A New Kind of Science, by Stephen Wolfram

Everyone who hacks their way beyond algebra in math class winds up having to slog through lessons on some peculiar math tool — first it was the slide rule, then for my generation it was the graphing calculator. You’d see older kids with the tools of the trade in their backpacks and wonder what the hell you were in for in high school. These days, the common Red Badge of Math Dorkiness for high-school and college students has become virtual — it’s a computer program called Mathematica, a software tool that can actually do math problems for you. Literally — you can type in your math problem, with all the little x’s and weird math symbols, and it will chug away and give you the answer. This ain’t no calculator, that only speaks in numbers — this Mathematica thing will give you back answers to algebra problems, with letters and fractions and all the other little funny math symbols.

That would have been nice back in math class, eh? Well, don’t feel too jealous of today’s math students — I guarantee there’s an unbathed college student out there right now pulling an all-nighter to finish a Mathematica-based homework assignment. But it really is a remarkable piece of software. And the man behind it all is just the kind of genius you’d expect — at the age where you and I were pulling all-nighters to write that book report on Brazil, Stephen Wolfram was wrapping up his PhD in physics. He won a MacArthur “genius” grant in 1981 at age 22, barely old enough to drink alcohol, and before the end of the decade he’d founded the software company Wolfram Research to sell Mathematica to students everywhere. After turning this rare double-play (world-class pure research and making it big in the business world), he went off on his own for ten years to create a completely new scientific research path. (Or so he claims — more on that later.) He essentially spurned publication of his research in traditional scientific journals, instead working in this new area in scientific seclusion until releasing a summation of it all in a gigantic tome called “A New Kind of Science”. If that isn’t enough, he can bench-press 420 and used to play drums for Peter Gabriel.

Alright, I don’t actually know what Stephen Wolfram bench presses (though I would love to see someone have the huevos to ask him), but isn’t it enough for a man to create from scratch an entirely new area of science? It’s a branch that (if you were to draw all of science as a tree) would stick out somewhere in between computer programming theory, mathematics, chaos theory, and statistical mechanics. It also happens to be a branch that existed before Wolfram’s book, that started with a fun little mathematical curiosity called “cellular automata”. This is one of the more blatant examples of a name being way too intimidating for the concept it embodies — cellular automata are very simple, fun, easy to play with, math puzzles not much more complicated than a game of Battleship.¹ Think of a game of checkers, played on a big grid of squares (what they call a “checkerboard”, to use the technical jargon), where the little pieces jump around and make each other disappear. Now, tweak the rules a little bit, start playing fast and loose with the rules. Say, give permission for checkers to reappear on the board, reanimated from the dead, in certain situations. Maybe, what the hell, make all the checkers just one color, instead of two opposing colors. Invent some different rules for when checkers have to be taken off the board, like maybe a checker has to go bye-bye if two other checkers wander up next to it. And that’s basically it – a “cellular automaton” is just a big checkerboard grid, with checker pieces appearing and disappearing over time, based on whatever rules you want to make up for each turn.

That’s it? Why the fancy name? Well, it turns out that if you pick your rules just right, all kinds of strange things happen on the board. Most of the time, nothing much special happens — for typical sets of rules you pick, what happens over the course of the game doesn’t look that interesting — sometimes all the checkers ‘kill” each other off until the board is empty, other times checkers multiply like rabbits until every single space gets filled and nothing else can happen. Occasionally, though, you stumble across a set of rules for playing the game where the checkers create intricate patterns that propagate across the checkerboard like ripples on a pond.  The patterns move around on the board, interact with each other, spawn new patterns — in effect, start to look temptingly like little life forms.

By far the most famous of these is a rule set created by John Conway, called the “Game of Life” since it makes such startlingly lifelike creatures. Here’s how you play the game:

• Start with a normal game of checkers, but play with only one color. There’s no competing white vs. black.
• Throw down some checker pieces at random, wherever you want.
• At each turn, look at the pieces on the board, and decide which piece lives or dies according to how many neighbors it has. If a checker has no neighbors or just one neighbor, it “dies of lonelieness” and is taken off the board. Four or more neighbors, and it develops an intense hatred for life in the crowded slums, and it leaves the board. Two or three neighbors, and the checker is happy, and stays on the board.
• For any empty space on the board that has three neighboring spaces with checkers, you get to reanimate a dead checker, and put it in the empty space to wander the earth as an undead zombie checker, forever doomed to feed on the brains of chess pieces.

