lewis wrote to Arthur C. Clarke in 1943: ‘but a race devoted to the increase of its own power by technology with complete indifference to ethics does seem to me a cancer in the Universe.’” (Aeschliman, p. 61). Also, he comments on “the advance, and increasing application, of science. As a means to the ends I care for, this is neutral. We shall grow able to cure, and to produce, more diseases, to alleviate, and to inflict, more pains, to husband, or to waste, the resources of the planet more extensively.” (God in the Dock, p. 312).

Lewis was aware of a tendency to apply science in ways that caused more problems that they solved. Today consider the problem of what to do with radioactive wastes. He even compared applied science to magic.

There is something which unites magic and applied science while separating both from the ‘wisdom’ of earlier ages. For the wise men of old the cardinal problem had been how to conform the soul to reality, and the solution had been knowledge, self-discipline, and virtue. For magic and applied science alike the problem is how to subdue reality to the wishes of men: the solution is a technique; and both, in the practice of this technique, are ready to do things hitherto regarded as disgusting and impious such as digging up and mutilating the dead. (The Abolition of Man, p. 77).

He even went so far as to say. “The evil reality of lawless applied science (which is Magic’s son and heir) is actually reducing large tracts of Nature to disorder and sterility at this very moment.” (Miracles, p. 179.

Lewis understood that the theories, models, or pictures that science uses are not reality but only useful to give us some glimpse or understanding of reality but that behind it all is mathematics. “If you start investigating the nature of matter, you will not find anything like what imagination has always supposed matter to be. You will get mathematics.” (Letters to Malcolm, p. 107). “The walls, they say, are matter. That is, as the physicists will try to tell me, something totally unimaginable, only mathematically describable, existing in a curved space, charge with appalling energies. If I could penetrate far enough into that mystery I should perhaps finally reach what is sheerly real.” (Letters to Malcolm, p. 104-5)

In discussing theories of the atonement he compares them to scientific theories.:

What [scientists] 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 that 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. (Mere Christianity, p. 54).

Earlier it was noted that Lewis believed that mathematicians and physicists were more likely to be religious that most other scholars. He even has his fictitious devil, Screwtape; advise another devil of the “danger” of science when tempting his human patient.

Above all, do not attempt to use science (I mean, the real sciences) as a defence against Christianity. They will positively encourage him to think about realities he can’t touch ands see. There have been sad cases among physicists. If he must dabble in science, keep him on economics and sociology; don’t let him get away from that invaluable ‘real life’. But the best of all is to let him read no science but to give him a grand general idea that he knows it all and that everything he happens to have picked up in casual talk and reading is ‘the results of modern investigation.’ (Screwtape Letters, p.14.

To summarize: Lewis had difficulty with and disliked arithmetic and elementary algebra but loved the logical structure of geometry. He appreciated the beauty of mathematics and understood the nature of higher mathematics and its relation to reality. He had realistic doubts about technology and scientism or the belief that the perpetuation of our species was the highest value. He used mathematics and science to illustrate spiritual ideas. Also, he appreciated the value of the disciplines of mathematics and science. Remember that this paper does not show his extensive use of space dimensions in his fiction and in his Christian writings. Therefore, to say that Lewis was poor at mathematics and simply leave it at that is to fail to appreciate his highly creative literary use of higher mathematics and his use to illustrate Christian principles.

I would like to end this paper on a lighter note concerning Lewis’ understanding of mathematics education, even in the United States. In Mere Christianity, in the chapter “Is Christianity Hard or Easy” he wrote,

Teachers will tell you that the laziest boy in the class is the one who works 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. (p. 197).

Then in a letter to one of his young American readers,
But beware of the Maths. master who over-marks the work. Generous marking is nice for 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 freshmen 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. (C. S. Lewis: Letters to Children, p. 83-4.

Does Mathematical Beauty Pose Problems for Naturalism?

Russell W. Howell
Prof. of Mathematics
Westmont College
Santa Barbara, California
July 28, 2005

The theme of this conference is Making All Things New: The Good, the True, and the Beautiful in the 21st Century. This paper will focus on features of truth and beauty contained in mathematics. More precisely, it asks whether aspects of mathematical theorizing, based mostly on notions of beauty and symmetry, and the subsequent success of mathematics in the natural sciences, cause difficulties for a naturalistic worldview. Several thinkers have raised these issues, at least indirectly, though not so much from the standpoint of mathematical beauty. We begin by reviewing some of their contributions.

