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Mathematical Siblings

In this interview, mathematical siblings Gregory and David Chudnovsky, who seem to almost think with one mind, banter about numbers theory, advanced computing, and the importance and aesthetics of mathematics. Learn why they call math an art and see formulas that they find particularly beautiful.


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David and Gregory Chudnovsky
David (left) and Gregory Chudnovsky have worked together throughout their long careers as mathematical researchers and teachers.
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© Noel Camardo

How we got into math

NOVA scienceNOW: How long have you been into math?

Gregory Chudnovsky: A long time. David is older so he actually started. He was always quite interested, so for me it was most likely because of David.

David Chudnovsky: My first choice was physics.

Gregory: David is still interested in physics. I am not.

David: Our parents were both engineers and they were very, very good at what they did. I think our father wanted us to be engineers, and it's too bad that we didn't.

Gregory: It actually requires more skill.

David: If you goof in math somebody will find a mistake and you will have an egg on your face. If you screw up a bridge, well, you are in much bigger trouble.

Gregory: Most likely it will be a mathematical mistake anyway.

David: Engineering requires a different set of skills and a different mindset. But mathematics is really a universal science. If you know math you can do absolutely anything.

The nature of math

In the first article Richard Preston wrote about you in the New Yorker magazine, he talked about how there is no reason why mathematics should be found in nature, but it is.

Gregory: Well, it's a philosophical discussion. There's a Platonic point of view on science in general, which assumes that mathematical objects, structures, and so on, are real. And everything else in a sense is a shadow. Some mathematicians do believe that constructs that they're working on are just as real as anything else. Math is more of a natural science than people sometimes think.

"The alphabet of science is mathematics."

What advice do you have for someone who wants to become a mathematician?

Gregory: First of all, everyone should study math as much as they can stand it.

Even those who aren't good at math, like me?

Gregory: It doesn't actually matter. First of all, what does "not good" mean? Many people—most people—will not and should not necessarily become a professional mathematician, like they shouldn't be a professional sportsman. But to say, well, unless you're really, really good at it, you should never do sports—this is a ridiculous statement.

David: Or unless you are very, very good at it you should never sing.

Gregory: Right, ridiculous.

David: Everybody has a capability. A lot of the problems with math go back to the fact that sometimes it's not taught well. A famous 18th-century mathematician, Leonhard Euler, said that if you are not able to explain the most complex mathematics to a 12-year-old you are a fraud as a teacher. Of course, there is a lot of horror surrounding math, but it doesn't necessarily mean that it has to do with the mathematics itself. Sometimes it has only to do with the way it's taught.

Chudnovskys' lab
The floor of the Chudnovskys' lab at Brooklyn Polytechnic University bears a spiral pattern of mathematical equations.
© Noel Camardo

Gregory: It's just the nature of fear that math is considered a hard subject. It's hard to imagine that math is really that much harder than good study of English grammar. Grammar can be as convoluted and require as much mental effort to learn as math.

David: But most people learn to speak English quite passably, and the same thing should be true of math. Of course, you can scare people stiff by reading them the fine points of grammar in a language they can't understand.

Gregory: People should just know that they should be confident. There's really no reason why kids shouldn't know math. These days math is taught reasonably well in high school. But if you haven't already grasped it to some extent by high school, it might actually be too late. By then you might just decide that it's just not going to work. And that's a shame. So, if you just study it from an early age according to the basics and with a good teacher, it's very simple.

Those who are interested in any kind of science, not necessarily math, really should just read as much as possible—math books or anything where math is being used. The silly idea is, "Well, I will never in my life use algebra." If it's really true that you never use any of the mathematical concepts in your life, then it really means that you are living a dull life.

David: There was a great book about mathematics. It's an old one, written by a friend of ours who already died. He was a great American mathematician, Herbert Robbins. It's called What Is Mathematics? It's still in print today. It's a wonderful book. I remember that when I was about 10 years old the book really influenced me greatly when I read it.

Mathematics is the language

So everyone should try to have some proficiency in math?

David: In the 19th century at Yale University, which was great in math and still is, the administration decided to axe mathematics as a subject that has to necessarily be taken. At that time, at Yale, there was a great American mathematician, Josiah Willard Gibbs. He was a man of very few words, very, very shy man. And when they decided to vote to axe math at Yale as a requirement for students he got up and—they wanted to replace the math requirement with Latin or something like that—and he said "Mathematics is the language." So I can only repeat was Gibbs said: Mathematics is the language.

