By the early 1800s several mathematicians, including Gauss, had independently developed the idea of geometrically representing complex numbers. Still, it was a major leap, and such leaps are generally far from obvious before they’re made. Indeed, it had eluded even Euler’s grasp. Although he was familiar with the idea of vectors, there’s no evidence that he visualized complex numbers as vectors that could be manipulated on a 2-D plane to represent calculations.
One indicator of the advance’s importance is that it blew away the air of impossibility that had long surrounded imaginary numbers. In effect, Wessel and his fellow explorers had discovered the natural habitat of Leibniz’s ghostly amphibians: the complex plane. Once the imaginaries were pictured there, it became clear that their meaning could be anchored to a familiar thing—sideways or rotary motion—giving them an ontological heft they’d never had before. Their association with rotation also meant that they could be conceptually tied to another familiar idea: oscillation. Eventually the formerly confusing will-o’-the-wisps came to be seen as solid players in physics and engineering for, among other things, representing phenomena that involve regular, back-and-forth patterns. (One such phenomenon is especially close to home: our very bodies oscillate daily via circadian rhythms.)
Pioneering electrical engineer Charles Proteus Steinmetz spearheaded the use of the imaginaries in calculations related to alternating current.* A diminutive hunchback, he’d fled his native Prussia as a young man and emigrated to America after being threatened with arrest for supporting socialist causes. Just months after arriving in the United States he began making fundamental advances that revolutionized the use of electricity. In 1892 he joined the newly formed General Electric and, soon after, published a landmark paper showing how to use imaginary numbers to greatly simplify analysis of AC circuits.
Delighted by animals that were typically shunned, Steinmetz kept a menagerie of peculiar pets in his Schenectady, New York, mansion, including alligators, rattlesnakes, and black widow spiders. School improvement was another of his avocations, and he pushed for the introduction of special classes for immigrants’ children. Once when Thomas Edison visited him, he delighted the aged, nearly deaf inventor by tapping out messages on Edison’s knee in Morse code. In later life he was dubbed “The Wizard of Schenectady” in the media, and at some point a more elaborate epithet was invented: “The Wizard Who Generated Electricity from the Square Root of Minus One.”
Euler’s general formula, eiθ = cos θ + i sin θ, also played a role in bringing about the happy ending of the imaginaries’ ugly duckling story. Even before the geometric interpretation shed new light on imaginary numbers, Euler worked out some remarkable things about them based on the formula. An example is the evaluation of ii, shown in Appendix 2.
The interpretation of eiθ as a rotating vector paved the way for constructing particularly elegant mathematical models of rotation and oscillation using this compact function. Such “exponential models” make it surprisingly easy to carry out calculations that, in many cases, would be considerably more difficult if trigonometric functions, the main alternative, were used instead. Recall from the last chapter how simple it was to derive eiπ/2 = i by thinking of eiθ as an angle sweeper and mentally rotating it. The use of exponential models also brings into play the ease of applying calculus to ex-based functions, mentioned in Chapter 2—eiθ offers a very similar user-friendliness.
Today, Euler’s formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It’s arguable that the formula’s ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century.
EULER’S GENERAL EQUATION stands out because it forged a fundamental link between different areas of math, and because of its versatility in applied mathematics. After Euler’s time it came to be regarded as a cornerstone in “complex analysis,” a fertile branch of mathematics concerned with functions whose variables stand for complex numbers.
But the special case, eiπ + 1 = 0, is mainly treasured because it’s beautiful. What makes it as exquisite as a great poem or painting?
I doubt that there’s a simple answer to this question that most of those who find the formula beautiful would agree with. In fact, some math lovers don’t regard it as particularly beautiful, nor do they even find it very interesting—more on that momentarily—which goes to show that the eye-of-the-beholder issue will always arise regardless of whether we’re contemplating art or mathematics. (This is why aesthetics strikes me as both endlessly provocative and fundamentally absurd.) But despite the risk of getting mired in a morass of conflicting opinions, I feel obliged to address the beauty question—after all, the word beauty is in the title.
First, let me frame what I’m calling beautiful. It’s not simply the equation’s neat little string of symbols. Rather, it’s the entire nimbus of meaning surrounding the formula, including its funneling of many concepts into a statement of stunning brevity, its arresting combination of apparent simplicity and hidden complexity, the way its derivation bridges disparate topics in mathematics, and the fact that it’s rich with implications, some of which weren’t apparent until many years after it was proved to be true. I think most mathematicians would agree that the equation’s beauty concerns something like this nimbus.