It looks kind of boring, might get to be a pain in the ass after a while to be counting neighbors. But if you code it up on a computer, and let the computer work out all the details, you can just watch the game go by in real time. If you do this, you’ll see all these funny little shapes made from clusters of adjacent checkers, moving around the board and bumping into each other. A bunch of shapes are possible — they’ve been named “gliders”, “spaceships”, “boats”, based on their shape and what they seem to do. There are even “guns”, persistent shapes that burp out gliders and other shapes.   Check out the example shown here (borrowed from the excellent Wikipedia page).  It’s frankly startling the menagerie of creatures that inhabit the Game of Life.  Lots of really smart people spend their free time exploring the world, cooking up interesting examples, and generally competing at out-nerding each other with what they can get the game to do.

 

So — back the Wolfmeister, and his gigantic book. Stephen was involved in the early years of cellular automata, published a few papers on the subject, founded a journal — in all respects was a major player in the field in the early eighties. Then he took his ball and went home. By the time Vanilla Ice was ruling the radio, Stephen Wolfram had disengaged from the research world and stopped publishing his work. From that point on it was more or less him and his computer (running Mathematica) doing work in what he believes to be a new theory of the universe based on cellular automata and similar simple “games”. Instead of incrementally publishing his work for everyone to chew on (as everyone else does), he saved it all up for ten years and released it in one giant dollop. And he didn’t publish in respected scientific journals (despite that he could get published in the most prestigious journals on reputation alone), instead he went straight to the public with a giant book, written for laymen, cleverly called “A New Kind of Science”.

Maybe you’ve seen it in the bookstores. Don’t remember it? It’s that thing towering over the bookcases from the science section, visible even way over in the coffee bar. Look at this thing! Just shy of 1200 pages, all told — the *index* is longer than a typical PhD thesis, for criminy’s sake. This book is practically a cube.  In it, Stephen Wolfram takes us on a guided tour of the world he built up in his ten years of exile, showing literally hundreds of examples of cellular automata and other simple little mathematical toys, that sometimes produce surprisingly intricate patterns (much like Conway’s Game of Life).

This is most definitely not bathroom reading (unless you eat lots of cheese and fried dough). It’s a long trek to make it to the end, though it is a surprisingly easy read for a book that took 10 years to write — the first third of the book is a guided tour lightly skipping by lots of example cellular automata (and variations thereof), with not much theory or abstract concepts to slow you down. It’s a marathon, but the big hills don’t appear until the end. In fact I found myself skimming here and there, as he has the bad habit of repetition.

Speaking of repetition, Wolfram’s guilty of an offense common to many of these otherworldly minds who write popular science books — inability to calibrate the level of explanation right, not too hard or not too easy. The author’s complete failure to empathize with normal human thought processes results in key points being repeated way too many times throughout the book, peppered here and there with momentary lapses back to incomprehensible genius-speak. Chapter 1 is particularly brazen case of over-repetition — you could probably compress the chapter down to about four bullet points. Overall, you could (and someone should) wring out the excess fat from this book and cut the page count by 50%.

But what he lacks in conciseness he more than repays in audacity. I don’t think I’ve ever read a book that makes bolder claims — to find prose this proud of itself, you have to look beyond respected scientists and read the latest from pseudoscientific crackpots on the web. Granted, cellular automata really are pretty amazing — it is stunning that you can get such intricate, lifelike patterns from such simple little toy setups. Wolfram has discovered hundreds (probably thousands) of simple rule-sets like Conway’s Game of Life that start from games an elementary school student could grasp yet create a wide range of intricate patterns. Some have an artistically intricate regular structure to them (like a celtic knot or a persian rug), while others look to all judgment to be essentially random. This is reminiscent of Chaos Theory (a division of physics that has the all-time coolest name of any theory), where random-looking complicated behavior pops out of extremely simple situations. There’s no question that this area is very interesting, poorly understood, and potentially very important to understanding the world, but I think Wolfram’s let it get a bit to his head.

