One of the principal architects of modern discrete mathematics has an obsession.
It’s not a problem Ronald Graham came up against as the chief scientist for UC San Diego’s Qualcomm Institute, or that was asked by one of his computer science or math students.
It’s juggling. It fascinates the 82-year-old. He’s currently co-authoring a book about the mathematics of juggling, and can often be found attempting to keep five balls aloft on the grounds of his house overlooking Coast Walk.
Graham is world famous (well, in math, at least) for discovering a number so big, there is not enough space in the known universe to write down all its digits. He hit upon what is now called “Graham’s number” in 1971, while thinking about multidimensional ways to solve a math riddle.
For someone with such an advanced left brain, Graham’s right brain is no slouch, either. Graham is unexpectedly creative and funny, as when he introduces the other occupant of his house today. It’s not his wife of 35 years, Fan Chung, who is also a celebrated mathematician and who has collaborated on at least 100 papers with her husband.
It’s Roberto, a life-sized human doll seated upright at one of Graham’s many workstations.
“He’s working on something,” Graham explains. “Roberto works a lot.”
So how is juggling like mathematics?
In math, you never solve all the problems. In juggling, you can’t juggle all the patterns. There’s always one more ball. In fact, Claude Shannon, he was the father of information theory. But later in his life, he just cared about the theory of juggling, and he built machines that simulated juggling.
But there’s also something else. The trouble with computing is that the computer does exactly what you tell it. And the trouble with juggling is the balls go exactly where you throw them. You can’t blame the phases of the moon. Every throw is a little bit off and you have to make corrections, but you are responsible.
At age 82, has your ability to solve complex problems slowed at all?
Well, you don’t think so. But then, there are some absolute physical things — like if you’re a marathon runner, you can tell your time slows down, and with juggling, five balls is harder than it used to be.
Does having this beautiful view at least help you think?
Well, you get used to it. It’s like anything.
Can you explain Graham’s number in simple terms?
A three-dimensional cube has eight points. If you look at all possible lines joining the vertices, how many are there? Then you take a five-dimensional, six-dimensional, hundred-dimensional cube and the result is, if the dimension of the cube is large enough, you must always find at least four lines that lie in the same plane. And when is that dimension large enough to guarantee it? That’s the Graham’s number. It’s a bound. It says, by the time you get to that dimension, this has to happen. But once you get to 13 dimensions, it’s already too big to compute. How many dimensions are there? Two to the 13th.
Can you explain Graham’s number in simple terms?
OK, so how large is the actual number? Is it a one followed by a trillion zeros?
Oh, it’s way bigger than that.
What’s Graham’s number plus one?
Is it infinity?
No, it’s a number. Infinity’s not a number.
So how can one infinity be larger than another infinity then?
Well, OK, that’s a question that bothered mathematicians 150 years ago. But look at the numbers that are perfect squares — 1, 4, 9, 16, 25 — then look at the numbers — 1, 2, 3, 4, 5. Are there more numbers than perfect squares? Yeah.
Infinity is still infinity. All infinite sets go on forever.
But there are more real numbers than integers. The integers are a countably infinite set. You can count them. The set of real numbers is uncountable. There’s no way to pair the two infinite sets up. As soon as you try to pair one set of numbers up, you’ve left one out. They’re both infinite, but one’s a bigger order of infinity. The size of the infinite set of countable numbers, which is called alef-naught, is less than the size of the infinite set of real numbers,
The big question now is if there’s anything in between that’s bigger than alef-naught but less than C. That’s called the Continuum Hypothesis, and people worked very hard to try to solve it. But it turns out, it was finally proved that you can’t resolve this from the axioms of mathematics. There can be nothing in between or there can be a lot of stuff. So the question becomes what is the cardinality of the real numbers? It’s almost philosophy.
What is the most difficult math problem that hasn’t been solved that can be?
Probably what’s called the Riemann hypothesis, which is a conjecture from the 1850s about the distribution of prime numbers, which seem to behave kind of randomly. One of the problems for 2,000 years has been, can you always find pairs of prime numbers that differ by 2 — like 11 and 13, 101 and 103.
If math really represents reality, wouldn’t a universal relationship that exists everywhere — like pi — not result in a series of random, non-repeating numbers stretching to infinity?
Representing numbers in decimal form is kind of limited. It’s just one way to represent numbers. But there are other ways in which pi is much more regular. The square root of 2 doesn’t repeat in decimal form. But if you use continued fractions — a way of writing a number as a fraction that goes on forever, where you say 3 + ½ + 1/6 +… — then the square root of 2 is beautiful. So it just depends how you represent them.
Now, one of the real philosophical problems with mathematics — although mathematicians don’t really worry about it too much — is whether this whole system that people are using to prove things is really inconsistent. If you can prove something is true and prove that it’s false, well, you’ve got a problem, you don’t know what you’re doing. And they proved back in the ‘30s that there is no automatic way to prove that your system is consistent — that you won’t eventually end up with a contradiction. People argue that, ‘Well, we didn’t find it yet, so it’s probably OK.’ But you can’t prove that it’s OK.
Why do you think so many people think they’re bad at math? Are they really?
You know, you never hear people say, ‘I’m really not good at reading.’ They don’t say that, do they? Even if it’s true! Partly, I think they had a bad experience when they were young. A lot of teachers are not comfortable with math and they project that to the students, and the students can tell. There’s no math gene, where if you don’t have it, you can’t do it.