]]> In 1960 the physicist Eugene Wigner published “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” in Communications in Pure and Applied Mathematics. He begins his paper with a story about two friends talking about their jobs. One of them, a statistician, was working on population trends. He showed a paper to his friend. It started, as usual, with the Gaussian distribution, and the statistician explained the meaning of the symbols. His friend was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?”
“Oh,” said the statistician, “this is pi.”
“What is that?”
“The ratio of the circumference of a circle to its diameter.”
“Well, now you are pushing your joke too far. Surely the population has nothing to do with the circumference of the circle.”
Wigner uses that story to introduce two issues: (1) the surprising phenomenon that we have used mathematics so often to build successful theories; (2) the nagging question, “How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?”
Regarding Wigner’s first point, he concedes that much of mathematics, such as Euclidean Geometry, has been developed because its axioms were modeled on what appeared to be true of the world. But this is not true for all—in fact most—of higher mathematics. Take my field, complex analysis, as just one example. It deals with “imaginary numbers”—things like the square root of minus one. In the 1500’s such notions seemed odd to mathematicians because even negative numbers at that time were treated with some suspicion. There simply did not seem to be any physical reality corresponding to them, let alone their square roots. But that didn’t stop mathematicians from using their imagination and pressing forward. The process that began the acceptance of complex numbers can legitimately be placed in the mid-fourteenth century when Scipione del Ferro of Bologna, and then later Niccolo Fontana solved the depressed cubic equation, which was later extended by Girolamo Cardano to the solution of the general cubic equation. Real-valued solutions to some cubic equations were then obtained by using these methods, but their solutions only came by using complex numbers as an intermediate step. The story that details the entire development of complex numbers is quite intricate, and it wasn’t until the end of the 19th century that complex numbers became firmly entrenched. It is important to note, however, that complex numbers were studied because they were useful for mathematical and not physical purposes.
But complex numbers now play a pivotal role in helping physicists understand the quantum world. According to Wigner,
Quantum mechanics originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices. … Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. … The results were quite satisfactory. However, there was … no rational evidence that their matrix mechanics would prove correct under more realistic conditions. As a matter of fact, the first application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was … understandable because Heisenberg’s rules of calculation were abstracted from problems which included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics … was applied to problems for which Heisenberg’s calculating rules were meaningless. Heisenberg’s rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg’s rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, … [agreed] with the experimental data within the accuracy of the observations, which is one part in ten million.
“Surely,” Wigner concludes, “in this case we ‘got something out’ of the equations that we did not put in.”
I will not elaborate on the detail with which Wigner cites other examples including: Newton’s law of motion—formulated in terms that appear simple to mathematicians, but which proved to be accurate beyond all reasonable expectations; quantum electrodynamics; or the theory of the Lamb shift—a purely mathematical theory.
Wigner ends his paper with the remarks,
… The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious, and … there is no rational explanation for it. … The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
In 1980 R.W. (Richard Wesley) Hamming took up the effectiveness issue raised by Wigner, and offered four tentative explanations that account for the applicability of mathematics. Let me review them briefly.
First, we see what we look for. Mathematicians craft postulates so that they will produce theories that conform to their prior observations. The Pythagorean theorem, Hamming claims, drove the formation of the geometric postulates and not vice versa.
Second, we select the kind of mathematics to use. By this Hamming simply means that we select the mathematics to fit the situation. The same mathematics simply does not work everywhere. Because we force mathematics onto particular situations, it is not all that surprising that we subsequently find it applicable.
Hamming’s third comment is that science in fact answers comparatively few problems. To the extent that this is true, the less of a miracle the success of mathematics would appear to be. Wigner, as a physicist, certainly lived with mathematics as an indispensable tool. But other sciences do not share their reliance on mathematics, at least to the extent that physics does. Biology, it is said, has not been successfully dissected by the mathematical scalpel.
