It’s tempting to think of Alexander Grothendieck, arguably the greatest mathematician of the 20th century, as the Syd Barrett of mathematics: a genius who saw too much, too fast – who reached for the secret too soon. You may have read at The Guardian or The New York Times that Grothendieck passed away last winter, having lived his last couple of decades as a recluse somewhere in the French Pyrenees. I’d like to tell you something of Grothendieck and describe in nontechnical terms a strikingly useful paradigm called category theory that is firmly rooted in his oeuvre.
First, why is this good fodder for a humanities blog? Well, in a unequivocal sense that we’ll discuss below, Grothendieck’s legacy embodies vastly more than the math problems he solved. His methodology during the years he was active in the math community, before his retreat from society, was that of a purist, an idealist, searching for universal truth. For example, instead of solving problems computationally, as might be expected of a mathematician, he famously embraced his penchant for recasting problems in a way so naturally capturing their essence that their solutions just fell right out.
Mathematicians habitually hone in on advantageous levels of abstraction and generalization, but Grothendieck’s approach was radically abstract, almost to a fault, according to some accounts of his interactions with colleagues. And even though Gothendieck’s standpoint has proved exceptionally insightful in modern times, first contact with his work can sometimes engender concerns that too much information is being lost, as if the meat of the problem or theory on which one is working is in danger of being abstracted away.
To give you, brave reader of this post, a better feeling for the dynamic in the world of pure math, consider the following analogy. At the lowest level we have various sets, on which we do arithmetic, consisting of numbers: the natural numbers: {0, 1, 2,…}, the integers {…,-2,-1, 0, 1, 2,…}, the rational numbers (fractions), the real numbers, the complex numbers, etc. Next, we introduce variables, with which we do algebra, that represent generic numbers; then functions (e.g., the canvas for calculus) which map between sets of numbers or, more generally, between sets of variables. From here we might consider even more general “functions” whose inputs and outputs are the functions from the previous level. And so on.
As we move higher through more and more general intellectual scaffolding, we “abstract away” unessential details of the objects we are interested in studying. A feature of the way this plays out in mathematics, is that we start getting interaction between the different levels. The machinery that Grothendieck built outputs numbers from certain abstract functions spaces, for example. Once we have a rich enough scaffolding, we start building mathematical gadgets that help us see what’s out there, and harvest information on the structures of interest.
What Grothendieck was famously known for, and led to his winning in 1966 the highest honor in math, the Fields medal, was his ability to see very deeply into connections between whole areas of mathematics. The breadth and depth of the enormously powerful scaffolding he built represented radical new connections between algebra and geometry, and topology.
A late-19th early-20th century paradigm shift in math and physics was centered around redefining objects of interest in such a way as to facilitate studying them via their intrinsic properties. To illustrate, picture surfaces like spheres or multi-handled tori (donuts, possibly with more than one hole, and hollow). In your mind’s eye, these surfaces are likely appearing already embedded in 3-space.
Now suppose we figure out that the solutions of some important set of equations – having to do with designing an airplane wing, say – always form a surface. In our effort to understand all possible behaviors of the wing in flight, we might benefit from knowing, in general, how many different kinds of surfaces there are. Well, we just need to visualize, as we did above, all the various surfaces that can occur, right? The problem with this is that there are objects out there that should rightfully be called a surface (since they look locally like a disk in 2-space) that don’t fit in 3-space without self-intersection. An example is the Klein bottle which can be embedded in Euclidean space of dimension 4 or higher but not smaller. Even in the case of surfaces, our intuition has some pretty serious limitations. The way to handle this situation is to figure out how to define surfaces via intrinsic characteristics, independently of extrinsic artifacts stemming from, for instance, how they can sit in Euclidean space. (By the way, once the definitions were correct, mathematicians were able to prove that any surface can be embedded into 4-space, and similar results for higher dimensional shapes.)
The early and mid-20th century was a golden age in that mathematicians, physicists and scientists were making huge connections between different areas and disciplines that had been built independently. In the mid-1950s, and for the fifteen or so years following, Grothendieck worked on rebuilding algebraic geometry, one of the deeper and more highly structured areas, which had years earlier more or less collapsed do to faulty foundational work earlier in the century. It was during these years that he made huge intuitive leaps, his scaffolding scaling vast structure previously unobserved.
But, and this is where his story starts to diverge from the world of pure mathematics, he resisted non-illuminating artifice and wasn’t too interested in working on famous problems. Instead he focused on developing a deeper understanding of structures underlying the central objects of study, particularly with regard to the interplay between algebra and geometry. In fact, his later writings (see the NYT obituary above) evidence his using mathematics as a device to explore structure grander yet.
Last Fall, my colleague in Shanghai, the philosopher Thomas Carroll, and I team-taught a philosophy of science course. Taking advantage of our diverse backgrounds and that of our students (we had students from the humanities as well as the sciences, engineering, etc) we worked in some basic category theory and infused a traditional, historically-grounded philosophy of science treatment with some categorical thinking.
One day in early November, I posted this picture of a cloaked Grothendieck, which had shown up on the perennial “anyone heard as to the whereabouts of Grothendieck?” post on a math news-feed. I usually don’t click on such posts since invariably no one has seen or heard from him. It was only a week or so later that Tom told me that Grothendieck had passed away. He’d been living in a small hideaway in Lasserre. Tom had heard of his passing via a philosophy blog.
