Gary Li Mathematician
We can try to use math to predict the weather. We can try to use math to predict the stock market. We can try to use math to study the motion of planets around the sun. And so on. There are a lot of “useful” applications of math, and a lot of mathematicians study such things. I focus more on what are called the “pure” aspects of math. I don’t really like to use the word “pure” here because it perhaps implies that the applied aspects are somehow “impure,” but it’s unfortunately kind of the standard name. The two are really not that different, and they interact with one another in lots of ways. The main difference is that a “pure” mathematician’s primary aim is not necessarily to do something that is practically useful, though it may end up being practically useful in a few decades or centuries.
I am mainly interested in geometry, which I mean quite broadly. When one typically thinks of geometry, one probably thinks of shapes like triangles and squares and circles. Or maybe things like spheres and cubes. And then there are concepts that one can use to study and distinguish such objects, such as angles, distances, areas, volumes. One can ask about the symmetries of such objects. What are the symmetries of an equilateral triangle? What about the symmetries of a square? There are other tools one can use to study geometric objects. For example, consider all of the ways of drawing a loop on the surface of a sphere. It turns out that any such loop can be deformed into any other such loop. Think about it! Or consider all of the ways of drawing a loop on the surface of a donut. Is it still true that any such loop can be deformed into any other such loop? The answer is no–think about it!
Since your interests in geometry are primarily “non-practical,” would it be accurate to say that they might be at least in part aesthetic? Mathematical geometry is sculpture abstracted and infinitized, is it not?
My interest in geometry is definitely aesthetic in a way, but I need to qualify what I mean by aesthetic. I mainly focus on something called algebraic geometry, and the objects I study usually cannot be drawn or realized in any accurate way, first of all because they live in a non-real and abstract world. Also, these objects don’t necessarily live in one or two or three dimensions. Still, they’re not some mysterious, weird, crazy, obscure, esoteric things. They are concrete things described by algebraic equations, although they are equations with potentially lots of variables. These geometric spaces are called algebraic varieties.
So what is it that makes these things interesting or beautiful? Algebraic varieties can have interesting and surprising properties. For instance, we know that if we draw two distinct points on a flat piece of paper, there is a unique straight line which passes through those two points. We also know that if we draw three random points on a flat piece of paper, there probably isn’t a straight line which passes through all three of the points. This fact is nicely illustrated by John Baldessari’s Throwing Three Balls in the Air to Get a Straight Line. Such facts are themselves simple and beautiful; and they are not necessarily obvious facts. We can also try to ask analogous questions for geometric spaces which are not flat pieces of paper. What would be the analogue of a “straight line”? How many such “lines” pass through two distinct points on these geometric spaces? Once we answer these questions, we can ask even more interesting questions with possibly interesting answers. For example, how do these geometric properties correspond to the algebraic properties of the equations which define the geometric spaces? And then by studying these questions about geometric objects more abstract than a flat piece of paper, we can shed light on the question of what is so special about a flat piece of paper that makes it have the geometric properties that it has. So by going into an abstract world, we can learn more about our real world.
Is there a part of mathematics which is cognizant of what it doesn’t know? Or are you pretty confident that there’s a basic mathematical order to things?
Math is very much comprehensible. It is simple. It is elegant. That’s why I like it. In a sense, it’s probably easier than everything else in life. There is a lot of mystery in mathematics, but perhaps not in the way that you might imagine–for instance, read about Eugene Wigner’s famous article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The mystery of mathematics is in the fact that it works, and works really, really well. There are lots of nice theorems, and another mystery is how these theorems so often admit such beautiful and elegant proofs.
Is that a compulsion that motivates you–the base desire to unlock riddles–or is it a fascination with the very fact of their unknown?
My motivation is simply that I like to learn new things. I like to explore new territory. But I also enjoy making sense of known things or finding an explanation for known things. Sometimes when we explore, we find something unexpected. But this is not always how it works. Sometimes we can vaguely see the top of the mountain, which is where we want to get to, but the rest of the mountain is covered in fog. We know, vaguely, what’s at the top of the mountain. We can sort of see it. But we don’t know how to get there. We have to make sense of the thing that we see at the top of the mountain. We have to find a path up to the top. Once a problem is solved and understood, it becomes less interesting to me. It has already been conquered. We’re always trying to move forward, always trying to advance. All the same, solved problems can lead to more interesting problems. When we reach the mountaintop, we can see more mountains that we were not able to see before.
Can you speak on a broader level about the role, if any, for wonder and the imponderable in mathematics? Do mathematicians even think in such terms?
Wonder plays an absolutely huge role in mathematics. Imagination is very important. How do you discover something new? You have to come up with a new idea. How do you come up with new ideas? You play around. You imagine things. You wonder and you ponder. You ask questions. Mathematics is a very social activity. You ask lots of questions, you try to use all of your tools, and if they don’t work, you try to invent new tools. It’s just like solving any other problem.
There is also a part of math, more specifically logic, which is devoted to answering questions like whether we can be sure that the mathematics we do is completely correct. Or asking how we can be sure that mathematics contains no contradictions. Or whether there are things that are true but not provable.
What you describe sounds like a mathematical ethics. What are some of the ideas that logic has raised in regards to your own work and interests in pure math and geometry?
More or less by definition, all of math involves logic. All of the steps that we make in math are supposed to be logical steps. Whereas mathematicians in general use logic, I guess that you could say that logicians study logic itself, or various other foundational things in math. It is–or can be–a sort of meta-mathematics.
For example, is it possible for us to prove both a statement and the negation of the statement? If this were possible, it would be pretty disastrous, at least from a certain point of view. It would mean that math is inconsistent. A single such example would call into question the validity of all mathematics, actually, which leaves us in an interesting situation. Is mathematics consistent, or is it not? We don’t know–perhaps we can’t know! On the other hand, it doesn’t necessarily have to be a concern, and it isn’t a very big concern for me personally. I mentioned the unreasonable effectiveness of mathematics in the sciences earlier–even if math is at the end of the day logically inconsistent, that doesn’t take anything away from the overwhelming effectiveness of math as a tool for studying the real world.