But what makes the nimbus beautiful?
Dictionaries define beauty as qualities that give pleasure or deep satisfaction to the senses or the mind. That’s nice, but it only leads to another question: Why does Euler’s formula induce pleasure?
A good starting point on that is a much-cited observation by the great British philosopher and mathematician Bertrand Russell:
Mathematics, rightly viewed, possesses a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
There’s ample food for thought in this eloquent declaration, but the first sentence bothers me. Russell’s reference to a cold, austere, sternly perfect quality that doesn’t appeal to our weaker nature strikes me as reinforcing sadly common negative stereotypes about math: that it’s dry, forbidding, and rock-hard. Further, he seems to be denigrating what might be called the soft power of music, painting, and other arts to elicit feelings—the gorgeous trappings by which they presumably enchant us. That is, he’s implying that the Dionysian realm is a lesser thing than the stricter, purer Apollonian realm, and that only the latter is associated with the beauty of mathematics and the greatest art.
It’s true that feelings are gorgeously messy and inextricably linked with subjectivity, which is seemingly the opposite of math’s stern objectivity. But it is also true that the limbic system (the brain system most closely associated with emotions) is crucial to the way our minds work. I suspect that if its activity were somehow greatly dialed down so that a person could experience genuinely cold, austere pleasure when contemplating, say, Euler’s formula, the result would be a mental state resembling that of a weirdly robotic Spock.*
Thus, while feelings may be the essence of subjectivity, they are by no means part of our weaker nature—the valences they automatically generate are integral to our thought processes and without them we’d simply be lost. In particular, we’d have no sense of beauty at all, much less be able to feel (there’s that word again) that we’re in the presence of beauty when contemplating a work such as Euler’s formula. After all, eiπ + 1 = 0 can give people limbic-triggered goosebumps when they first peer with understanding into its depths. (I’m one.)
But what accounts for this limbic-mediated thrill? I think it springs from a combination of things, including the equation’s seriousness, generality, depth, unexpectedness, inevitability, and economy—qualities that prominent twentieth-century English mathematician G. H. Hardy singled out as key ingredients of mathematical beauty. (Hardy was much concerned with the nature of mathematical beauty because he held that “there is no permanent place in the world for ugly mathematics.” That isn’t quite as strong as the celebrated Keats formula, beauty = truth, but it’s pretty close to it.)
There’s also the formula’s elegance, a word that mathematicians use to designate a clever mix of concision and sophistication. And then there’s the cool way that disparate ideas cunningly fit together in its derivation—we get essentially the same kick from this aspect of mathematical beauty that people get when they puzzle out how to snap together elaborate Lego windmills or spaceships at a certain age (like 46, when the kid gets a big Lego set for Christmas). Homo mechanicus are us.
But I think Euler’s formula most importantly invokes a rarer, deeper thrill: the feeling of exaltation that we get from an encounter with an example of our species outdoing itself. I get pretty much the same thrill from the creations of Michelangelo, Beethoven, Jane Austen, George Eliot, and W. H. Auden; from the discoveries of Charles Darwin, Marie Curie, and Albert Einstein; from Lincoln’s managing to hold the union together while ending slavery in the United States; from Helen Keller’s incredible achievements; from the monumental persistence of Elizabeth Cady Stanton, Susan B. Anthony, and the other marathoners of the spirit who secured basic rights for women; and from Nelson Mandela’s world-changing magnanimity. This is what Russell’s second sentence is about, and he put it beautifully despite his outdated use of “Man” to refer to all of us—his phrase “the sense of being more” really nails it.
What I’m getting at here is closer to what’s called the sublime in aesthetic theory than to beauty. Ideas about the sublime go back to Longinus, a first-century Greek thinker who described it in terms of grand, awe-inspiring thoughts or words. Later thinkers posited that the sublime and the beautiful are different; the former is supposedly associated with feelings of horror-tinged awe, such as those felt by an astronaut looking at the receding earth while hurtling toward the moon, while beauty is mainly about pleasure. While this distinction is interesting, what I find most salient is the idea of exaltation from the sense of being more.