More so than chemistry or biology or physics?

Gregory: Well, all those subjects are extraordinarily important. But in any good study of chemistry, biology, physics, the language is mathematical.

David: You will have to use math.

Gregory: Your sentences, your approach, the underlying structure is math.

David: The alphabet of science is mathematics. It is really the key to the kingdom. Computers would never be possible without mathematics. People don't realize that. Any computer is nothing but a hardwired set of mathematical rules. And that is very often forgotten.

Chudnovskys' supercomputer
The Chudnovskys built their own supercomputer with mail-order computer parts.
© Noel Camardo

Gregory: Any science, engineering, practically anything, you will use mathematical equations. Practically everything, you will have no choice.

David: Mathematical ratios, mathematical algorithms. Whatever.

Gregory: It's just like saying, "I like to read or write but I don't care about letters or sentences and grammar." You still need some sort of background on which everything can be built. Now, do you want to go way above this background? That depends on your interest. Math is as interesting as any other subject. You can push it as far as you want and as far as your interest extends. Kids would have a lot of fun if they would just embrace mathematics.

David: You should not be scared. There is no problem that you couldn't understand. Just try to make sense out of it. Don't be scared. If you don't feel like it, don't push it. It's like with English. You can be forever very mediocre at language and it won't hurt you. You're not going to be Shakespeare, but you owe it to yourself to learn the discipline well.

Beyond pencil and paper

How would you explain what it is that you do to someone who doesn't know what number theory is?

Gregory: We study the properties of numbers.

David: Number theory is a very basic part of math. People use number theory every day. The moment you use your cell phone, you are using a lot of number theory and algorithms. You watch TV, you browse the Internet. It's really mathematics in the form of digital signal processing. There are great engineering contributions to digital signal processing, but without the underlying crunching of numbers—mathematics—modern communication won't be possible.

Gregory: Basic number theory plays an important role in the algorithms of communication. But it also can be completely self-contained. It has played a self-contained role as just "number theory" in many of the most important problems mathematicians have worked on for over 2,000 years.

These days, is number theory always being done on a computer, or do you sit down with a piece of paper and work with numbers?

Gregory: Both, but computers do really help a lot.

David: It's really an experimental science.

Gregory: Yes, it's experimental. It always starts with an individual observation and people, which many times these days starts on a computer.

"Sometimes equations look very simple, but just try to make them work!"

David: A computer is a tool like physicists use accelerators, biologists use microscopes, whatever. Sometimes these tools are very useful and sometimes not. That's all they are, tools.

Gregory: These days computers are more than just tools for number crunching. As computers are getting more sophisticated, they are starting finally to do what they were intended for 50 years ago. There's not yet artificial intelligence, but there is sufficient computer power to allow mathematicians to start using them as development tools. We are beginning a kind of symbiotic relationship with computers. And they are moving in the right direction.

Fifty years ago people assumed that mathematicians would one day be able to formulate a statement and then ask their computer to check whether it's right or not. That's not here yet and we can't say for sure whether this will ever come to fruition. But more than 30 or 40 years ago, mathematicians exclusively used pencil and paper. There was a famous joke, "What is the cheapest department in a university?" The answer was mathematics, because you only needed to provide the math department with paper and pencils with erasers. (Actually, there's a cheaper department, the joke goes, philosophy, because an eraser is not needed!) Today the pencil and paper model is no longer the norm. Mathematicians use computers, and they play a very important role. There are sometimes very hard mathematical problems that can really only be proved with the help of a computer.

David Chudnovsky
David moves around the lab in blue plastic booties, which protect the equations printed on the flooring.
© Noel Camardo

Parallel universe

What are you working on these days?

Gregory: It's a good example of the diverse things that mathematicians do: we are working on a computer chip, which will be very powerful, a parallel computer on a chip. It's a very interesting problem. If you want to make a really, really, really good chip in a very small area, you have to use all sorts of mathematical equations. You have to optimize the logic, the placement, the wiring.

David: It uses status quo mathematics for a very unique result.

Gregory: It's something for which mathematics was used for many, many years. If you think about the planning of a chip, because of its scale, the small size of its features, you are tasked with optimizing the plans for a little digital city the size of a chip. You have to do optimization of various mathematical algorithms to achieve this kind of thing. Optimization of movement, of planning—this is a classic mathematical problem, all done on the scale of a small chip. The smaller the features get the more complicated the mathematics gets.