It seems to me, parenthetically, that this position is not obviously correct. A great deal of mathematical effort has been focused as of late on biological questions. A colleague of mine, for example, is currently looking at a Coxeter Groups as a model for DNA similarity. Also, knot theory, a newer branch of mathematics that deals with topological invariants, has had some success in the classification of DNA strands according to how they crinkle up under certain conditions. Even if we grant the argument, however, the success of mathematics in physics is something that cannot simply be dismissed by pointing to slower progress in other areas.
Finally, Hamming posits that the evolution of man provided the model, meaning the model for why humans are able to mathematize the physical universe. This is an interesting claim, but is not fleshed out beyond Hamming’s remark that, “Darwinian evolution would naturally select for survival those competing forms of life which had the best models of reality in their minds—‘best’ meaning best for surviving and propagating.” It is interesting to note that Hamming concludes with,
If you recall that modern science is only about 400 years old, and that there have been from 3 to 5 generations per century, then there have been at most 20 generations since Newton and Galileo. If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics.
I do not find this refutation compelling. Just as an inclined block needs a critical slope to overcome its friction and begin sliding, and once the sliding starts it proceeds rather rapidly, so too one might argue that, once science started it progressed quickly, but the evolutionary development that occurred before this explosion cannot be discounted.
But evolutionary accounts have problems as well. Let’s briefly look at three such explanations. The first can be called the sexual selection hypothesis as argued by Geoffrey Miller. He claims that excessive capacities or acquisition of resources of any kind is basically a sexual display. If you’ve got the energy or time or intrinsic capacity to do things that don’t have direct adaptive value—carrying around a set of antlers that are so big they are more of a detriment than a defense, or a peacock walking around with a big colored tail, or possessing artistic or mathematical brains that don’t contribute to reproductive success—then that energy or time or intrinsic capacity by itself attracts mates.
Of course, physical attributes may well have some role in mate attraction, and artistic brains may as well insofar as they enable people to make attractive artifacts for display. The argument for mathematical brains, however, does not seem to hold up as well. Miller has some ways of dealing with this problem. For example, he states, “The healthy brain theory suggests that our brains are different from those of other apes not because extravagantly large brains helped us to survive or to raise offspring, but because such brains are simply better advertisements of how good our genes are. The more complicated the brain, the easier it is to mess up.” But how would a larger brain be evident, and how would one somehow deduce that this is evidence of good genes? Such speculation seems to be forcing a theory when there may be no good evidence to support it.
Next is what we might call the module approach as argued by Stephen Mithin. Mithin writes from the perspective of an anthropologist, and has an enormous amount of archaeological data on which to draw. His thinking is that integrative and higher level (meta) cognitive processes grew out of the unification of specific evolutionary modules such as a module for tool use, or a module for interpersonal relations. He further argues that only in humans do we find a structure on top of modules—call it general purpose rationality.
This last approach has been extensively debated. For example, Alvin Plantinga’s “Evolutionary Argument against Naturalism” claims that rationality is very unlikely a quality produced by survivability. Plantinga’s approach, as he himself acknowledges, is similar to that found in C.S. Lewis’s Miracles. Lewis’s argument, incidentally, was recently enhanced by Victor Reppert in his book C.S. Lewis’s Dangerous Idea. The thrust of Lewis’s and Reppert’s thinking is that you cannot get rationality out of a causally closed system that works solely on the basis of physical interactions operating in accordance with the laws of nature.
This brings us to the byproduct hypothesis, as exemplified by Pascal Boyer, who argues against Lewis’s and Reppert’s view. His main thesis is that many higher cognitive functions (mathematics, art, religion, ethics, etc.) are not evolutionary adaptations at all. Instead, they are byproducts of things that are adaptive, and just piggyback on the adaptiveness of these other capacities. Some form of mathematical or quantitative ability is adaptive, Boyer argues, and as a byproduct of this we get the capacity to do higher order mathematics, the naked capacity of which at the time of its development wouldn’t have been adaptive (or evolution wouldn’t have known it was adaptive), though it may have turned out to have been adaptive.
But I can’t find any compelling evidence that would support Boyer in his contention, try as he might to produce one. His claim reminds me of scaffolding theories that are used to refute “Intelligent Design” arguments. If one is going to argue against something using an evolutionary framework, it behooves that person to supply a detailed model or story that will support such a refutation. Otherwise, the “God of the gaps” charge normally levied against design theorists can be turned around into, if you will, a “natural selection of the gaps” counter charge against the person arguing for blind chance natural selection.