Sooner or later, anyone who studies physics gets seduced by chaos theory. I won’t go into detail on it here, but the cool aspect about chaotic systems is how they can create complex, intricate, random-looking patterns from very simple scenarios. For example, a simple little mathematical equation called the logistic map is so easy to work out that you could compute it on a watch calculator (remember those?)², and yet it spits out a string of numbers that look completely random. They *look* random, but they’re not — if you spend the afternoon cooking up the first 1000 numbers from the logistic map, and your eighth cousin Helmut in Germany does the same thing next week, you’ll both get exactly the same numbers. Compare this to the last 1000 lottery numbers that popped up in the North Dakota state lottery — these numbers really *are* random, and couldn’t be predicted or duplicated by any process, much less something as simple as the logistic map.

And here’s the tempting bit. You occasionally stumble across simple little procedures (such as cellular automata or the logistic map) that generate very complicated, random-looking patterns. You also happen to see complicated, random-looking patterns all around you every day in nature — in the swirls of milk in your coffee, the pattern of stars in the sky, even the last 1000 lottery numbers in the North Daktoa lottery. So… any chance that some of these are being created by simple little setups like a cellular automaton? Could something that looks random (like the stock market, for example) really be completely determined by some unknown but simple little equation, that we just don’t happen to know? Maybe the randomness of the world would resolve itself, become predictable and understandable, if only we could discover what simple mechanisms are creating it all.

It’s a tempting bit o’ reasoning, but so far there’s no real evidence that it’s true on a grand scale — things out in the world that look complicated and random might really be just complicated and random. But it’s the central argument of Wolfram’s book. The key argument, then, seems to be that 1) some simple cellular automata can make complex-looking patterns, 2) nature has lots of complicated patterns, so 3) cellular automata must be behind it all.  I pulled a hamstring trying to keep up with that syllogism.  What’s more, the series of real-world examples he walks us through to support his argument ratchet up from completely mundane to breathtakingly ambitious faster than I’ve ever seen.  In the span of a few chapters, he lays out scientific whiplash in three steps:

Step 1: hey, look at all these cool-looking patterns that I get when I do this little computer program!

Step 2: Whaddya know, these little pictures kind of look like some things you see in nature, like snail shells, tree branches, and the veins in leaves.

Step 3: I wonder if this could be responsible for the entire universe?

In other words, Wolfram proposes that the entire universe, all of existence, is one colossal cellular automaton. Based on the fact that occasionally you see patterns in nature (like the stripes on a snail shell) that sorta look like his automata. If you squint.

To me, one of the most genuinely impressive examples in the book is *not* his attempts at a theory of everything, but those simple mollusc shells. The pattern on a sea mollusc shell isn’t particularly remarkable by itself, but I’ll be damned, it really *does* look like the cool patterns that some of Wolfram’s cellular automata spit out. This is one of the few cases where my betting money is on a cellular-automaton mechanism like he describes being responsible for the coloration on the shell. I dunno about his patterns behind a universal theory of everything, but I think he may have the mollusc-shell problem all wrapped up.

To be fair to Wolfram (I sense him, as he reads this, pausing before pressing the “release the hounds” button), it isn’t completely preposterous that a cellular automaton could create a universe as complex as ours. Back before WWII, Alan Turing proved that a very simple device, much simpler than the cellphone in your pocket, can in principle emulate any other computing device and hence perform *any* computation, no matter how complex. This device (now called a Turing machine) consists of a simple printer that can print and/or erase a single letter at a time, on a simple piece of paper “tape” that shuffles along under the printer. It operates by reading the tape and deciding what to do next given the value of the current symbol. It’s basically the tickertape device you see spitting out stock prices (one of the official Stereotypes used by movies and TV)³, except it can back up and rewrite. If you’re clever (i.e. nerdy) about it, you can set up this contraption to chug through any possible calculation whatsoever — even an incredibly realistic simulation of the entire universe and everything in it. It’ll take a while (and yards of tape), but you could do long division on it. Another mile of tape, and you could check your gmail.

So if time and tape supply are no object, you can do anything you want on a Turing machine — all those G5 pentiums just save you a pile of tape. And if a little tape-chewer can do whatever a supercomputer can do (albiet inefficiently), that means that very simple mechanisms can be set up to do arbitrarily complex patterns. (Sound like I’m repeating myself? At least I’m not as bad as Wolfram.)  You don’t in principle *need* anything more complicated than a Turing machine, since a Turing machine can be set up to emulate anything more complicated that you might need. It’s only for convenience that we bother to build computers and electronics — anything, from your TV, to a Cray supercomputer, even the most complex computer that conceivably could be built in the future. Pretty remarkable.