Chinese philosopher Tsang Lap-Chuen is a leading modern exponent of the idea that the sublime involves this kind of experience. In The Sublime: Groundwork towards a Theory, published in 1998, he wrote that the sublime “evokes our awareness of our being on the threshold from the human to that which transcends the human; which borders on the possible and the impossible; the knowable and the unknowable; the meaningful and the fortuitous; the finite and the infinite.” In his view, there is no single essential common property possessed by sublime works or sublime natural objects, nor is there a single emotional state evoked by all of them. But he argues that there’s a common thread in experiences of the sublime, which is that they take us “to the limit of some human possibility.”
Euler’s formula may seem elementary to modern mathematicians, but many of them still feel that it’s extraordinarily beautiful. I think this may largely be because they’ve retained a lively sense of it as emblematic of the sense of being more—it represents the true story of how an all-but-supernatural genius reached beyond what once seemed possible to come up with a deep, almost miraculously concise truth. Thus, their familiarity with it breeds no contempt. To them, as to me, Euler’s formula is a joy forever.
This exalting, sublime-related kind of beauty is relatively rare, and of course the word beautiful applies to other sorts of things. But as Russell noted, great math and great art both possess it. And that points to a longstanding conundrum in aesthetics: How can it be that certain works are prized as beautiful/sublime across many generations, somehow defying the incessant zigzags of fashion and the continual rethinking of what’s deeply satisfying to the senses or mind? A number of Paleolithic cave drawings, for example, are widely regarded as sublimely beautiful some 30,000 years after their creation—I’ve seen some of them up close in France and can testify that they are hair-raisingly, impossibly superb. Euler’s formula has similarly retained its charismatic appeal to mathematicians across generations. The brain-scan study that was mentioned in the first chapter, involving mathematicians’ neural responses to certain equations, suggests why: the durability of beauty in mathematics and other realms is based, at least in some cases, on ubiquitous aspects of the human mind. Beauty may be in the brain of the beholder, but beholders’ brains (including their limbic systems) seem to react similarly when confronted with truly sublime rarities, such as Euler’s formula.
AND YET SOME PEOPLE assert that Euler’s formula is much overrated. It’s obvious, they say—there’s no mystery, no paradox, no reaching into the depths of existence.
I’ve seen variations on this theme in a number of online commentaries. One blogger even claimed that Euler’s formula is so simple that “small children” understand its meaning. French chemical engineer, writer, and amateur mathematician François Le Lionnais struck a similarly jaded stance: the formula “seems, if not insipid, at least entirely natural,” he wrote. One skeptical respondent in the 1988 survey that ranked Euler’s formula as mathematics’ most beautiful result commented that it’s “too simple” to classify as supremely beautiful, and another apparently gave it a low rating because it’s simply “true by virtue of the definition” of its terms.
My take on these opinions probably won’t surprise you—they strike me as abusing hindsight.
True, Euler’s equation lacks the charm of novelty or the charisma of a major unanswered question in mathematics. And its derivation is straightforward compared to, say, the celebrated proof of Fermat’s Last Theorem by mathematician Andrew Wiles—published in 1995, it required over 150 pages of complex mathematics.* But I think those who assert that Euler’s formula is obvious, or that the connections it reveals are less than amazing, are displaying a lack of historical perspective as well as a sadly blunted sense of wonder. They seem to believe that understanding and wonder are fundamentally incompatible, and that people who marvel at Euler’s equation are either pretentious or mathematically naïve. The same logic would imply that marveling at Michelangelo’s David is incompatible with understanding exactly how it was carved out of marble.
Perhaps those who assert that the formula is borderline boring are implicitly claiming that after you’ve mastered the main math concepts that are combined in the equation, and know how to derive it, you should find it no more interesting or profound than 2 + 2 = 4. To me, however, this would be akin to a champion hurdler declaring, “Hey people, leaping over a bunch of three-and-a-half-foot hurdles while sprinting as fast as you can is really no big deal. I can show you how ridiculously simple it is in nothing flat. Even small children can do it.”
