Vladimir Kazakov began his talk at ICTP-SAIFR this week with a variant of Tolstoy’s famous opening to the novel Anna Karenina: “Happy families are all alike; every unhappy family is unhappy in its own way.” Kazakov flipped the order of the quote, stating that while “Un-solvable models are each un-solvable in their own way, solvable models are all alike.”
In talking about solvable and un-solvable models, Kazakov was referring to a concept called integrability, the idea that in certain quantum field theories it’s possible to avoid the messy approximations of perturbation theory and instead jump straight to the answer. Kazakov was observing that these integrable systems seem to have a deep kinship: the same basic methods appear to work to understand all of them.
I’d like to generalize Kazakov’s point, and talk about a broader trend in physics.
Much has been made over the years of the “unreasonable effectiveness of mathematics in the natural sciences”, most notably in physicist Eugene Wigner’s famous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. There’s a feeling among some people that mathematics is much better at explaining physical phenomena than one would expect, that the world appears to be “made of math” and that it didn’t have to be.
On the surface, this is a reasonable claim. Certain mathematical ideas, group theory for example, seem to pop up again and again in physics, sometimes in wildly different contexts. The history of fundamental physics has tended to see steady progress over the years, from clunkier mathematical concepts to more and more elegant ones.
Some physicists tend to be dismissive of this. Lee Smolin in particular seems to be under the impression that mathematics is just particularly good at providing useful approximations. This perspective links to his definition of mathematics as “the study of systems of evoked relationships inspired by observations of nature,” a definition to which Peter Woit vehemently objects. Woit argues what I think any mathematician would when presented by a statement like Smolin’s: that mathematics is much more than just a useful tool for approximating observations, and that contrary to physicists’ vanity most of mathematics goes on without any explicit interest in observing the natural world.
While it’s generally rude for physicists to propose definitions for mathematics, I’m going to do so anyway. I think the following definition is one mathematicians would be more comfortable with, though it may be overly broad: Mathematics is the study of simple rules with complex consequences.
We live in a complex world. The breadth of the periodic table, the vast diversity of life, the tangled webs of galaxies across the sky, these are things that display both vast variety and a sense of order. They are, in a rather direct way, the complex consequences of rules that are at heart very very simple.
Part of the wonder of modern mathematics is how interconnected it has become. Many sub-fields, once distinct, have discovered over the years that they are really studying different aspects of the same phenomena. That’s why when you see a proof of a three-hundred-year-old mathematical conjecture, it uses terms that seem to have nothing to do with the original problem. It’s why Woit, in an essay on this topic, quotes Edward Frenkel’s description of a particular recent program as a blueprint for a “Grand Unified Theory of Mathematics”. Increasingly, complex patterns are being shown to be not only consequences of simple rules, but consequences of the same simple rules.
Mathematics itself is “unreasonably effective”. That’s why, when faced with a complex world, we shouldn’t be surprised when the same simple rules pop up again and again to explain it. That’s what explaining something is: breaking down something complex into the simple rules that give rise to it. And as mathematics progresses, it becomes more and more clear that a few closely related types of simple rules lie behind any complex phenomena. While each unexplained fact about the universe may seem unexplained in its own way, as things are explained bit by bit they show just how alike they really are.