Someone once said maths is like love — a simple thing that tends to get complicated. Yes, maths is simple, but only if one understands it. It must be very simple, for example, for Srikar Varadaraj, the 14-year-old who held Viswanathan Anand to a draw in simultaneous chess at the International Congress of Mathematicians (ICM) in Hyderabad last month.
It was evidently complicated for the bureaucrats who got Anand’s citizenship wrong; for them, the details in the world chess champion’s passport just didn’t add up to make him Indian.
But sometimes even the brightest of mathematicians get it wrong, the likes of those who presented papers at the ICM — a 113-year-old event, held in India for the first time, and in a developing country for only the second time. And when these geniuses goof up, the consequences can be devastating, puzzling or plain comical.
Approximating art
At the root of the problem is the mathematician’s obsession with beauty and elegance. But why are numbers — and symbols — beautiful? The eccentric Hungarian mathematician Paul Erdos put it thus: “It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you.” Broadly, it means that equations are beautiful, especially if they lead to some blissful truth. But then, some beautiful equations, like some beautiful women, can be deceptive.
Peter Woit, a mathematical physicist at Columbia University and author of the acclaimed Not Even Wrong, says: “The use of sophisticated mathematical techniques that few people understand can sometimes obscure basic problems.”
That is the constrained language of a mathematician, and Woit is a very good one. In layman’s language, elegant but improper use of maths can screw up the world.
A formula for chaos
It sounds straight out of Ripley’s Believe It or Not, but much of the chaos of the global financial crisis can be traced to a single formula — the Gaussian copula. The name sounds bovine, but is far from benign.
The Gaussian copula is a grand theme. It wanted to be in finance what the string theory wants to be in physics — a ‘theory of everything’. It was a formula created to “model default correlations, so that collateralised debt obligations (CDOs) could be accurately priced”, says Felix Salmon, the American financial journalist and analyst who brought the formula to the attention of the wider world in a groundbreaking article in Wired last year (CDOs are a type of sophisticated financial tool to represent different types of debt and credit risk).
If you unscramble it, what Salmon means is that the formula was trying to figure out if different types of bonds were correlated, and whether they were sailing or sinking together.
For Wall Street, the Gaussian copula acted as the proverbial philosophers’ stone. Minting money, traditionally, had never been without sweat and grime. But here at last was a shortcut to reap untold riches from the fuzziness of the financial world. All it required was variables like bonds, stocks or securities to travel from one end of the formula and congeal into hard currency by the time they reached the other.
For a time it worked. Spectacularly. So much that even Moody’s, the credit rating agency, factored the Gaussian copula into its methodology for evaluating CDOs. This happened in 2004. But within four years, the model had taken over, and then taken apart, the financial system.
The idea of the formula was simple: to predict the pattern of risk. The man who published it in 2000, David Li, is considered a genius even now (though, since the financial crisis hit, he has melted into the vast homeland, China). But it did not factor in unpredictable or risky events of great magnitude, like millions of Americans defaulting on home loans. “In the case of the Gaussian copula, I believe what got obscured was certain very large sources of risk,” Woit says.
The beauty myth
“Just because a formula is complex does not mean that it is right, and is a good model,” says Nigel Goldenfeld, a physics professor at the University of Illinois and founder of the investment software firm NumeriX. Indeed, an elegant formula can actually mislead the practitioner by its beauty.
“The elegance means that the practitioner can be deluded, and the complexity... the arcane mathematics means that the practitioner (the trader in the financial context) does not understand what is going on — and therefore takes appalling risks.”
Topping the ranks of the people who get besotted with the beauty of elegant formulas are physicists. Sometimes the beauty of the maths underlying the physics is so overwhelming that it very much overtakes reason. When this happens, physics stops being science, as Woit points out in Not Even Wrong, Lee Smolin illustrates in The Trouble With Physics, and Lawrence Krauss suggests in Hiding In The Mirror.
Take string theory, for example. Smolin, a founding member of the Perimeter Institute for Theoretical Physics, Waterloo, Canada, writes in The Beauty Myth, the second chapter of his book: “The most cherished goal in physics, as in bad romance novels, is unification. To bring together two things previously understood as different and recognise them as aspects of a single entity... is the biggest thrill in science.”