It took mathematicians centuries to make the conceptual advances that led to the equation and its geometric interpretation. These ideas were unknown to many brilliant mathematicians before the nineteenth century. The reason is clear: they aren’t obvious, period. And understanding how to derive the formula only highlights further loci of wonder. Consider again eθ = 1 + θ + θ 2/2! + θ 3/3! + …. When 1 is plugged in for θ, the equation becomes e = 1 + 1 + 1/2! + 1/3! + …, which can be rewritten as e = 1/0! + 1/1! + 1/2! + 1/3! + … (since by definition the factorials 0! and 1! both equal 1). Despite knowing how to derive this last equation from basic principles, I still find it beautiful and surprising. It shows that e, which at first glance seems to be a messy, endlessly chaotic irrational number (based on its definition as the limit of (1 + 1/n)n as n goes to infinity), is a perfect paradigm of order and simplicity—in fact, just a slightly dressed-up version of 0, 1, 2, 3,…—when viewed from another perspective.
Keith Devlin, the mathematician who compared Euler’s formula to a Shakespearean sonnet, has summed up what such surprises seem to be telling us. “Surely,” he wrote, such a formula “cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view.” Those who say that Euler’s formula is ho-hum imply that they can perceive the whole of this pattern and all of its implications. Perhaps they really are incredible geniuses and have managed to do that. I’ve not seen convincing evidence that they have.
But my main objection to the killjoy view of Euler’s formula has to do with a practical matter. The most inspiring teachers I’ve known possessed the gift of infectious enthusiasm—they communicated intellectual excitement about their subjects by seeming to regard them with the fresh eyes of impassioned novices. While explaining things, they often appeared to be re-experiencing, or at least to be re-enacting, what it was like when they first fell in love with their specialties. I’ve tried to do the same thing in this book. The killjoys dismiss such approaches as simple-minded. While there’s no disputing taste, I must confess that I wouldn’t want them as my kids’ math teachers. I’m convinced that mathematics seems boring to many people largely because they’ve never caught on to the fact that it’s full of beauty and surprise. The killjoys seemingly want to keep that under wraps, as if to prove that rigor = rigor mortis.
I SUSPECT THAT INTELLIGENT BEINGS on a distant planet would discover many of the number- and logic-based relationships contained in our math textbooks, perhaps including Euler’s formula. This belief reflects my view that mathematics is grounded in patterns that exist independently of our minds and thus is concerned with objective truths. This intuition has helped shape my thinking about Euler’s formula. In a way it has been the elephant in the room throughout this book.
The idea that statements like 1 + 1 = 2 and eiπ + 1 = 0 express truths that exist independently of human thought is called mathematical Platonism. G. H. Hardy was one of the most prominent modern math Platonists. In an essay on his life’s work, he wrote, “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onward… .”
While I lean toward a form of Platonism, Hardy’s purist version doesn’t appeal to me. Harvard mathematician Barry Mazur nicely described the kind of ambivalence I experience on this topic in his book Imagining Numbers (Particularly the Square Root of Minus Fifteen): “On the days when the world of mathematics seems unpermissive, with its gem-hard exigencies, we all become fervid Platonists (mathematical objects are ‘out there,’ waiting to be discovered—or not) and mathematics is all discovery. And on other days, when we see someone who… seemingly by willpower alone, extends the range of our mathematical intuition, the freeness and open permissiveness of mathematical invention dazzle us, and mathematics is all invention.”
Such ambivalence is seldom voiced, however, and debates about math’s basic nature have long been a philosophical version of the sweet science, replete with lots of complicated bobbing, weaving, and punching. Sometimes the fights have even spilled over into the pages of nontechnical publications. For instance, two celebrated heavyweights—math popularizer Martin Gardner and mathematician Reuben Hersh—clashed about it in such venues as The New York Review of Books in the 1980s and 90s. Gardner, who died in 2010, was a mathematical realist (math realism is basically the same as Platonism). He argued that “if all intelligent minds in the universe disappeared, the universe would still have a mathematical structure, and that in some sense even the theorems of pure mathematics would continue to be ‘true.’”
Hersh, also a very distinguished writer (his 1981 book, The Mathematical Experience, co-authored with mathematician Philip J. Davis, won a National Book Award), countered that mathematics is a human cultural construct that has no reality independent of people’s minds. Its statements are invented “social objects” like institutions and laws, in his view. Thus, it’s wrong to speak of math as true in any timeless sense—even statements such as 2 + 2 = 4 lack infallibility. And although he says that mathematics is objective, he interprets the word objective to mean “agreed upon by all qualified people who check it out”—not “out there” in some sense. “Saying [mathematics] is really ‘out there,’” he adds, “is a reach for a superhuman certainty that is not attained by any human activity.”