Why is it then that Grothendieck is highly revered by philosophers? A possible answer is categorical logic. But I don’t think that’s it, although there is a very interesting ultra-modern effort to recast foundational mathematics in terms involving homotopy theory (read about it here: A new foundation for mathematics). One has to understand that, though category theory was developed initially by mathematicians Eilenberg and Mac Lane in the context of their work during the 1940s in algebraic topology, Grothendieck is one who with almost otherworldly insightfulness used it to punch out the boundaries of what we can know.
Category theory can seem a bit cryptic at first glance, not because it was fleshed out by mystics like Grothendieck, but because it almost looks too simple. Here’s the definition of a category.
A category consists of objects A, B, C, … and morphisms f, g, h, …. Each morphism has an associated initial object and terminal object which we indicate by writing, for example, f : A → B. If A, B and C are objects such that there exist morphisms f : A → B and g : B → C, then there must also exists a morphism g ⋅ f : A → C called the composition of f and g. For each object A there must exist a identity morphism 1A : A → A. In addition, the following two conditions must hold.
- Whenever f : A → B, g : B → C, and h : C → D we have (h ⋅ g) ⋅ f = h ⋅ (g ⋅ f); and
- For every morphism f : A → B we have f ⋅ 1A = 1B ⋅ f.
This definition might remind you of something you were bored stiff by in about sixth grade: mapping the domain of say all sixth graders to their favorite flavor of ice cream. And it’s true that we can take sets as objects and functions from sixth grade as morphisms and we get a category. It’s OK to think of set theory as a sort of prototypical example of a category, but realize that in the definition above the objects can be anything and the morphisms need not be functions. Importantly, categories are more structured than just objects since they include morphisms which, in turn, are often defined so as to capture even more structure.
We don’t have space to work out more technical details, but there are a couple of considerations that math folks are generally cued in to and that are encoded into the definition above.
First, most objects of interest in say math, physics, science, engineering admit some flavor of algebraic structure. The way the universe is built makes us want to combine things. The mathematician wants to add or multiply, the chemist wants to compose reactions, the computer scientists sends the output of one program to the input of another, the bioinformatician composes processes. Composition is modelled as generally as possible in the definition of a category.
Second, we are typically interested in pairs of objects and structure-preserving maps between them as opposed to just individual objects. In fact, an important mantra among category theorists is that it’s the morphisms that really matter. Imagine a string theorist trying to “see” a string that lives in a dimension higher than 3. How does she study those elusive objects? By studying all morphisms of similarly structured objects into and out of the string in question. Similarly, the high-energy physicist studies yet unobserved particles by colliding other particles. We study the objects in which we are interested by placing them in categories and studying their relationship with similar objects.
Grothendieck pioneered using the paradigm of category theory to study mathematical structure. Lawvere, in the 60s, worked out some deep implications regarding logic. Since then, applications have popped up all over: in linguistics, cognitive science, philosophy, functional programming, dynamical systems, theoretical physics, control theory, to name a few.
The real magic happens when one considers categories whose objects are categories. The morphisms in a category of categories are called functors, which are defined so as to preserve all structure in sight. Now we can study an object of interest (which may be a whole category) by studying functors from its category to related categories. Working on the level of categories of categories turns out to be wildly enlightening and is the reason that Eilenberg and Mac Lane developed the theory in the first place. We can work on the level of entire theories, picturing one inside another. In modern times, one studies the strings, which are certain higher dimensional shapes, in string theory by pushing them into cobordism theories and turning loose the machinery of algebraic topology.
Then, of course, we can take categories of categories of categories, leading to topos theory, to which Grothendieck contributed greatly. At this stage we should pause and think about all of this from a philosophy of science perspective. This business of layering ever more “categories of” takes on the flavor of Godel’s work wherein we have to keep tweaking the axiom systems we use to study things, or some sort of generalized uncertainty principle making it impossible to tell which theories are factual, if there even is such a thing.
I tend to think of this “folded” way that science progresses as more of a feature of the way humans pursue knowledge, rather than a drawback. We discussed this at length in our philosophy of science class last Fall in the context of Kant and the others. If we start thinking about patterns in music and art, and aesthetics, we realize that category theory is, at the very least, a formal tool in with which to organize and communicate ideas. Perhaps this is partly the reason philosophers are interested.
Contemplate the following toy example for a moment. Analogies are the morphisms in the category of ideas. Like the fourth and fifth paragraphs and eighth paragraph above. It seems as if categorical thinking delivers some ontological heft, possibly because we can use it as leverage at the edge of what is knowable by humans.
The question arises: what was Grothendieck doing holed up all those years? Answer: writing – much of it inward looking – tens of thousands of pages on a wide range of subjects including ecology, politics, and his dreams.
Upon his death, mainstream press releases described Grothendieck as an enigma. Mathematicians David Mumford and John Tate, themselves legends, wrote an obituary, containing an explanation of Grothendieck’s work profound in and of itself, for Nature which was initially rejected (you can read the original here) as too technical but later published in edited form. They recount an “intense” and “complex” man of uncommon will with exceptional drive and talent, “unique in almost every way”, who “could be very warm” as a friend and whose legacy is omnipresent, even today.
They write that when Grothendieck resigned from his position at the Institut des Hautes Etudes Scientifiques in 1969 for “reasons not entirely clear” and “plunged into an ecological/political campaign he called Survivre”, he did so with “a breathtakingly naive spirit (that had served him well doing math)” believing “he could start a movement that would change the world.”
The greatest of visionaries though he was, Grothendieck was indeed, also, all too human.
Dr. Simmons is an associate professor of mathematics at Drury University.