Perhaps, though, evolutionary theory will eventually come up with a plausible explanation of our rationality. If so, any such theory that also attempts to promote a naturalistic world view would still run up against the arguments of Mark Steiner, author of The Applicability of Mathematics as a Philosophical Problem. Strictly speaking, Steiner’s argument attempts to refute “Anthropocentrism” rather than Naturalism. But if Steiner is correct the naturalist should not take comfort. As far as I can tell, and Steiner shares this opinion, any form of Naturalism is defacto non-anthropocentric in that it would disallow a privileged status for humans in the scope of the universe. If, as Steiner argues, the success of mathematics can be shown to put humans in such a position, then naturalism has problems.
And just how does the success of mathematics put humans in a privileged position? For Steiner, it is not so much the success of any one particular mathematical theory in an area of science. After all, there have been many, many failures of mathematics in addition to its successes, and in this respect Steiner agrees with Hamming’s third point and is thus critical of Wigner’s approach in citing specific success examples from physics while ignoring error stories. The use of pi by the statistician in Wigner’s opening line ignores all the failures, for example, in attempting to predict population trends. What Steiner is talking about is the success of mathematics as a grand strategy. It is a strategy that takes, for example, the raw formalisms of complex Hilbert space theory and boldly uses them as tools to make predictions about the quantum world, predictions that subsequently seem to be born out via experiment. And how is this phenomenon anthropocentric? Let me give an analogy. Most cultures use a base ten number system. No one is 100 per cent sure why this is the case, but the general consensus is that it has to do with our having 10 fingers. (Some primitive cultures use base 20, and to many this confirms the appendage hypothesis.) Now, what if successful theories of how the universe operates were based on multiples of 10? That would be anthropocentric in an extreme, as the only reason the number 10 is special to us is due to how we appear to ourselves.
Now suppose that, not only did the number 10 have special significance, but time and time again human aesthetic criteria played a significant role in understanding the universe. Such occurrences, when looked at from a meta-level, would surely make one wonder why such privilege seems to fall on the human species. Yet this situation is precisely analogous to what mathematicians and scientists actually do when they rely on human notions of beauty and symmetry in the development of their theories.
In fact, such activity has been a long standing and consistent strategy. Galileo, for example, pursued this tactic even though the best empirical evidence at the time did not support—indeed, it tended to disconfirm—his heliocentric theory. He adopted it because it seemed much more elegant than the Ptolemaic model. Most physicists generally admit that elegance, beauty, and symmetry hold primary sway in theory development. As Brian Green says in The Elegant Universe, “Physicists, as we have discussed, tend to elevate symmetry principles to a place of prominence by putting them squarely on the pedestal of explanation.” G. H. Hardy argues that mathematics itself, at least what constitutes good mathematics, is driven primarily by aesthetic criteria such as economy of expression, depth, unexpectedness, and seriousness, qualities that also seem to form standards for good poetry. The theories in mathematics that Hardy deems “important” are precisely the ones that satisfy these standards.
Regarding aesthetics, Steiner’s book contains several examples of beautiful mathematical systems being used in applications to the physical world, including the use of complex analysis in fluid dynamics, relativistic field theory, and thermodynamics. Let’s quickly examine two additional instances.
First, consider Schroedinger’s use of the wave equation. He begins with the equation , where he makes an assumption that energy is constant so he can eliminate it by differentiating. After a series of manipulations he gets , and then successfully uses his solution in situations where Energy is not constant. As Steiner says, this is “a perfect example of allowing the notation to lead us by the nose.”
In the interest of time, I will skip Steiner’s powerful examples from Quantum Mechanics (already alluded to when citing Wigner), where the tinkering of raw mathematical formalisms has often led to predictions about the quantum world that have subsequently panned out. Instead, I will focus on what at least one person considers to be a weak argument of Steiner’s: Maxwell’s anticipation of a physical reality. As you may know, Maxwell noted that the experimentally confirmed laws of Faraday, Coulomb, and Ampere, when put in differential form, contradicted the conservation of electrical charge. By working with Ampere’s law and adding a term to it, Maxwell got the laws to be consistent with, and indeed to imply, the conservation of charge. With no other warrant, Maxwell made the indicated changes and baldly predicted that his new term corresponded to some physical phenomenon. Ten years after his death Heinrich Hertz demonstrated the reality corresponding to this term—electromagnetic radiation.