There is much to commend in Hersh’s writing. His 1997 book, for example, includes a highly readable account of the history of thinking about the basic nature of mathematics. But I doubt that he and other advocates of “social constructivist” theories about mathematics have won over many Platonists. In a 1997 review of such theories, for instance, Gardner ardently reiterated his opinion that “the world out there, the world not made by us, is not an undifferentiated fog. It contains supremely intricate and beautiful mathematical patterns.… It takes enormous hubris to insist that these patterns have no mathematical properties until humans invent mathematics and apply it to the world.”
In 2000, cognitive scientists George Lakoff and Rafael E. Núñez launched another provocative attack on mathematical Platonism in their book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. They acknowledge that Hersh “has long been one of [their] heroes.” But they disagree with the most radical implications of social constructivist theories, emphatically stating that they themselves “are not adopting a postmodernist philosophy that says that mathematics is merely a cultural artifact.” (Emphasis in the original.)
Lakoff and Núñez posit that mathematics—or, at least, basic arithmetic—is grounded in our sensory and motor experiences. For instance, as children we map the idea of a measuring stick’s length to the size of a number. Such “grounding metaphors” anchor basic math in the objectively knowable, real world, which includes our neural and other bodily apparatus—thus, their term “embodied” mathematics. In their view, “Two plus two is always four, regardless of culture,” because such statements are based on real-world objects, which possess culture-independent qualities such as stability, consistency, generalizability, and discoverability.
More complex mathematical ideas, such as Euler’s formula, are based on “conceptual metaphors” that link and blend math concepts, according to their theory. Conceptualizing the sine and cosine functions as infinite sums based on terms like θn/n!, for instance, entails blending conceptual metaphors about functions as numbers, numbers as wholes that are the sum of their parts, and limits, they explain.
They argue that math Platonists are wrong because mathematics mainly consists of layers upon layers of conceptual metaphors that exist only in human minds. In their view, the belief that math possesses some sort of “transcendent” reality is a mystical doctrine with no empirical support. They further assert that mathematical Platonism is central to an elitist culture in math that “rewards incomprehensibility” and “has contributed to the lack of adequate mathematics training in the populace in general.” (See what I mean about the throwing of punches?)
THE SURPRISING CONNECTIONS that frequently crop up in mathematics bear on this debate. Do they show that mathematical patterns exist independently of our minds, and that we sometimes only gradually perceive and piece them together into a gestalt—like flashes of reflected light seen through clouds from an airplane that turn out to be Lake Erie? Or do they spring from the fact that mathematicians sometimes invent conceptual metaphors that are so intricately interlinked that it often takes a long time for people to see all the implied connections?
And what about the many math advances that have seemed at first to be no more than the products of an abstract, rule-based game with no bearing on the real world, and then later have clicked into place as amazingly well-suited for representing physical phenomena? Euler’s general formula is an example—long after Euler derived it, engineers found it to be just what they needed to help model AC circuits. Such uncanny coincidences often resemble fantasy-novel plot twists: “Suddenly Sam realized that the mysterious ornament he’d taken from the ancient Egyptian mummy was actually a key for unlocking the door leading to the hyperconium reactor’s control room.”
Physicist Eugene Wigner famously dubbed this phenomenon “the unreasonable effectiveness of mathematics in the natural sciences.” On my Platonist-leaning days, I see this effectiveness as suggesting that mathematics is out there, and true independently of us. Not surprisingly, Lakoff and Núñez disagree: “Whatever ‘fit’ there is between mathematics and the world,” they aver in Where Mathematics Comes From, “occurs in the minds of scientists who have observed the world closely, learned the appropriate mathematics well (or invented it), and fit them together (often effectively) using their all-too-human minds and brains.”
In a review of Lakoff and Núñez’s book, mathematician and author John Allen Paulos, who wrote the bestseller Innumeracy: Mathematical Illiteracy and Its Consequences (1988) and other entertaining books, offered a “quasi-Platonist” compromise that I find appealing. “Arithmetic may… be transcendent in the sense that any sentient being would eventually develop the metaphors that ground it and be led to its truths, which can thus be said to inhere in the universe,” he wrote. (Such hypothetical beings have come up frequently in the debate—Hersh, for example, has allowed that “little green critters from Quasar X9” may do mathematics, but has argued that their math could well be totally different from ours.)
