Posted on 06/09/2002 5:39:56 PM PDT by aculeus
AMONG a small group of very smart people, the publication of ''A New Kind of Science,'' by Stephen Wolfram, has been anticipated with the anxiety aroused in literary circles by, say, Jonathan Franzen's recent novel, ''The Corrections.'' For more than a decade, Wolfram, a theoretical physicist turned millionaire software entrepreneur, has been laboring in solitude on a work that, he has promised, will change the way we see the world. Adding to the suspense, the book has been announced and withdrawn as the artist returned to his garret to tinker, ignoring the bad vibes and hexes cast by jealous colleagues hoping to see him fall flat on his face.
Now, weighing in at 1,263 pages (counting a long, unpaginated index) and 583,313 words, the book could hardly be more intimidating. But that is the price one pays for a first-class intellectual thrill. While experimenting with a simple computer program 20 years ago, Wolfram stumbled on something rather eerie: ''the beginning of a crack in the very foundations of existing science.'' Ever since, he has been following it deeper as it widens into a crevasse.
The normal thing would have been to dispatch regular reports from the field -- unreadable papers published in fashionable zines like Physical Review Letters or Physica D. Instead, Wolfram decided to do what Darwin did (and he would not shun the comparison). He is springing loose his vision all at once, in a book intended for nonscientists and scientists alike.
From the very beginning of this meticulously constructed manifesto, the reader is presented with a stunning proposal: all the science we know will be demolished and reassembled. An ancient error will be corrected, one so profoundly misguided that it has led science down the wrong avenue, until it is approaching a cul-de-sac. The mistake (as everyone who hated calculus will be happy to hear) is trying to capture the richness of the universe with mathematical equations -- Newton's, Maxwell's, Einstein's. All are based on an abstract, perhaps dubious idea -- that time and space form a seamless continuum. Whether dealing with an inch or a second, you can chop it in half and the half in half, ad infinitum. Thus things can be described with unlimited, infinitesimal precision.
This conceit works fine for simple phenomena like a planet's trajectory around the sun or a weight falling from the Leaning Tower of Pisa. But as scientists try to explain systems of greater complexity -- a hurricane, the economy of Portugal, a human or even a reptilian brain -- the calculations become ever more elaborate until one is left with an unwieldy array of symbols that do not explain much at all.
Wolfram believes that even his own field, theoretical physics (he got a Ph.D. from Caltech when he was 20), suffers from the problem. Equations can capture characteristics of individual particles with breathtaking precision. But put three or four particles together and the complications begin to overwhelm. The problem, he proposes, is that equations are the wrong tool for the job. They should be replaced with computer programs -- more specifically, the little snippets of software called algorithms.
That sounds absolutely ridiculous. Programs are just human inventions, marching orders for a machine. They serve well as a quick and dirty means of tricking a computer into approximating the smoothness of nature, roughing out reasonably good facsimiles of a scientist's perfect equations. But computers understand nothing but 1 or 0, with no gradations in between. Algorithms can mimic reality's grain as finely as the engineers can manage, but the simulation can never be as sharp as the real thing.
Wolfram contends that this, the common wisdom, gets things upside down: the algorithm is the pure, elemental expression of nature; the equation is an artifice. That is because the continuum is a fiction. Time doesn't flow, it ticks. Space is not a surface but a grid. A world like this is best described not by equations but by simple step-by-step procedures. By computer programs.
The universal operating system Wolfram imagines is not something horribly complicated like Windows. The key idea in the book is that simple, byte-size programs have the surprising ability to produce endlessly intricate behavior. His most basic example is a group of elegant little algorithms with a clunky name: cellular automata.
These have been kicking around in the popular science press for years. Start with a row of squares (the cells), some white and some black. Then transform the pattern according to a mindlessly simple rule. Here is an example: if either of a cell's neighbors is black, then make the cell itself black in the next round; otherwise, make it white. That is the whole program. Print each new generation below its progenitor and a pattern unfolds like a piano roll. Automate the procedure with a computer and watch what scrolls down the screen.
Most of these experiments -- Wolfram has tried them all -- settle into numbing repetition, churning out the same configuration again and again. But every now and then a rule takes flight and soars. What Wolfram calls Rule 30 sounds about as dull as can be: if a cell and its right-hand neighbor are white, the next time around make the cell the same color as its left-hand neighbor is now; otherwise, make it the opposite. Apply the rule to a single black square and the pattern that emerges looks every bit as random as the snow on a television tuned to an empty channel. You have to see it to believe it, and Wolfram obliges with stunning illustrations (including the book's goldenrod endpapers, spattered with output from Rule 30). The implication is that some computation like this may be the engine of entropy in the universe.