Richard Carrier, a freelance writer who received his M.Phil. in Ancient History from Columbia University, is unimpressed by this episode, saying that what Maxwell did is entirely consistent with Naturalism. First, Maxwell’s putting laws in differential form conforms to the naturalistic observation that nature works in continuous, not broken, processes. Second, Maxwell took a logically sound hypothetical step: if charge isn’t being conserved, then it must be going somewhere. Carrier then states, “Maxwell rightly picked the simplest imaginable solution first, which due to human limitation is always the best place to start an investigation, and which statistically is the most likely [as] simple patterns and behaviors happen far more often than complex ones. [Thus] Maxwell’s moves [that] anticipated EM radiation [were] therefore a natural conclusion from entirely naturalistic assumptions.”
But with such language Carrier plays into Steiner’s hands. Picking a simple solution in accordance with human limitations is precisely analogous to using the number 10 as a means of unlocking secrets to the universe. It is anthropocentrism in the extreme. I wonder, therefore, if it is difficult for people who were not trained in science to appreciate how absolutely uncanny is the continued use of mathematical formalisms by physicists. Green thus agrees with Steiner’s main point: at least unconsciously, physicists have abandoned a raw naturalism in favor of a theory formation method that has principles of beauty imbedded in its core. If they are correct, this approach certainly appears to be an anthropocentric—and thus non-naturalistic—strategy.
Or could it be naturalistic after all? Might it not be argued that plausible evolutionary models can be devised that would explain, for example, the human preference for symmetry? I certainly think such constructs are likely, especially considering symmetries that might be adduced in examining our DNA code. But even if evolutionary thinking can explain our preference for symmetry, how can a “blind chance” form of such thinking explain why such preferences are successful?
Three strategies seem possible at this point. The first is to argue for some kind of probabilistic “weighting” that would drive physical processes towards the production of sentient life forms, and in such a way that their preferences for beauty coincide with the actual mechanisms of the universe. The second involves reverting to some kind of a primal basic position: it just so happens that the universe evolved in such a way that our notions for beauty happen to work. Finally, a thoroughgoing Postmodernist might argue (along the lines of Wigner’s second question) that what we call successes came only because humans have invested a great deal of energy into science over the last 500 years. Who is to say that, if similar energies had been funneled in a different direction, there would be operating today a totally different paradigm yet with the same degree of “success”? The success is due to effort, not necessarily some amazing connection humans have with reality. Thus, mathematical beauty poses no problems for Naturalism whatsoever.
Time is running short, so let me give three very quick responses. First, with respect to the probabilistic weighting hypothesis, I wonder where the evidence is for this weighting. As Keith Ward comments, “A physical weighting ought to be physically detectable, … and it has certainly not been detected … In this sense, a continuing causal activity of God seems the best explanation of the progress towards greater consciousness and intentionality that one sees in the actual course of the evolution of life on earth.” Next, although primal basic explanations are needed at some level, invoking them in an effort to explain the apparent privileged status for humans in the universe—they just do—appears akin to pulling a rabbit out of a hat. Likewise, the claim that our constructs of success are ad hoc appears to be an objection without any realistic alternative suggestion. It’s almost like saying, “Well, your theory makes sense, but only if one buys into some of your commonly accepted cultural notions. Other—unspecified—theories will be able to show that the success you claim is really arbitrary, and thus not privileged.”
I would suggest, in summary, that a theistic explanation here is the more plausible one in accounting for the continuing successes of mathematical theories that ultimately grow out of aesthetic criteria. In assessing these arguments it is hoped my listeners will adopt an approach similar to Reppert’s in his defense of C.S. Lewis: There are, of course, valid points to be made on the side opposing these ideas, which should be looked at not as final answers, but as a spur to think through the relevant issues. It seems to me that human aesthetic values, and their subsequent use in successful physical theories, dovetail nicely with a Christian view that we are created in the image of God. Whatever being in God’s image exactly entails, it seems to include a rational capacity reflective of his that enables humans to understand and admire his creation. While not a final answer, such a perspective seems to me very plausible, and one that a thinking person can confidently put in the marketplace of ideas for appraisal.