I’m drawn to Paulos’s quasi-Platonism largely because it fits with my hunch that the brains of far-away sentient beings, if they exist, would probably have been shaped by an evolutionary process that works like the one that gave rise to human intelligence. Basic quantitative and abstracting abilities could confer a Darwinian edge in many situations. For instance, such faculties would be invaluable for beings that band together in environments with limited resources, and that exchange goods and services. Based on this logic, evolutionary thinker Haim Ofek has theorized that resource exchanges helped drive the explosive growth in brain size and cognitive abilities that led to modern humans. As he has observed, “Exchange requires certain levels of dexterity in communication, quantification, abstraction, and orientation in time and space—all of which depend (i.e. put selective pressure) on the lingual, mathematical, and even artistic faculties of the human mind.”
After such math-enabling brain structures come into play, competitive pressures from equally brainy types could result in the kind of positive feedback that occurs in arms races, rapidly amping up such brainware. At some point, this process might lead to brains capable of registering sophisticated math-related patterns.
These speculations are consistent with findings by cognitive scientists. For instance, studies on the emergence of a basic number sense in infants, as well as brain-imaging studies that have delineated the neural bases of mathematical ability in adults, suggest that we’re born with elementary arithmetical abilities. Stanislas Dehaene, a professor at the Collège de France in Paris who has conducted influential research on this topic, theorizes that we possess brain circuits that evolved specifically to represent basic arithmetic knowledge. Such circuits can also support high-level mathematical reflection. He and colleague Marie Amalric showed via brain imaging that high-level math thinking in mathematicians activates a brain network that appears to be largely dedicated to mathematical reasoning.
Importantly, this math-related network is distinct from more recently evolved language centers. That is, humans’ mathematical ability appears to be evolutionarily ancient, which suggests that it’s the kind of faculty that natural selection fosters early on when conditions are right for human-like intelligence to evolve. This may explain, among other things, why we can often intuit the truth of mathematical statements without being able to articulate exactly why they’re true. (Dehaene observes, for instance, that a typical adult can quickly decide that 12 + 15 is not equal to 96 without much introspection on how this cognitive feat is performed.) When this sort of thing happens to mathematicians, they can find themselves obliged to devise proofs for theorems or formulas that they’ve already sensed must be right. No one has illustrated this phenomenon more vividly than the astounding, self-taught mathematician Srinivasa Ramanujan.
In 1913, Ramanujan, who had lived in poverty during most of his early years in India and had twice failed as a college student, sent off a 10-page manuscript to Cambridge University’s G. H. Hardy containing a set of formulas he’d intuited. After perusing the manuscript with growing amazement, Hardy commented that some of the formulas “defeated me completely; I had never seen anything in the least like them before.” Although Ramanujan hadn’t included proofs, Hardy concluded that most of the strange formulas must have been true because, as he said, “if they were not true, no one would have the imagination to invent them.” Hardy was soon telling colleagues that he’d discovered a new Euler in India—like the great Swiss mathematician, Ramanujan had an uncanny ability to sense hidden connections.*
In 1914, Hardy arranged for Ramanujan to come to Cambridge to collaborate with him and his colleague, mathematician J. E. Littlewood. But Ramanujan’s ability to dream up jaw-dropping formulas rubbed some math traditionalists the wrong way. They regarded formulas without proofs as probable twaddle, and those who spouted them as something like con artists. Ramanujan also suffered from increasing health problems after coming to England, including tuberculosis and a severe vitamin deficiency that led to his hospitalization. He died in 1920, soon after his return to India.
One time Hardy came to visit his protégé in a London hospital and experienced an example of Ramanujan’s brilliance that has become one of the most famous anecdotes in math history: “I had ridden in taxi cab 1729,” Hardy recounted, “and remarked [to Ramanujan] that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’” Ramanujan was referring to the fact that 1,729 = 13 + 123 = 93 + 103, which he’d recorded in one of his notebooks.
Hardy and Ramanujan were both mathematical Platonists, but they personified a schism within the Platonist camp. Hardy was an atheist, and thus to him the independent existence of mathematical truths had nothing to do with divine revelation. Ramanujan believed that his mathematical insights were gifts from the Indian goddess Namagiri, and he famously stated that “an equation has no meaning for me unless it expresses a thought of God.”
