Beautiful Mistakes

Every January, several thousand mathematicians converge in a convention centre for a meeting of minds, rotating through the cities of San Diego, Baltimore, San Antonio, Seattle and Atlanta. This year, back in San Diego, the theme of the massive Joint Mathematics Society (attendees mostly comprising members of the American Mathematical Association and the Mathematical Association of America, plus a decent Canadian showing) was the “Mathematics of Planet Earth.” Keynote talks addressed the math of climate change and the melting polar ice caps. And the screening of a new film, Darwin’s Extra Sense, raised questions about how mathematical models can help us make sense of nature’s complexity—the film’s title deriving from Darwin’s lament that he had not worked harder at mathematics and gained the “extra sense” he believed mathematicians lent to comprehending the world.

That mathematics serves as the so-called handmaiden to science rests on its power to probe fundamental truths. Its power, in turn, lies in its beauty—its ability to tap into aesthetic laws rooted in the age-old tenets of symmetry, harmony and unity.

When the mathematician Paul Erd?s came upon a particularly beautiful piece of mathematics, elegant and simple, he declared, “That’s straight from The Book.” (God’s manual of creation, as it were.)

Or, as John Keats suggested, “‘Beauty is truth, truth beauty,’—that is all / Ye know on earth, and all ye need to know.”

The Keats quotation is how applied mathematician and writer David Orrell opens Truth or Beauty: Science and the Quest for Order (his previous books include Apollo’s Arrow: The Science of Prediction and the Future of Everything and Economyths: Ten Ways Economics Gets It Wrong). The latest subject he has chosen is one of perennial interest. Other recent titles exploring this territory include This Explains Everything: Deep, Beautiful and Elegant Theories of How the World Works, a collection of essays by scientists edited by their agent John Brockman, Ian Stewart’s Why Beauty Is Truth: A History of Symmetry, as well as Truth and Beauty: Aesthetics and Motivations in Science by Subrahmanyan Chandrasekhar. Orrell sets himself apart with his use of the conjunction “or”—a small distinction that suggests his unease with truth and beauty being bedfellows in science. He argues that mathematicians’ and scientists’ obsession with beauty and with equating beauty with truth has led them astray. And that our worldview—of economics and the environment, for instance—has suffered as a result.

Orrell charts the course of his argument by following the history of Homo sapiens’s attempts to make sense of the world and universe around us, starting with the ancients (yet another title on the subject was a collection of essays published in 2006, Mathematics and the Aesthetic: New Approaches to an Ancient Affinity, edited by Nathalie Sinclair, David Pimm and William Higginson). First, there was Pythagoras’s Harmony of the Spheres, proposing that the universe is based on a mathematical harmony not unlike musical harmony. Ten was the perfect number, and so the cosmos consisted of ten heavenly bodies. There was the Earth, the Moon, the Sun, the stars (thought to be a single outer layer) and the five planets known at the time. But that totalled nine, so in order to reach the harmonious ten Pythagoras created the “counter-earth,” which was opposite our Earth and thus out of sight. “The Pythagoreans were well ahead of their time in believing that the Earth could be in motion,” writes Orrell. “Their invention of the counter-earth, on the other hand, was an early example of scientists inventing a phenomenon—in this case a planet—in order to meet the requirements of an elegant theory.”

Similarly, Johannes Kepler’s geometric model for the solar system, nesting the then six known planets within the Platonic solids in a way that inscribed their circular orbits, was beautifully simple and elegant. But subsequently Kepler could not get the data to match his circles. He admitted that, unappealing as it was, the orbits were instead ellipses, and this was the first of his three laws of planetary motion.

From there, Orrell maps a sweeping history of the physical sciences, a history riddled with the search for harmony, unity, symmetry—and a search that was at times very successful. James Clerk Maxwell’s law united the electric and magnetic forces, which set the stage for Einstein’s E=mc², one of the most beautiful equations in physics, unifying space, time, matter and energy. And around the same time Emmy Noether showed that energy conservation laws are associated with symmetry principles. Of the latter, Richard Feynman said, it is “a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.”

When we arrive at the present day, however, Orrell portrays the search for unity, harmony and symmetry as, once again, not so successful. “The problem with The Standard Model, or supersymmetry, or string theory,” he says of the latest candidates for the much coveted Theory of Everything, “is not that they fail to achieve perfect unity, harmony, symmetry, and so on. It’s that they may be trying too hard.”

The Standard Model, for example, does not require the addition of an extra planet or two à la Pythagoras, it requires the tweaking of about 20 parameters to make the math work. As physicist Leon Lederman put it, “the drive for simplicity leads us to be very sarcastic about having to specify twenty parameters. It’s not the way any self-respecting God would organize a machine to create universes. One parameter—or two, maybe. An alternative way of saying this is that our experience with the natural world leads us to expect a more elegant organization … The problem is the aesthetics.”

“This wouldn’t matter much,” comments Orrell, “if it weren’t for the fact that—like characters stuck in a bad novel—other aspects of our lives and culture are being reshaped by the very same, rather corny aesthetic principles.” In taking their lead from physics, other facets of life have been cast as similarly reductionist, atomized and mechanistic, possessing component parts that can be deduced and explained.

In a section titled “The Standard Climate Model,” Orrell details how weather models have their intellectual roots in nuclear physics and quantum theory. But, he counters, “rather than view climate change as a kind of elaborate physics problem, it would be more realistic to adopt a medical analogy”—to treat it as a complex living organism.