And from this idea, the next cognitive leap was to consider the entire universe as kind of a “natural” computer. After all, if a Turing machine can simulate a computer that can simulate the universe, then basically you’re saying that a Turing machine could be *making* the universe. If we simulate the universe using a computer program, the little simulations of you and me would be programmed not to realize they were part of a computer program. And if you did the simulation on a Turing machine, the little simulated people wouldn’t realize they exist only as letters on a piece of tape. Now nobody seriously thinks that all our universe is done on a big steam-powered paper tape printer — but the cool thing is, the entire universe *could* be created by something just as simple, and we’d never know. Something, perhaps, like a cellular automaton?

Yep, so Wolfram claims. He expends quite a lot of effort proving that some of his cellular automata are just as good as a Turing machine, and can therefore perform any calculation whatsoever, no matter how complex. Setups like the Turing machine that are sophisticated enough to do any computation are called “Universal”, and he succeeds in proving that one of his cellular automata is indeed universal. This is actually a big acheivement in the field (though it seems that one of his research assistants did the work, not him).

Although, I gotta say, it left me a little underwhelmed to see how his particular cellular automaton (called “rule 110” in notation he invented) can be rigged up to be a universal computer. Rule 110 is very simple, basically just as simple as Conway’s Game of Life I talked about earlier (which also was proven to be Universal). And man is he proud of it — a simple little automaton, one of the 256 simplest possible automata, one dimensional, even… and it achieves universality? Sure — but look at what you have to do to make it universal! You have to take my word for it, on the Rube Goldberg-ian contortions they go through to show how to set up a universal computation scheme using this rule. On second thought, you probably shouldn’t take my word for it, since I skimmed through most of it. They in effect build up little gizmos and tools using the funny little structures that rule 110 creates, then build up more complex tools using the simpler tools, until they get what they need.

To me, it seems like a stretch to be boggling about how these simple rules are “universal”, when you have to go through such contortions to make them such. Even the Turing machine, the Old Wise Man of the universal computers, you’d have a hell of a time actually setting it up to do long division, much less simulate the universe. Should we really be that impressed?  If I told you that you that I made a working, functioning aircraft carrier out of Legos, would you be impressed at the simplicity of the Lego building block, or the sheer amount of effort I went through to arrange the Legos the right way? Wolfram’s rule 110 won’t compute for you out of the box — it takes a lot of human ingenuity to get it working for you, and maybe the surprising simplicity of the automaton is outweighed by the intellectual sweat you have to expend to get it up and running.

So now I’ve written a book review almost as long as his freaking book. As my brother would say, “to make a long story short…” ⁴  It all boils down to this question — in his ten years of withdrawal from the rigors of public research, did Wolfram really single-handedly create a new branch of science, or did he just disappear up his own ass? There is no doubt some cool stuff in this book, a couple of genuine scientific advances, and it is (mildly) fun to read. But I think Wolfram’s become too entranced with the admittedly addicting world of toy universes, where the work is mostly empirical exploring and the next Game of Life might be just around the corner, waiting to be discovered. If he’d published in normal journals (like anyone not living off Mathematica royalties would have to do), he probably would have been forced to trim the bloat from his narrative and would have made more progress faster.

For further entertainment, cruise the web for some of the other reactions to his book. My favorite is Cosma Shalizi’s review, entitled  A Rare Blend of Monster Raving Egomania and Utter Batshit Insanity    (http://www.cscs.umich.edu/~crshalizi/reviews/wolfram/). I couldn’t sum up my objections to Wolfram’s book any better than he does:

“I don’t even object to writing 1000 page tomes vindicating one’s own views and castigating doubters; I do object to 1000 page exercises in badly-written intellectual masturbation.”
Hear hear! Until the day Wolfram vindicates his theories by discovering the automaton that spontaneously produces better-written scientific essays, let’s all try to remember — Stephen Wolfram might be rich, but do NOT sit next to this man on a long plane flight…

 

 

A new kind of footnotes:

1.   You sunk my battleship…  using a gigantic book

2.  I had a calculator watch in 3rd grade – thought I was as cool as Michael Knight in Knight Rider.  Then the watch died one day in class, emitting all kinds of sad beeps as it went out.  It sounded like the KLF showed up in an Ohio elementary school class…

3.   Stored in the stereotype warehouse right next to French person with beret, thin mustache and red striped shirt

4.   And as we used to tell him, “Too late!”