Winter 2000–2001

Robert Kaplan teaches at the Mathematics Department at Harvard University and is the author of the best-selling book *The Nothing That Is: A Natural History of Zero*. Sina
Najafi talked with him over the phone to find out how a mathematician approaches games of chance, numbers stations, and other apparently random phenomena.^{[1]}

Cabinet: What is a random number and how does mathematics define randomness?

Robert Kaplan: The most important point is this: we haven’t even a good definition of random numbers. And there is no such thing at this point as a random number generator because however one tries to generate random numbers there is no guarantee that you will not find a pattern in the sequences.

Does seeing patterns that would allow you to predict the next number in a sequence offer any useful definition? Is that predictive ability different from the ability to find a pattern after a series has been finished?

The latter is so much easier, but both are difficult problems. But here is an interesting way of approaching the question. I’m going to give you a number, which is neither zero nor not zero. Ready, here it is. Zero point zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero, zero—bored yet?—zero, zero, zero, zero, zero, zero. Eventually you’ll say, “Are they all zeros?” and I’ll say “I don’t know. I’ve only gotten to the fifty-eighth decimal place.” Maybe after the ten-thousandth decimal place, it will turn out that the number isn’t zero after all but at this point this number has a weird status. It’s neither zero nor not zero. And this bears on the whole issue of an incomplete string of numbers. Because it is incomplete, you can’t look back on it and say “Ah, I see.” This is the position of people who think mathematics and numbers exist in time—that they are not timeless—that if I want to tell you one-half in decimal form, I can’t just say point five and all zeros. I’ve actually got to go through it and tell you all the zeros. How do I do that? I don’t have the time.

The best thing you can do faced with a given string of numbers is to try every single pattern you know on it. There are a lot of patterns around, but a point comes when for all intents and purposes, you say I can’t find a pattern there. But of course there might be one. Our definition of randomness is basically negative. Random means “no discernible pattern as far as I’m concerned.”

Relatively recently, the attempt to make a random number generator has been shifted to using the background radiation of the universe to produce pure randomness. But there too, if you hook up your computer to this radiation, we will find a Gaussian curve. They are waves; there’s nothing we can do about it. A Gaussian distribution is a bell-shaped curve where the peak of the curve shows us what will most often happen and the ends show us the rare and infrequent. I think this speaks to something very deep in the human mind—our obsession with pattern. We are creatures who survive by seeing, making, and thinking pattern.

In a perfectly random distribution, we would have a straight line as opposed to a Gaussian curve because every single point would receive the same number of hits. When you say that there is no such thing as randomness, are you saying that when we throw a die 15 million times, we will not get a perfect straight line?

If it’s a fair die, of course you would get a straight line. It is not the beginning of *Rosenkrantz and Guildenstern Are Dead*, where 89 throws of a coin have come up heads because Rosenkrantz and Guildenstern are dead. In pure randomness, one would get the odds so arranged that, yes, something would not happen more often than something else because of a preponderance built into the system. But in heredity, for
example, if A and B are mating, each with his or her plus and minus. You’re going to get a plus and minus, and a minus and plus, which will outweigh the two pluses, on the one hand, and the two minuses on the other. You get a 1-2-1 distribution. That’s the Gaussian curve. The Gaussian distribution is a sign of a kind of ruled randomness.

So the problem would then be that no throw of the dice is random; it’s just that we don’t have enough data about the exact position where the dice are released, the force it was thrown with, etc. If we did, we could predict accurately where the dice would end up every time. This is an optimist’s view of the world.

Of course. I’m an optimist by default. One cannot help but think causality. Kant describes this perfectly. Causality isn’t something we think about; it’s something we think with. Just because I have not been able to find a pattern, or even if no living human has found one, it does not mean that some incredibly clever, malign figure in the universe hasn’t hidden a pattern in there. So in that sense, I’m also a pessimist.

But it’s also an essentially paranoid view of the world, where beyond every event is a hidden guiding hand orchestrating all of it toward one specific end.

There are two aspects to that paranoia; one is teleology and the other is Missouri. Let me give you an example. Do you hear the sounds in the background over the phone? I think it would be very hard for you to guess what the cause of the sounds is, and if there is a pattern to them. It is in fact the sound of the mailman dropping in the mail through the slot. I can predict them because I can see what’s going on, but you can’t. As far as you know, these are random sounds of the universe but in fact there’s definitely a pattern to them. That’s why we are compelled to find a teleology. Now I see there is a letter from Cabinet here for me. It must be from you.

Actually, that’s no coincidence. I’ve coordinated everything with the mailman so you’d get it in the middle of our talk. If you open the envelope, you’ll find my next question in it.

You see, there’s nothing random in the world! The other thing is people from Missouri, who say “you have to prove it to me.” The feeling there is that despite my best feelings that there is a cause, I’m not going to believe it until you prove it to me. Both components are strongly at work in us.

There’s a website (www.fourmilab.ch/hotbits) that claims to produce completely random numbers based on a random decay box. I understand that computers are incapable of producing random numbers since they are operating according to an algorithm, albeit a complicated one. But has the idea of a random decay box also been discredited?