And similarly with the economy. One of the standard economic models is Homo economicus—“rational economic man.” “Rationality is a symmetry,” says Orrell, “because rational people with the same preferences will make the same decision given the same information. Stability is symmetry in time—if markets are in equilibrium, then the future looks like the past. And if market participants have similar power and other characteristics, then that means transactions are symmetric.”

“Using this imaginary being as the atom of the economy, economists argued that in a competitive market prices would be driven to a stable equilibrium via Adam Smith’s invisible hand,” says Orrell. “Just as Aristotle’s Theory of Everything was really a theory of stability, so mainstream neoclassical economics is based on the idea that the complex, dynamic, evolving, and crash-prone economy is best modeled by assuming it yearns towards a stable resting place.”

Thinking back to the 2007–08 economic debacle, Orrell recounts the explanation provided by chief economist at the International Monetary Fund Olivier Blanchard: “the mainstream ‘had converged on a beautiful construction’ to explain the economy, but now had to admit that ‘beauty is not synonymous with truth’.” While the fascination with elegant ideas and equations has been mostly harmless in areas such as weather forecasting or genetics, in economics it has done direct damage to people’s lives by helping to perpetuate a system that is unfair, unstable and unsustainable.

Orrell’s solution is that scientists, rather than striving for simple mechanistic explanation of all existence, should instead embrace the complex qualities and emergent behaviour of the universe, which he defines as follows: “properties which emerge from the system but cannot be predicted using knowledge of the system’s components alone.” We should wholeheartedly (and with the right brain, seeing as the left brain has dominated science up to now) embrace the shift from the Age of Reductionism to the Age of Emergence.

To this end, Orrell gives a sidebar cameo appearance to Princeton mathematician John Horton Conway, but not for Conway’s lifelong obsession with symmetry, his latest book being The Symmetries of Things. Instead he recruits Conway for his 1970 invention of the Game of Life, a universal constructor (a machine that can make any machine) with emergent properties. Based on three simple laws of Conway’s creation, this cellular automaton replicates itself and even evolves, much like life itself. Although the laws governing Life are known, its end result is not deterministic—it is impossible to predict, given any starting configuration, what Life forms will crawl out of the soup (such as the beehive, the blinker or the glider gun). Orrell notes that “the existence of emergent properties means that reductionism fails. As Aristotle said in Metaphysics, the whole is more than the sum of its parts.”

In contemplating the implications of reductionism versus emergence, Orrell also borrows from Aldous Huxley (at times Truth or Beauty becomes a complex curiosity cabinet of quotations that, fittingly, has an impressionistically emergent effect). According to Huxley: “It is fear of the labyrinthine flux and complexity of phenomena that has driven men to philosophy, to science, to theology—fear of the complex reality driving them to invent a simpler, more manageable, and therefore, consoling fiction … With a sigh of relief and a thankful feeling that here at last is their true home, they settle down in their snug metaphysical villa and go to sleep.”

And from Wittgenstein we get: “Man has to awaken to wonder … Science is a way of sending him to sleep again.”

And, one page later, from the Cubist artist Georges Braque: “Art is meant to disturb, science reassures.”

Exactly how scientists are to make the paradigm shift from reductionism to emergence Orrell leaves unstated. And in a sense it rather seems like proposing “Let’s try something other than capitalism for a while and see where that gets us.” It is a nice idea.

But on the other hand, Homo sapiens’s obsessive fixation with simplicity, symmetry, unity and harmony evolved for a reason. As Orrell admits, beauty is hardwired into the human brain—we seek symmetry in faces, and asymmetry is an indicator of genetic deficiency—and scientists, after all, are only human. Without this aesthetic sieve, this tool with which we have been endowed, we could never make sense of all the information coming at us.

This was the point made in various sessions at this past January’s massive math meeting in San Diego. Marjorie Senechal, a mathematician from Smith College, discussed her recent book, I Died for Beauty: Dorothy Wrinch and the Cultures of Science, about the renowned chemical theorist.

“The main intellectual tool we have is to find patterns,” says Senechal (elaborating after the fact by telephone)—“to find some idea behind the complexity. Nobody wants to look at complexity and just say, ‘Wow, that’s so complex!’ Even if it is emergent, you want to find out what is going on. And to that end, scientists rely on a back-and-forth dialogue between beauty and truth.”

“It’s a trial and error process,” says Senechal. “You see the complexity, you look for meaning, you find a pattern. Then you try it out to see if in fact it has the explanatory power that a pattern should have. Usually it doesn’t, but it gets you part of the way there, not the whole way … The dialogue begins when you recognize that your simple model gets you somewhere, but not all the way. That puts you back facing the complexity, but at a slightly higher level than before. You’ve seen through some of it, but you now have to deal with more complexities, of other kinds. So it’s back and forth and back and forth. Sometimes you hit on a real solution and sometimes you don’t.”

Dickinson similarly suggests that the dialogue between truth and beauty never really ends—it is an ongoing conversation, an endless iterative process.

Which might be what Orrell is getting at after all. The cover of his book features a Venn diagram: a blue circle labelled “truth” overlapping a yellow circle labelled “beauty” with a green “or” in the area of intersection. So to read the Venn interpretation, Orrell’s title could in fact be Truth and/or Beauty.

“Like the universe itself,” he concludes, “our models are impermanent, flawed, and delicately contrived—and that perhaps is the most beautiful thing about them.”