Will this chip that you are designing have commercial viability, or is it just an experiment?

Gregory: It's really a research chip.

David: It might at some point.

Gregory: But now it's just a research project in the sense that one has to prove that something like this works. It's an interesting problem involving mathematics and computer science. It's an extreme parallelization. We've basically predicted that the performance of computers will somehow double every 18 months. So using unconventional technology, we are out to improve the speed of individual chips. It all boils down to the question, Is it possible now to take standard applications for computers and do it in parallel?

Instead of running it faster it will simply do many parts at the same time. Parallelization of computing is kind of the next big frontier. Parallel machines will do more and more what we've always expected from computers. We will get a big boost in performance. That's what our project is all about. With new chips like ours, parallelization will start taking hold. This will be something new. Kids who are studying now in high school, by the time they get to college there will be an interesting question for them. They will need to start to learn how to take control of these parallel machines made out of different kinds of computer chips. Will they be able to?

"Most of the computers we use are completely underutilized."

Will parallelization have an impact on people's day-to-day lives, or is it just something that will make a difference in the lives of scientists?

David: Medical imaging has progressed a lot in the last half century, but there are inherent mathematical limitations to how well you can acquire the image and how well you can process it. And there are computing limitations on how fast you can process the image. What might have been sufficient years ago is no longer good enough. You want to see everything in 3-D and in motion. And you want to see it on the cellular level. Theoretically this is possible, but it presents a monumental processing problem. This is one of the applications we hope to use our new chip for. It will do image processing but really 3-D and really in motion. When you're dealing with 3-D, your complexity goes through the roof. You immediately run out of everything: storage, processing capability. So you have to have two things: You have to have horsepower and, most importantly, you need a smart mathematical algorithm.

Gregory: You do need a tremendous amount of horsepower. This was the problem with the unicorn tapestry. But there it was a reasonably mild problem because it's only two-dimensional. So we knew we could work with that. If we're dealing with something that's 3-D, you have a really big problem on your hands. Computation like that has become extremely hard. It's not so much an issue of when we can do it, it's even whether we can do it at all.

Will parallel computing help with these problems?

David: Maybe.

Dürer's etching, <i>Melancholy Octahedron</i>
In addition to the equations, the Chudnovskys have a massive copy of this Dürer etching, Melancholy Octahedron, on the floor of their lab. It contains a lot of hidden math, they say.
© Noel Camardo

The art of math

Do you have a favorite math equation?

Gregory: Sorry, no.

David: No.

Gregory: There is no such thing as the formula of the month. Interesting idea though. But sorry.

But what about all those equations on the floor of your lab? You must at least like those?

Gregory: We like those equations.

David: The underlying structure is interesting and graphically equations are quite stunning. They're like art. But the real beauty is the machinery behind them. But unfortunately, sometimes that machinery doesn't look that cute. Sometimes equations look very simple, but just try to make them work! Like E = mc2. Suddenly you have an H-bomb! There's some heavy-duty stuff behind equations. Sometimes equations get people excited, but the real excitement is behind the scenes.

Gregory: Sometimes a final equation is all about showing off.

Does the complexity of these equations make them similar to or different from great works of art?

Gregory: All great works have complexity behind them. That being said, I'm not sure you want to know a great deal about Da Vinci just by gazing at the Mona Lisa.

David: We have a huge Dürer etching on the floor of our lab, Melancholy Octahedron. The images hide a lot of very sophisticated math. They really have multiple levels of meaning. And lots of intrigue. There is no mathematical consensus about what the heck is shown in the picture.

Gregory: But if you look at Dürer's etching and you take it out of cultural and historical context, it's really not that great. If you look at his famous Four Horsemen of the Apocalypse, you don't know what it's about. It looks weird, scary, more like a page out of a comic book than anything else. Like something from Night of the Zombie. So sometimes you have to know what it is that you're even looking at. There's a parallel to math there.

David: It's a well-known story that there was once an anthropologist who visited a tribe in the rain forest somewhere and showed the tribesmen their photographs. These people could not even recognize themselves with a mirror. It takes quite a bit of practice and assimilation into culture for a person to be able to recognize your own image in 2-D. There is a similar thing going on in math and art. You need certain preparation before you can get meaning out of things you see.

