The story of G. I. Taylor and the atomic bomb is well established in the folklore of applied mathematics, and has become a staple of introductory courses on dimensional analysis. As commonly told, it runs something like this.
After the end of the Second World War, the US military wanted to show off their new atomic toys without giving too much away, so they declassified a number of photos of the Trinity tests but kept details, including the energy yield of the bomb, secret. When the first batch of photos appeared, the great Cambridge mathematician G. I. Taylor realised that he could use them to estimate the radius of the fireball as a function of time, and thus by dimensional analysis deduce the energy yield. Thinking that this made a nice illustration of an underrated mathematical technique, he wrote a happy little letter to the Times pointing this out; and he was startled to be woken at 3am the next morning by a number of serious gentlemen with American accents who wanted to know how he’d got hold of top secret information and who he was working for. It took Taylor a great deal of effort to persuade them that all he’d done was mathematics rather than espionage.
As Michael Deakin points out in a recent article [Int. J. Math. Ed. Sci. Tech. 42(8): 1069-1079, 2011], this telling involves a mixture of inaccuracy and sleight of hand. Not only Taylor, but also John von Neumann and Leonid Sedov were carrying out similar calculations in the early 1940s; both Taylor and von Neumann were doing so at the behest of their governments with direct application to the atomic bomb — as, quite probably, was Sedov. Furthermore, dimensional analysis of the blast-wave problem can’t alone give the energy yield: it leaves a dimensionless coefficient undetermined, which must be found by constructing an appropriate solution to the full dynamical equations. It does seem that Taylor was the first to use the Trinity test footage released in 1947 to estimate the energy yield and to publish the result, but the midnight visitation is apparently fictional: G. K. Batchelor’s account was that Taylor was “mildly admonished” but no worse.
Accuracy aside, the folk version is clearly the better story, and is liable to continue propagating more widely and enthusiastically than the accurate one. I think it’s interesting to consider why this is: it isn’t the only similar story that mathematicians tell about ourselves, or that is told about us, and so I suspect that it reveals something about the way we’d like to see ourselves and our discipline.
The classic story with which the Taylor legend seems to share a pattern is that of Archimedes and the siege of Syracuse — told, for example, in E. T. Bell’s Men of Mathematics with his usual polemical zest. Archimedes, in this tale, is the archetypal pure mathematician: readily distracted from all sublunary concerns by a nice piece of geometry, and willing to demonstrate his astonishing mechanical discoveries to the vulgar masses only at the request of King Hieron. Nevertheless, when a Roman invasion fleet approaches, he snaps into action, constructing a battery of “ingenious devilries” to greet them, of which “super-catapults” and the claw of Archimedes are the best remembered. (In popular imagination, the story is further refined by the addition of Archimedes’ mirror — an anecdotal detail that Bell manages to resist.) The invaders are devastated; the city saved; and it’s only through the foolishness of the vulgar that Syracuse is betrayed to a Roman sneak attack and Archimedes killed.
Both these stories are evidently examples of the old “lone genius” trope, and the effect of both relies on the contrast between the technological might but relative stupidity of the military and the figure of the mathematician armed with nothing but his brain. (The basic tale may be older than classical mathematics, judging by Ecclesiastes 9: 13-15.) I think, though, that it’s possible to place these stories in a more specific genre, which might be called the mathematician as magus.
The point of the magus is that his esoteric knowledge gives him access to the fundamental nature of the world, and thus power over it. (One of the beauties of Ursula K. Le Guin’s Earthsea books is that they capture this idea perfectly by making the mage someone who knows the “true names” of things and people.) What we’d now think of as the wizard is one example of this; so, in various ways, are the alchemist and the astrologer, the latter of course frequently indistinguishable from the mathematician for much of history, as an infamous rendering of Augustine reminds us. John Dee is perhaps the most vivid historical example of the overlapping of mathematician with magus, and his popular reputation today probably rests on both foundations: his mysticism borrows prestige from his status as a mathematician, while his appeal to mathematicians is enhanced because it enables us to mention casually that this mathematician was “probably” the inspiration for Shakespeare’s Prospero.
The identification of maths with magic, though, isn’t restricted to those like Dee who themselves saw mathematics as an intermediary between the natural and the supernatural. (See J. M. Rampling, BSHM Bull. 26: 135-146, 2011, for a fascinating discussion of the positioning going on in the famous “Mathematicall Praeface” in which Dee makes this claim.) The Taylor story is recognisably a modern version of the diviner who by subtle calculations reveals the secrets of the great from afar; the only difference is that his knowledge is attributed — with a little finessing — to a respectable mathematical technique rather than to communion with spirits or knowledge of the heavenly influences. (The popular telling of Turing versus the Enigma code, currently receiving saturation publicity in the UKs, is surely another version of this legend.) The Archimedes story sees the magus summoned from his books to unleash the hidden potentials of nature upon his enemies; the later version involving the mirror seems even more blatantly symbolic, with the enlightening power of the sun employed to blast the ignorant. Why does this pattern appeal so much to us?