Quantum mechanics predicts, a strange word to use here, behavior well. So did a lot of medieval theories, as did the theory that the sun goes around the earth. If you only add enough epicycles to it, you can get the right prediction. Quantum theory has two basic positions: either Heisenberg’s, where he says there may be no randomness there but that every time we look, our looking makes it unpredictable; or Bohr’s position, which is that there is randomness. After all, why should we, with our puny, causal minds, reflect the way things are, especially at the quantum mechanical level. But any definition of randomness is still one we make with our causality-drenched minds. It’s a wonderfully paradoxical position. One wants to feign utter chaos. When Einstein says to Bohr that “God does not play dice,” he’s responding to precisely these questions.

There could be two versions of your position also. Is your argument that there is randomness in the universe but we cannot see it because of the human desire to see structure, or the stronger claim that there simply is no randomness in the universe?

I would not even use the word ‘desire.’ I would put it in Kantian terms. Kant says just as space and time are not out there but are our ways of making jigsaw puzzle pieces that our perception can put together, so too causality is our way of taking those space-time pieces and fitting them together. These causal chains have neither beginning nor end, so arguments for a first cause or a God will always fail. To think of randomness is terrifying to us; the difference between structure and randomness corresponds to the Kantian difference between the beautiful and the sublime.

When does mathematics first begin to take randomness as a formal problem?

It begins with Pascal. He comes up with Pascal’s triangle, which indicates how things will fall out given a distribution of chances. Buffon, the French naturalist, found that if you take a needle (it’s called Buffon’s needle problem) of, say, one unit and draw a series of parallel lines on a board that are just a little more than one unit apart, and throw the needle down on the board and count the number of times it lands across a line, as opposed to the number of times it doesn’t, you get a remarkable ratio which is a fraction of pi. It turns out to have to do with the radius of a circle. After this, theories of randomness are developed in England by nineteenth-century scientists and mathematicians interested in statistical behavior and hoping that if you can’t see how individuals are behaving, at least you can see rules to the mass.

The best examples in mathematics of
randomness are prime numbers, which are the building blocks of our numbers, and have defied our understanding. We simply don’t know the pattern of the primes. Given one, we simply can’t predict the next. It’s been a problem for two thousand years. When I say, we don’t know the pattern, I’m assuming there is one, but there might be none. We do know something statistically, something which Gauss discovered in the early nineteenth century. As you go out into higher and higher numbers, the number of primes gets closer and closer to *n* over log *n*, the natural logarithm of the number. That gives a statistical grasp of what we still have to understand individually.

But if you ask a mathematician about primes, he’ll probably say, “We just haven’t found the pattern yet.” Do you know about the Chudnovsky brothers who are counting pi to enormous lengths? They converted their home in New York into a giant computer lab and their only purpose is to find the
next decimal place of pi. Pi is an irrational number, which means that it not only
goes on forever but that it does not have a repeating pattern. They now have billions of decimal places.^{[2]}

I know you’ve been researching the anonymous number stations on shortwave radio. Is there a method to their madness?

These aren’t random numbers. There are many conjectures as to their purposes: that they are for spies communicating (These transmissions continue before, during, and after the so-called Cold War.); that these are smugglers communicating information; that it’s diplomatic traffic. It is conjectured that the Federal Emergency Management Agency (FEMA) may be responsible, as are the National Communications Agency and the KKN, which is apparently part of the State Department. There are hundreds of them; some of them are live voices, and some of them are generated voices.

Apparently, if you find countries in which the Voice of America is broadcast, there are sometimes number stations on a frequency close to the Voice of America frequency, or sometimes, as in the case of Liberia, they will be packaged in on the same frequency.

Without actually deciphering the code, how can you be sure that they are not random numbers?

For a number of reasons. They are expensive to produce, and expensive means probably government support, which means purpose.

There are also certain stations that broadcast single letters again and again. They are called SLHM, Single Letter High-Frequency Monitoring. One conjecture is that they are keeping the channel open for possible later transmission of code, because there are apparently occasional bursts of code. A second conjecture is that they are navigational markers. Another conjecture is that they are weather data transmitters, or measurements of water levels.

So you would wager on a pattern being there because you think there’s an “author”
behind the stations. It reminds me of the film* Pi. *Did you see it?

No, I was told to avoid it.

- Numbers stations are shortwave stations which consist solely of apparently random strings of numbers being read out. There are hundreds of such stations in many different languages. No one is certain as to who is responsible for these stations or what their function is, but most listeners agree that they are related to espionage. A four-CD set
of recordings is available from Irdial under the name
“The Conet Project.” Their very informative website at www.ibmpcug.co.uk/~irdial/conet.htm [link defunct—Eds.] also has some sample recordings posted. Simon Mason’s book
*Secret Signals, The Euronumbers Mystery*is now out of print but is available at www.btinternet.com/~simon.mason/page30.html [link defunct—Eds.]. Donald Schimmel’s book*The Underground Frequency Guide: A Directory of Unusual, Illegal, and Covert Radio Communications*(Solano Beach, CA: Hightext, 1994) is in print and can be ordered through the web. - The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of
*pi*. An article called “The Mountains of Pi” on the Chudnovsky brothers appeared in*The New Yorker*issue of March 2, 1992.

Robert Kaplan teaches at the Mathematics Department at Harvard University and is author of the book *The Nothing That Is: A Natural History of Zero*. In 1994, with his wife Ellen, Robert Kaplan founded the Math Circle, a program open to the public for the enjoyment of pure mathematics. His forthcoming books include *The Math Circle* (with Ellen Kaplan), *Accesible Mysteries* (with Ellen Kaplan), and *Inspired Guessing*.

Sina Najafi is co–editor-in-chief of *Cabinet*.