Gregory: Some equations are just pretty and nice to show to kids. One is e^(i*pi) = -1.

David: But the math behind this is very, very sophisticated.

Tips of the trade

Have you always worked together?

David: We have subjects where we work separately and subjects where we work together. But we've always worked together to some extent.

Do you argue?

David: Yes.

When you get mad at each other, how long does it last?

Gregory and David: It depends.

What's the longest it's ever lasted?

Gregory: I don't think we've counted something like that.

Brothers Chudnovsky
The mathematical pair show the power of collaboration through their work together as well as with scientists outside their field.
© Noel Camardo

Can a smart teenager get everything he or she needs from a hardware store and just go up to the attic and build a supercomputer like you two have from mail-order parts?

Gregory: A smart guy can go to a hardware store and then assemble some of the parts. He can order the rest online. But most of the computers we use are completely underutilized. If a smart kid actually has something that he wants to do, then right now he can just assemble a computer from standard parts. He doesn't need a supercomputer. If the kid is a little bit older, he can get this chip we are working on when it shows up and try to figure out how to make parallel computing work.

"A computer isn't a substitute for ignorance."

David: The key is not the device. The key is the algorithm, your project, and what you want to do with it. Jascha Heifetz, the violinist, once told this story: Someone said to him, "Your violin plays beautifully." Then he put it down and said, "Now how does it play?" It's all about the person who is using an instrument like a computer and how he or she is using it. A computer is a dead piece of metal.

Gregory: But if you don't have a tool you are in trouble.

David: If you have only a stone it's a problem. An axe is better. A laser is still better. It has to be adequate to the task.

Gregory: Simple computers, like PDAs, are quite adequate to the task of what people are using them for, like listening to MP3s and sending e-mail. If you already have a laptop or a desktop you are not using it to its potential.

David: Every desktop now is what a supercomputer was 20 years ago. They were the mainstays of the U.S. Department of Defense and everything. Now, 12 hours a day it is gathering dust...

...when it could be flying the space shuttle?

David: Yes! The computer flying the space shuttle is basically like your PDA.

Gregory: The Apollo mission would not have occurred without computers. But the computer that was operating it was worse that the processor of a calculator you can buy in a supermarket. Have we really changed the requirements of computing for what we want to do? Not really. This is what kids should be working on figuring out.

Think big

What about the idea of distributed computing?

Gregory: It's a very interesting idea and good for research purposes, but it takes 10 years to gather enough research to understand anything. People have to figure out better ways of doing big computing projects.

David: We have friends who are geneticists, and they want to start modeling every gene in the body, and they ask us if we can make them a computer. The answer is yes, we can give you the computer, but it is not going to do anything for you. Unless you are in the wizard business, you can have several football fields worth of computers and programmers all working on the same problem, but you would need a wizard to sit in front of them and give them commands to make it all work. You have to know the appropriate questions to ask. A computer isn't a substitute for ignorance.

Gregory: This is one of the problems of parallelization. Just learning how to bring chips together is not enough. We have to know what to ask them to do. There's a lot of mathematical work and conceptual work that will have to be done.

David: There is a big problem with weather modeling or climate modeling. First of all, we don't know the equations which describe atmospheric properties or how things in the atmosphere at different heights interact with each other conceptually. This is a huge problem. The other is even probably more pernicious. We don't have enough data with which to crunch. You cannot make reliable predictions if you don't have data.

Gregory: Eventually somehow it will happen. People will make it happen.

David: Mathematics by itself won't do you any good. You really have to know what you are doing.

Do you think people are starting to rely too heavily on computers?

Gregory: No, they take them for granted.

Has the U.S. fallen behind in science or math?

David: No. America is number one. If we start believing we have fallen behind we will fall behind. So I reject that mentality. Kids from all over the world, even places like France and England, they all want to come here to study. If we have fallen behind I don't know why that would be the case. They are coming here to Brooklyn Polytech! There are a lot of smart kids here.

Gregory: It's possible that in the future we will have problems. But not right now.

Maybe people will read this interview or see your segment on NOVA scienceNOW and want to go into math.

Gregory: That would be nice.

David: That would be great! People with a good education can do anything.

National corporate funding for NOVA is provided by Draper. Major funding for NOVA is provided by the David H. Koch Fund for Science, the Corporation for Public Broadcasting, and PBS viewers. Additional funding is provided by the NOVA Science Trust.