Classic Article: The Art of Math

They say that a picture is worth a thousand words—but how many equations might be saved with the right choice of image? Mathematician John Baez argues that if we want to solve some of the deepest puzzles in physics today, then we need to go back to the drawing board—literally.

Baez, who is based at the Center for Quantum Technologies in Singapore, champions rewriting—or rather redrawing—quantum physics using a pictorial branch of math called "category theory." It’s a system that mathematicians know well, but which physicists rarely exploit. "This math goes back to some very basic ideas and uses them to do some very new things," he says.

This yen for un-cluttering physics and simplifying it in an approachable way developed during Baez’s childhood, when his uncle, Albert Baez—father of well-known folksinger Joan Baez—visited the house. Albert had set up physics departments in Algeria and Iraq and so he was constantly devising cheap ways to demonstrate physics on the bench-top. He also wrote a physics textbook for college students which he gave to his eight-year-old nephew. "That made me decide early on that I wanted to do physics," says Baez.

Category theory is a very visual form of math that allows you to keep track of individual particles by linking objects in diagrams using arrows. If you ever studied set theory at school, you may remember grouping items that had something in common (objects you own that are items of clothing, say) into a large set, and then splitting them into subsets (socks, gloves, hats) and drawing Venn diagrams of intersecting circles to represent how some of the objects shared qualities (the color blue, say), while others did not.

For a more mathematical example, you could draw a set of all the positive integers (1, 2, 3, 4...) and another of all the positive even integers (2, 4, 6, 8...) and draw an arrow representing the function linking the elements in the two sets together (in this case, the function would be "multiply by two.") Category theory does much the same thing, but on a more abstract level, depicting connections between more mathematically sophisticated objects. If Venn diagrams can be thought of as the equivalent of kids’ crayon drawings, then category theory is more akin to the works of Picasso.

Multi-dimensional Lego

"There are papers on category theory where the diagrams are so large they go over several pages!" says another fan, John Barrett, a mathematical physicist at Nottingham University, UK. Barrett describes using category theory to click algebra together in space-time "like multi-dimensional lego."

The lego analogy is apt, because the power of "categorifying" a physics theory lies in the fact that it boosts the theory’s dimension by one. That can be helpful in the same way that constructing a 3-D model provides a better way to visualise what a building will look like, than if you are restricted to just looking at 2-D blueprints.

"Categorification helps you understand conceptually what’s going on. You can stack up the algebra in spatial directions as well as time directions and ensure all the equations come out correctly," says Barrett.

This also means that if the physics that you are trying to understand is too tough to figure out in three-dimensional space, you can work out a simplified version in two or one dimensions, and then categorify it. Hey presto! You have your 3-D physics theory.

The Flow of Time

Category theory could also help illuminate an ancient saying about the nature of time. The Greek philosopher Heraclitus famously mused about the notion of change and the passage of time, stating that you cannot step into the same river twice. Why not? "Well, people usually say it’s because the river would be different later: all the water molecules in it will be different," says Baez. "But the writer Borges pointed out that you will be different too." It won’t be the exact same river, and it won’t be the exact same you, but you and the river will be the same in a way. "The Greeks found this idea very troubling: they thought things were either equal or not equal," says Baez.

It’s a puzzle that faces mathematicians and physicists every day too, whenever they look at equations and think hard about what they really mean. On one hand, the two sides of the equation look different, but on the other hand, the "equals sign" tells us that they are the same. Categorification is a fancy name for coming clean on this issue: instead of merely saying that two things are the same, it specifies a way of regarding them as the same, says Baez.

Specifically, category theory replaces equations with isomorphisms. For example, if you have seven objects in one category and seven objects in another category, you can match each object in the first category with a partner in the second category and then conclude that the collections are isomorphic to each other. The two objects in each pair are not the same object, but they are the same in some specific mathematical way, and isomorphisms explicitly detail that way, says Baez. "One of the main ways that isomorphisms show up in nature is when you’re thinking about time, about how things change but also stay the same," says Baez.

Thank you Mr. Feynman

Category theory may help our understanding of ancient Greek philosophy. But any sceptical physicist who does not want to change her mathematical ways will ask: Has categorification ever actually been shown to be useful in physics?

The answer is a resounding "yes." Most people don’t realise that Feynman diagrams—the squiggly representations that physicists routinely use to analyse what will be produced when subatomic particles collide—are a form of category theory.

"In my view, Richard Feynman invented categorical physics," says FQXi member Louis Crane, a mathematician based at Kansas State University. Crane had originally introduced Baez to category theory when Baez was a postdoc at Yale in the 1980s. Crane recalls the conversation they had in a corridor near the library: "John was getting excited about string theory and I told him not to waste time on that, there’s much better stuff."

String theory’s loss was category theory’s gain. Crane told Baez that this alternative mathematical framework could be useful to the hunt for a theory to unite quantum mechanics with general relativity. Baez "took off on that and has been hooked on it ever since," says Crane. Today, Baez and his students, Chris Rogers, John Huerta and Christopher Walker, are busy categorifying quantum theory and classical mechanics, with the help of a $131,865 grant from FQXi.

Baez has also popularized category theory with physicists through his blog , The n-Category Café. It’s not just about explaining finished work: "We’re asking questions publicly and getting help from some good people," he says. (And if tackling quantum gravity isn’t enough, Baez recently started another blog, Azimuth, to find ways for scientists to help save a planet in environmental crisis.)

Most importantly, Baez is a bridge builder. "John can write maths that other mathematicians understand, and can talk to physicists and try to understand what they are doing in a mathematical way," says Crane.

Barrett is optimistic about the strategy. "I expect that one day there will be a good category theory explanation of fundamental physics," he says. "It’s definitely a useful avenue to explore."

This article first appeared on October 20, 2010.