The aim of string theory is to seek the unification of quantum mechanics (the mathematical explanation for the behaviour of matter at the atomic and subatomic scales) and the general theory of relativity (the mathematical formulation of gravitation). String theorists say theirs is the best bet to correlate the physics of the very small and the physics of the very big. The quest is for a single formula, like Einstein’s E=mc^2, which is the unification of energy (E) and mass (m), showing they are essentially the same thing and can be converted into each other (c is the speed of light). Such a formula doesn’t exist for quantum mechanics and gravitation.
What feeds physicists’ optimism about arriving at such a formula is experience: Galileo and Newton unified rest and motion (the principle of inertia, or Newton’s first law of motion), Maxwell unified electricity and magnetism (Maxwell’s equations), and Einstein joined rest-and-motion with electricity-and-magnetism into the special theory of relativity.
Physics, in a nutshell, is about four fundamental forces — the strong nuclear force (which holds the nucleus of atoms together), the weak nuclear force (which determines radioactivity) the electromagnetic force (the force between charges and the magnetic force), and gravitation. String theory tries to reconcile gravitation with the other three fundamental forces.
But it does this on the basis of mathematical speculation and the arbitrary use of extra dimensions of space without any support of experiment.
The most common defence string theorists mount in defence of their hypothesis is that since its mathematical formulation is elegant and beautiful, it must be true. But as Smolin points out, the assertions of string theory are hardly testable because they emanate from a theory which comes in an infinite number of versions — all resulting from complex, esoteric maths.
“With such a vast number of theories, there is little hope that we can identify an outcome of an experiment that would not be encompassed by one of them. Thus, no matter what the experiments show, string theory cannot be disproved. But the reverse also holds: No experiment will ever be able to prove it true.”
In other words, string theory is ‘not even wrong’ — the name of Woit’s book and also blog, but originally a phrase coined by the physicist Wolfgang Pauli about scientific arguments which can be neither proved nor disproved by experiment.
Woit says: “In the case of string theory what gets obscured is that choosing an extra six dimensions introduces so much complexity and so many possibilities that the theory becomes useless.”
Science beyond logic
Smolin writes: “If string theorists are wrong... Theirs will be a cautionary tale of how not to do science, how not to let theoretical conjecture get so far beyond the limits of what can rationally be argued that one starts engaging in fantasy.”
An example of such fantasy is the anthropic principle, which says the universe is fine-tuned for the benefit of human life. Its chief proponent is Leonard Susskind, one of the founders of string theory. Susskind is an atheist, but is his theory any different from the theory of intelligent design, which says the universe is too complex to not have been made by a creator? One of the critics of the anthropic principle, the influential American evolutionary biologist, the late Stephen Jay Gould, said the theory seems to reverse known instances of cause-and-effect.
He said the theory is like saying ships were invented to house barnacles. Gould showed through compelling biological evidence that life adapted to the cosmos, to the laws of physics, and not the other way round.
When cricket creeps in
Salmon goes so far as to say that economists, physicists, and academicians from other disciplines wield mathematics inappropriately not just often, but almost always. One may take exception to such an extreme claim, but not if one is a cricket lover. Because whenever maths has been used to tinker with the game, the results have been ridiculous.
Take the Duckworth-Lewis method, for example. It is supposed to be a mathematically perfect way to set the target for the team batting second when a match is interrupted for reasons like inclement weather, but the targets thus set often seem to militate against commonsense.
Last month, two statisticians announced the result of a similar exercise to rank batsmen by factoring in the vagaries of cricketing eras, but it was so bizarre that it challenged sanity. While Donald Bradman emerged on top, Sachin Tendulkar was dumped deep, beyond even the figure that makes up the entire team line-up.
A mortal matter
The question arises, how can maths, the ‘queen of the sciences’ (as the ‘prince of mathematicians’ Carl Friedrich Gauss used to refer to it) get it so wrong sometimes. This cannot be answered unless one is able to understand what maths is. Reuben Hersh, co-author of the landmark book The Mathematical Experience and an emeritus professor at the University of New Mexico, says: “Mathematics is a science, like physics or astronomy; it constitutes a body of established facts, achieved by a reliable method, verified by practice, and agreed on by a consensus of qualified experts.
“But its subject matter is not visible or ponderable, not empirical; its subject matter is ideas, concepts, which exist only in the shared consciousness of human beings. Thus it is both a science and a ‘humanity’. It is about mental objects with reproducible properties.”
In the book, Hersh and co-author Philip J Davis, in answering how maths works and what gives it its power, demonstrate that “mathematical truth, like other kinds of truth, is fallible and corrigible”.
In short, maths is just another human pursuit — with warts and all. Maths, indeed, is much like love.
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