Monday, June 17, 2013


Excerpt from Scott Aaronson's Who Can Name the Biggest Number?

Turing continued to explicate his machine using ingenious reasoning from first principles. [A paper tape], said Turing, extends infinitely in both directions, since a theoretical machine ought not be constrained by physical limits on resources. Furthermore, there’s a symbol written on each square of the tape, like the ‘1’s and ‘0’s in a modern computer’s memory. But how are the symbols manipulated? Well, there’s a ‘tape head’ moving back and forth along the tape, examining one square at a time, writing and erasing symbols according to definite rules. The rules are the tape head’s program: change them, and you change what the tape head does. 

Turing’s august insight was that we can program the tape head to carry out any computation. 

Just as we can classify words by how many letters they contain, we can classify Turing machines by how many rules they have in the tape head. Some machines have only one rule, others have two rules, still others have three rules, and so on. But for each fixed whole number N, just as there are only finitely many distinct words with N letters, so too are there only finitely many distinct machines with N rules. Among these machines, some halt and others run forever when started on a blank tape. Of the ones that halt, asked Rado, what’s the maximum number of steps that any machine takes before it halts?

Conclusion? The sequence of Busy Beaver numbers, BB(1), BB(2), and so on, grows faster than any computable sequence. Faster than exponentials, stacked exponentials, the Ackermann sequence, you name it. Because if a Turing machine could compute a sequence that grows faster than Busy Beaver, then it could use that sequence...The Busy Beaver sequence is non-computable, solely because it grows stupendously fast—too fast for any computer to keep up with it, even in principle.

In 1984, A.K. Dewdney devoted a Scientific American column to Busy Beavers, which inspired amateur mathematician George Uhing to build a special-purpose device for simulating Turing machines. The device, which cost Uhing less than $100, found a five-rule machine that runs for 2,133,492 steps before halting—establishing that BB(5) must be at least as high. Then, in 1989, Heiner Marxen and Jürgen Buntrock discovered that BB(5) is at least 47,176,870. To this day, BB(5) hasn’t been pinned down precisely, and it could turn out to be much higher still. As for BB(6), Marxen and Buntrock set another record in 1997 by proving that it’s at least 8,690,333,381,690,951. A formidable accomplishment, yet Marxen, Buntrock, and the other Busy Beaver hunters are merely wading along the shores of the unknowable. Humanity may never know the value of BB(6) for certain, let alone that of BB(7) or any higher number in the sequence.

We’ve seen that progress in notational systems for big numbers mirrors progress in broader realms: mathematics, logic, computer science. And yet, though a mirror reflects reality, it doesn’t necessarily influence it. Even within mathematics, big numbers are often considered trivialities, their study an idle amusement with no broader implications. I want to argue a contrary view: that understanding big numbers is a key to understanding the world. Imagine trying to explain the Turing machine to Archimedes. The genius of Syracuse listens patiently as you discuss the papyrus tape extending infinitely in both directions, the time steps, states, input and output sequences.

At last he explodes. "Foolishness!" he declares (or the ancient Greek equivalent). "All you’ve given me is an elaborate definition, with no value outside of itself." How do you respond? Archimedes has never heard of computers, those cantankerous devices that, twenty-three centuries from his time, will transact the world’s affairs. So you can’t claim practical application. Nor can you appeal to Hilbert and the formalist program, since Archimedes hasn’t heard of those either. But then it hits you: the Busy Beaver sequence.

You define the sequence for Archimedes, convince him that BB(1000) is more than his 10^63 grains of sand filling the universe, more even than 10^63 raised to its own power 10^63 times. You defy him to name a bigger number without invoking Turing machines or some equivalent. And as he ponders this challenge, the power of the Turing machine concept dawns on him. Though his intuition may never apprehend the Busy Beaver numbers, his reason compels him to acknowledge their immensity. Big numbers have a way of imbuing abstract notions with reality.

Indeed, one could define science as reason’s attempt to compensate for our inability to perceive big numbers. If we could run at 280,000,000 meters per second, there’d be no need for a special theory of relativity: it’d be obvious to everyone that the faster we go, the heavier and squatter we get, and the faster time elapses in the rest of the world. If we could live for 70,000,000 years, there’d be no theory of evolution, and certainly no creationism: we could watch speciation and adaptation with our eyes, instead of painstakingly reconstructing events from fossils and DNA. If we could bake bread at 20,000,000 degrees Kelvin, nuclear fusion would be not the esoteric domain of physicists but ordinary household knowledge. But we can’t do any of these things, and so we have science, to deduce about the gargantuan what we, with our infinitesimal faculties, will never sense.

Who can name the bigger number? Whoever has the deeper paradigm. Are you ready? Get set. Go.