The simplest reason, of course, is that it inverts the usual picture of the mathematician as the slightly contemptible outsider — there’s an echo of everything from David and Goliath to Rudolph the Red-Nosed Reindeer, and who doesn’t want to identify with the little Hebrew rockchucker or the noctilucent ungulate? I think, though, that the appeal may be more specific than that.
I think it’s possible that the idea of an occult dimension to reality, accessed through the cabbala of mathematics, agrees nicely with the streak of Platonism that almost all mathematicians carry in our souls. However naïve we may know it is, the Platonic myth of mathematics does capture an aspect of our experience — the resistance of mathematical ideas to being merely what we want them to be — which is almost uncommunicable to the uninitiated. And, once one has this sense of a Platonic world brooding just behind one’s shoulder, it becomes natural to see the “unreasonable effectiveness of mathematics” in an almost animistic light. A belief that the world is densely populated by spirits, who may by appropriate rituals be induced to act benignly toward us, seems to be one of the most basic and universal religions, and I don’t imagine that the few centuries in which we’ve taught ourselves to despise it have entirely ejected it from our worldviews.
In a society that genuinely does depend on mathematics more than it realises, and in which elementary mathematical proficiency is often treated as a mystery open to only a chosen few, there are plenty of opportunities for a mathematical magus complex to be reinforced. It’s interesting that attacks on mathematical modellers (for example, the recent book Useless Arithmetic) often describe them as a “priesthood”. The only real difference of implication I can see here between “priest” and “magus” is that in a supposedly secular world we’re meant to assume that priestly rituals are ineffective, whereas those of magi belong to a more fictional universe where the possibility that they might work remains open. At a less serious level, the Mathemagician of the Phantom Tolbooth might have been created principally by wordplay, but would the joke have worked so well with any other discipline? (When Douglas Hofstadter took over Martin Gardner’s Mathematical Games column and anagrammatised it to Metamagical Themas, he was deploying a closely-related pun.)
Maybe we can even see the same pattern in the popularity of “tricks” as a technique for communicating maths to a wider audience. The mathematical card trick, of which Prof. Persi Diaconis is perhaps the most famous exponent, is one example; the mesmerising achievements of mathematical jugglers are another; and even humble “think of a number” tricks inherit their format and appeal from the art of conjuring. Of course, these tricks have great pedagogical strengths, not least that they harness the power of surprise to open a mind to new or reshaped ideas, and I’d defend their use in the last ditch of mathematical education (which may, in fact, be where I currently work). To ask the kind of question too rarely asked in pedagogy or popularisation, though: without denying that mathematical magic is a good thing, what dangers ought we to be alert for?
The obvious danger is that when we let the figure of the magus shape our self-image, and the image of our subject that we project, we will inherit the negative aspects of that figure. Instead of mathematics lying open to the sun, available for scrutiny by anyone who is prepared to devote their time to testing its reasoning and its coherence, we may suggest that it is a cult — or even a shadowy universe — to which only initiates have access, and the rules of which may not (or even must not) make sense. Instead of being moderately respected, we may be slightly feared. Instead of holding open the gates of mathematics, we may be seen to be guarding the entrances against invasion. (There is a school of thought that believes this has already happened, and every time I have to work through a pile of homework scripts in which the ritual performance of algebraic manipulations has clearly driven out all actual thought, I renew my membership in this school. Lest that be thought too cynical, note that Shlomo Vinner [J. Math. Behavior 26(1):1-10, 2007] takes the ritual performance of mathematical procedures as the entry point for some interesting speculation about mathematics and the meaning of education.)
Or, perhaps, we should just embrace the myth. My notation is probably already as obscure as, say, Dee’s (although my spelling is a bit more regular); I can just about picture myself in a robe adorned with cabbalistic symbols; and I’d love to conduct tutorials with an appropriate visual aid or two — say, a human skull — prominently displayed upon my desk. Whatever the drawbacks for the profession as a whole, the only disadvantage I can see for myself is that mathematics is rarely as effective at the everyday level as a respectable magus might like. I’m more likely to end up like the mathematician in the joke who finds himself locked in a friend’s basement with a large supply of canned food but no can-opener. When they discover three or four days later he’s standing, emaciated and swaying slightly, with a tin clutched weakly in one hand, and repeating endlessly the formula “I define this tin to be open”…