ESSAY

When Even Mathematicians Don't Understand the Math

New York Times, May 25, 2004

Susan Kruglinski

To most of us, smudgy white mathematical scrawls covering a university blackboard are an icon of incomprehensibility. The odd symbols and scattered numerals look like a strange language, and yet to read it, neurologists tell us, we would have to use parts of our brains that have nothing to do with what we normally think of as reading and writing.

Math and physics writers can be thought of as interpreters of this unconventional language. Books about math meant for a general audience have trickled into the public consciousness, as with The Golden Ratio or The Millenium Problems, and the bestseller lists are occasionally topped by books about physics, as with Stephen Hawking's phenomenal hits and Brian Greene's current bestseller, Fabric of the Cosmos. And yet much of this subject matter is virtually impenetrable, as mathematicians and physicists cultivate its power to calculate that which is inconceivable, undetectable, non-existent, and even impossible. So when we read about the strings and branes of the latest physics theories, or the Reimann surfaces and Galois fields of higher mathematics, are we to believe that these are accurate depictions of abstract concepts? And what does it mean when an understanding of our physical reality is based on stuff that even the scientists who calculated them cannot comprehend? Despite the best metaphors and analogies, it is hard to imagine something is not lost in the translation.

"It is a bit like trying to explain football to people who not only have no understanding of the word 'ball,' but are also rather hazy about the concept of the game, let alone the prestige attached to winning the Super Bowl," wrote Ian Stewart, professor of mathematics at the University of Warwick and author of Flatterland: Like Flatland, Only More So, in an e-mail correspondence. Asked if there exist mathematical concepts that defy all explanation to a popular audience, Stewart replied, "Oh yes -- possibly most of them. I have never even dared to try to explain noncommutative geometry or the cohomology of sheaves, even thought both are at least as important as, say, chaos theory or fractals."

Keith Devlin, a mathematician at Stanford University and author of The Millennium Problems, a book that attempts to describe the most challenging problems in mathematics today, admits defeat in his last and most impenetrable chapter, where he is forced to interpret something called the Hodge Conjecture. He writes, "If you [the reader] find the going too hard, then the wise strategy might be to give up...This [problem] caused me by far the greatest difficulty."

The Hodge Conjecture deals not only with cohomology classes, a complicated grouping construct, but involves algebraic varieties, which Devlin describes as generalizations of geometric figures that really don't have any shape at all.

"Those equations represent things that not only can we not visualize, we can't even imagine being able to visualize them," Devlin said. "They are beyond visualization." Devlin believes this difficulty points to a math truism that ultimately framed his entire project. "What the book was really saying was, 'You're not going to understand what this problem is about as a lay person, but neither will the experts.' That is in fact the story in this book. The story is that mathematics has reached a stage of such abstraction that many of its frontier problems cannot be understood even by the experts."

And yet it is this higher math that guides us to an understanding of our reality. As an end product of mathematics, physics attempts to explain the existence and composition of the world around us. But how can it make sense that a nearly unintelligible language can explain the physical world?

Columbia University physicist Brian Greene painstakingly distills the confounding calculations of quantum physics and extravagant equations of string theory in his new book, invoking images of flowing time and textured space that should not necessarily be taken literally. "These translations by design suppress a huge amount of technical machinery that underlies the everyday English description," Greene said. "I would say that there's absolutely always something lost in the translation."

"We have the familiar model of the atom," said Devlin, "which has a nucleus and protons and neutrons surrounded by these electrons that orbit around. We've known since the beginning of the 20th century, with quantum theory, that atoms are not at all like that, and yet physicists can successfully use that image of the solar system model with its rotating billiard balls. It's the same with string theory. I mean, give me a break -- they're not little loops of string! For one thing, they're in eleven dimensions."

Like Devlin, Greene is straightforward with his readers about the impossibility of explaining certain abstractions. But unlike Devlin, Greene thinks there is enough graspable material in the mathematics of physics to almost accurately depict just about anything.

"We go pretty far out in physics," he said, "but we have an anchor in the real world, because we really ultimately only consider something interesting if its going to tell us something about the universe that we live in. And that's perhaps why we have an easier job of explaining our work to a general audience." For example, he said, "the equations that govern a violin string are pretty close to the equations that govern the strings we talk about in string theory. So although the notion of strings is metaphorical, it's pretty close."

Optimistically Greene added, "I suspect that the overarching aim of most every mathematical study can be described, even if you can't get to the guts."

This may be true. But if science writers described breakthroughs in genetics or zoology in terms of "overarching aims" and not concrete facts, readers would question the foundations of that field. The fact that readers see higher math as acceptably inconprehensible is evidence of its singular nature. According to some of its experts, mathematics is not in the same category as biology, astronomy or geology. It is difficult even to simply explain what math is, let alone what it says. Math may be seen as the vigorous structure supporting the physical world or as a human idea in development. While there are empirical systems of experimentation and discovery in most of the sciences, some might say mathematicians rely on something more intuitive to further their peculiar area of study.

"It isn't science," said John Casti, mathematician and author of Five Golden Rules: Great Theories of 20th Century Mathematics and Why They Matter. "Mathematics is an intellectual activity -- at a linguistic level, you might say -- whose output is very useful in the natural sciences. I think the criteria that mathematicians use for what constitutes good versus bad mathematics is much more close to that of a poet or a sculptor or a musician than it is to a chemist."

And just as one cannot define what it is that makes a moving phrase played on a violin moving, it seems the essence of the superb equation dashed across the classroom blackboard is also ineffable.

"Abstraction is the feature of math that distinguishes it from the other sciences," said Devlin. "Algebraic-looking language is used to describe an abstract world that is entirely created by the human mind. And really, there is nothing in everyday life that you can really latch onto to help you understand that abstract world."

This makes for a frustrating human dilemma: our brain has the ability to compute the abstract mathematics it created to construct theories about reality, and yet it may never be smart enough to comprehend those theories, let alone explain them.

Despite his and his colleagues' tireless efforts to fight against this paradox, Greene concedes that it all ultimately makes sense. "Our brains evolved so that we could survive out there in the jungle," he said. "Why in the world should a brain develop for the purpose of being at all good at grasping the true underlying nature of reality?"