It seems they could only have agreed to differ on this issue. Indeed, I suspect that the positions that people stake out in the debate about the basic nature of mathematics—like those of Hardy and Ramanujan on secular versus religious Platonism—generally have more to do with axiomatic beliefs than with the close examination of well-established facts. And that’s at least partly because the mental processes underlying mathematical intuitions typically aren’t directly accessible to the conscious, articulating parts of our minds. Even the most sophisticated brain-imaging studies tell us little about them, and so at this point we can only speculate about why mathematical truths seem possessed of a special kind of inevitability. Still, I like to think that my quasi-Platonist speculations, while only a first step toward an explanation, are on the right track—at least they are consistent with what we know about evolution and the human brain.
BUT WHAT ABOUT INFINITY? We never encounter infinitely large numbers of things, or infinitely small objects, in the real world, and so how might metaphors grounding this crucial mathematical concept arise in sentient beings? Of course, religious-minded Platonists have no problem explaining our notions of infinity: such ideas come to us, according to their basic beliefs, as we channel the thoughts of an infinite-minded God. But I’d prefer an explanation that fits with my secular quasi-Platonism.
It isn’t difficult to support the idea that metaphors related to Aristotle’s potential infinity bear the stamp of common real-world experiences. For example, the standard Q&A during long car trips with children—”Are we there yet?” “Not yet.”—followed after a while by lamentations such as, “It seems like we’re never going to get there,” suggests a near-universal ability among earth’s young sentient beings to conceive things that “never give out in our thought,” as Aristotle put it.
But actual infinity seems different. I, for one, can’t really get my mind around it. (I’m not referring to the modern mathematical conceptualization of infinity in terms of limits. Rather, I mean the metaphysical monster—the Thing—that visits terrible paradoxes upon us when it’s not carefully wrapped up inside the clever evasions of that math framework.) At most, I can summon up an extremely crude idea of actual infinity based on my experiences with large distances (driving across the United States), vast collections of things (sand grains on a beach), and teensy objects (very small specks of dust).
However, the quasi-Platonism I favor doesn’t require perfectly exact mappings of independently existing patterns into the mental realm. Even if actual infinity doesn’t exist in the real world (and I’m not claiming that it doesn’t exist, in some sense; I’m remaining agnostic about that here), independently existing patterns (all those sand grains, etc.) can suggest the idea of it. Thus, I see no reason to exclude actual infinity, and theorems based on it, from the set of concepts that may possess the limited kind of transcendence that I ascribe to mathematics—metaphors for actual infinity would almost certainly occur to the mathematicians of Quasar X9, just as they have on earth.
Ironically, Lakoff and Núñez’s theory can be construed as supporting this quasi-Platonist view. They propose that people conceptualize actual infinity by picturing endless processes “as having an end and an ultimate result.” That is, we metaphorically blend the idea of completion with process-based potential infinity to conceive actual infinity. They also theorize that mathematicians always use this “Basic Metaphor of Infinity” (BMI) to conceptualize cases of actual infinity that arise in math, such as infinite sets and limits of infinite sums. Their BMI sounds to me like the kind of idea that all sentient beings would probably come up with as they went about both inventing and discovering mathematics.
ALTHOUGH I PART WAYS with Lakoff and Núñez on some issues, I couldn’t agree more with the important pedagogical thrust of their book—Where Mathematics Comes From makes a strong case for paying more attention to metaphors in math education. Like them, I think we come to understand new things largely by making connections between unfamiliar novelties and familiar mental constructs. Metaphors, broadly defined, help us infer things about novel ideas based on ones that we’re familiar with. As Lakoff and Núñez make abundantly clear, mathematics is chock full of such metaphors.
Math’s standard terse communication style—all those symbols—makes it possible to compress many conceptual metaphors into a single equation or theorem. That enables great elegance and economy. To students, however, it can seem like a sadistic device dreamed up by a spiteful Numerocracy. Even people who know a lot of mathematics can find it very challenging to understand one of math’s intricate concept mashups upon first encountering it. Metaphoric elaboration, as Lakoff and Núñez call it, can be extremely helpful to math learners as they grapple with such high-end mashups.
To show how it’s done, they devote a 70-page section of their book to an impressive explication of none other than Euler’s formula. But such explanations (including the one offered in these pages) must always be incomplete in an important way, for they can’t tell us how Euler sensed the hidden trails that led to the formula. Of course, the proofs he devised provide some clues. But, as suggested by Ramanujan’s story, I think mathematical proofs generally represent ex post facto glosses on intuitive processes that are mostly inaccessible to our conscious thoughts. A memorable observation by German physicist Heinrich Hertz bears on this point: “One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own,” he wrote, “that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.”
