Category Theory and the extraordinary value of abstraction
Bob Coecke has received a grant of over $111,000 from the Foundational Questions Institute to continue his work on a graphical language to describe quantum mechanical processes. The work is based on category theory, a branch of mathematics that focuses less on the mathematical objects themselves, and more on the maps that transform them. The institute has an article on his work here.
The quantum mechanical world of subatomic particles is conceptually difficult. There is little dispute over the validity of the mathematical description of this world. It has a highly precise predictive value. But the mathematical grasp of this reality deviates from our experience more than any of the previous models of our universe which physics has proposed. As I heard Frank Wilczek once say, in the quantum mechanical world empty space is a turbulent medium of fluctuations, of particles that come to be and pass away.
We all have difficulty seeing this world, not because the mathematics is complex (although it is), but because we can’t quite grasp what the mathematics seems to be telling us. There is still disagreement among physicists about whether the wave function, the mathematical object that determines the probability of different outcomes of measurements, reflects the incomplete knowledge of an experimenter, or actually tells us that the state of a physical system is never one way or another. The other fairly popularized problem in physics is the incompatibility of quantum mechanics with general relativity.
Our description of the tiniest reality creates a problem for our description of the really large one. Since it is the mathematics that seems to have created the difficulties, the hope for resolution is in mathematics as well. String Theory is one of these hopes.
But I was struck today by what category theory has to offer, and I became interested particularly because category theory is a shift from objects to processes and is yet another level of abstraction.
Category theory began as a study of the processes that exist in mathematics and then became a mathematical subject in its own right. A category is a collection of objects together with what one would call the structure preserving maps that exist between them, like a function that turns real numbers into other real numbers. Since category theory doesn’t depend on other mathematics, it can be used as the foundational structure of the discipline (an alternative to set theory). Its advantage is that it provides for the discussion of the properties of maps in the abstract. Maps are actions more than things. In their abstraction, the maps are sometimes called arrows (and in diagrams they are arrows). About the graphical scheme that he uses, Coecke says that it’s a much more natural way to write things.
That means you can go on to describe the relations between different, and complicated, quantum processes in a clear and simple way—an especially useful trick when you want to represent some of the strange features of quantum theory, such as entanglement, nonlocality and teleportation.
The processes are represented by boxes and their interaction by wires between them.
The diagrams provide a clear, visual way to see at a glance how things change in a system, as time progresses across the diagram, depicted by the changing connectivity of the boxes. Difficult quantum calculations can be reduced to simple changes to the picture, retaining quantitative information.
To understand the advantage of the scheme, Coecke refers to a common example encountered by computer programmers. “If someone gave you a computer program written in zeroes and ones, there’s no way you could see what it does,” he notes. But if, instead, someone gave you a pictorial flow chart representing the algorithm you are hoping to recreate, you would immediately understand what the program does. “For us, these diagrams are a high-level language to reason about physics,” says Coecke.
Some physicists are optimistic that these alternative representations may also help reconcile quantum theory with general relativity by contributing to a quantum theory of gravity. But, in keeping with the theme of this site, it’s worth noting that Coecke’s scheme can be used to analyze word and sentence meaning, to get at something that language is doing.
Surprisingly, the way words interact to make up a sentence is similar to the way quantum processes interact. Google takes no notice of the order of words on a page, but actually the ordering can completely change a sentence’s meaning. Coecke has used his graphical approach to connect individual words in a sentence so their meaning can be extracted according to both the content of each word and its positioning. This is quite an achievement: most models of human language either focus on individual words or grammatical rules, not both. “Our categorical model blows away the existing language processing models,” says Coecke.
The author of the article, Sophie Hebden, makes an important observation:
The fact that Coecke’s high-level approach to understanding quantum information has such power when applied to other diverse fields, including linguistics, may point to a higher truth: that there are structures common to all layers of reality.
But I would also add the following: It’s significant, I think, that what may be this ‘higher truth’ is brought to light when the mathematics focuses on process rather than object. And this is particularly interesting if we agree to see cognition itself as biologists Maturanna and Varela see it, “not as a representation of the world “out there,” but rather as an ongoing bringing forth of a world through the process of living itself.” While these new worlds are being detected by the very creative use various instruments (our extended senses), they can’t be given shape without mathematics.