I don’t see this as an endorsement of Platonic mysticism. It merely highlights the fact that the full meaning of formulas like Euler’s is tied to the very deep mystery of how the human mind works. And until that mystery is solved in a detailed way, burning issues about the basic nature of mathematics will probably continue to burn. In any case, no one, as far as I know, has better articulated the basic intuition behind math Platonism (and quasi-Platonism) than Hertz did in this single sentence. He also managed to convey the same sense of wondering delight about the depth, surprise, and beauty of formulas like Euler’s that I’ve experienced, and that I’ve tried to bring out in this book.
WHILE WRITING ABOUT Euler’s formula, I recalled a sculpture I’d seen at the Boston Museum of Fine Arts that struck me as beautifully related to the subject. A kind of epiphany I had while thinking about the sculpture makes a fitting close to the book.
Created by American artist Josiah McElheny, it employs semi-transparent, two-way mirrors to produce reflected rows of bottles, decanters, and other glassware that seem to extend infinitely into the depths of the piece. This visual suggestion of infinity brought to mind the exquisite patterns of the infinite sums that Euler showed are coiled up inside eθ, sin θ, and cos θ.
As I gazed into the piece’s visual depths in my mind’s eye, however, it occurred to me that the sculpture also represents a concrete metaphor for the sublime profundities that tend to escape our notice as we hurriedly go about our daily routines—until the day, say, that an infant son or daughter is placed in our arms for the first time, or that a beloved person or animal dies, or that one of the most beautiful minds in history reminds us that all the time we’re madly rushing around in the unit circles of our days, the infinite, that fantastic figment that can feel so real to us, is quietly lying just beneath the surface.
* Alternating current is electricity that reverses direction many times a second, giving it the character of a pendulum that rapidly swings back and forth—it oscillates. AC’s voltage can be readily ramped up and down by transformers, like the ones cased in metal boxes on telephone poles. That enables its efficient long-distance transmission from generators at very high voltage; the voltage is then reduced for relatively safe home use by local transformers. AC’s ability to be efficiently transmitted all over the place is why it is used in the electric grid instead of direct current, the unidirectional electricity that flows from batteries.
* Actually, I think that Spock, for all his stern, austere logic, had a perfectly functional, human-like limbic system. (After all, he was half human—his father was Vulcan.) Note that he had little trouble interacting with the earthlings around him. Thus, despite his ostensible lack of emotions, he was not at all like a severely autistic person who was essentially flying blind when it came to others’, as well as his own, feelings, and who therefore could barely operate in society. Indeed, my sense is that Spock had a secretly high emotional intelligence, and his fastidious pretense of being an affective doofus was exactly what made him so charismatic. Whenever he was in a scene, I, for one, found him upstaging the other Star Trek characters, and it wasn’t simply because of his pointy ears, amused ironic wit, or precise manner of speaking—it was largely because, as a supposedly very-low-emotional-intelligence being who was paradoxically competent in social interactions, he enigmatically defied my instinctive sense of what is psychologically possible.
* In 1637 French mathematician Pierre de Fermat conjectured that no positive integers a, b, and c satisfy the following equation (make it true) if each number’s exponent, n, is a whole number greater than 2: an + bn = cn. This became known as Fermat’s Last Theorem, and mathematicians tried to prove it for 357 years before Wiles finally did so. For n=1 or n=2, however, it’s easy to find values of a, b, and c that work. For instance, 32 + 42 = 52.
* Here’s just one of Ramanujan’s many provocative formulas: (1 + 1/24) × (1 + 1/34) × (1 + 1/54) × (1 + 1/74) × (1 + 1/next prime number4) × … = 105/π4. The infinite product on the left side of this equation is based on successive prime numbers raised to the 4th power. Primes are integers greater than 1 that are evenly divisible only by themselves and 1. Thus, 3 is a prime, but 4 isn’t because it’s evenly divisible by 2. The first nine primes are 2, 3, 5, 7, 11, 13, 17, 19, and 23. The primes go on forever, which accounts for the ellipsis at the end of the product in Ramanujan’s formula. This formula shows a deep connection between π and the prime numbers.