Other rules have the opposite effect: seed them with a random jumble of cells and, after a few iterations, they begin generating complex order. Some of the output resembles intricately varied stalactites; some looks like tracks of colliding particles in a high-energy accelerator lab. Think of stars and galaxies emerging from the confusion of the Big Bang, or life from the primordial sea.
Most pleasing to the eye are rules generating nested patterns like those of a crystal or a snowflake, or the markings on a seashell, the branching of a leaf, the spiral of a pine cone. Other patterns swirl like clouds, smoke or turbulent streams of water.
Wolfram believes he has clinched the deal with what, for many scientists, will be the meat of the book: a proof that a simple cellular automaton can be programmed to perform any conceivable computation (making it equivalent to what the British mathematician Alan Turing called a universal computer). If you buy all this, then a simple algorithm like those described in the book could constitute the machine code of the universe, the platform on which all the other programs run.
One idea after another comes spewing from the automata in Wolfram's brain. Maybe it is not evolution but algorithms that generate biological complexity. Maybe, if everything arises from computations, it makes perfect sense to think of the weather and the stock market as having minds of their own. Maybe free will is the result of something called ''computational irreducibility'' -- the fact that the only way to know what many systems will do is to just turn them on and let them run.
All this is laid out clearly and precisely. Any motivated reader should be able to plow through at least a few hundred pages before the details become too burdensome. Then one can just marvel at the pictures. (It's evident why Wolfram, who adds depth to the term ''control freak,'' published this work himself. Some illustrations contain hundreds of checkered cells per inch, requiring ''careful sheet-fed printing on paper smooth enough to avoid significant spreading of ink.'')
Probably only scientists will read the 348 pages of notes (though these can be very amusing, providing us with Wolfram's thoughts on subjects like ''clarity and modesty,'' ''whimsy'' and ''writing style''). Many may already be thumbing through the index, whetting their knives. At least in the main text, Wolfram often gives the impression that he has the field -- sometimes called physics of computation -- all to himself. Some of his colleagues will find their work acknowledged in the notes; others may not.
Yet Wolfram has earned some bragging rights. No one has contributed more seminally to this new way of thinking about the world. Certainly no one has worked so hard to produce such a beautiful book. It's too bad that more science isn't delivered this way.
George Johnson contributes science articles to The Times. His new book, ''A Shortcut Through Time,'' will be published next year.
Of course, though I don't know what you are getting at. I very wisely stick to those subjects that I actually know something about when making assertions (which in fact does include the Bible).
Oh, then I guess we won't be hearing much from you on the subject of personal humility, huh?
Hardly stunning - it happens fairly often. This is the real, original meaning of "paradigm shift" and anyone who thinks science is locked in concrete hasn't been paying attention.
Well, I don't know. Just picking through the idea of algorithms vice equations, I can think of some real problems with this approach.
The algorithm approach, to be of any use whatsoever, requires knowledge of initial conditions, one set per element of the system being considered. And without something (like an equation) to tell us what is a proper initial condition, from whence would we gain such information? Beyond that, one has to think that Mr. Heisenberg would have some objections to the idea.
Also, the idea of using such algorithms for predictive work, not to mention theoretical work, introduces a plethora of additional difficulties. I think we'd be seeing Monte Carlo-based theories -- which would be rather hard to apply without some manner of equations to back it up.
Finally, I'm not all that impressed by the idea that Wolfram has it all figured out simply because his cellular autonoma "look like" other things. So do fractals. The real test would be in showing that those algorithms are in fact representative of the real mechanisms at work in nature. How would Wolfram propose we answer such questions?
He may well have answered these issues in his book; for example, he could set up his system elements with equations and just let them run, to see what happens.
Personally, I'm cynical enough to think he's out to sell his software. And, more importantly, to become a "guru" who makes big consulting bucks for seminars and gullible upper managers.
He may well have answered these issues in his book; for example, he could set up his system elements with equations and just let them run, to see what happens.Actually, he does claim to have found, by letting such simple programs just run, that simple rules can determine some structures found in nature in a way that he holds is general: The structures are explained comprehensively, and the programs do not predict structures that are not found in nature.
A man's gotta know his limitations.
Ok, let me take a stab at it.
I haven't read the book itself, but I've read a lot of descriptions, excerpts, and reviews, *and* I did a lot of experimentation/reading on cellular automota (this is what Wolfram is exploring) back around 1980 when Conway's "Life" automota was all the rage.
Short form: If you divide a space (planar, 3D, or oherwise) into an array of "tiles" (e.g. the squares on a checkerboard), and choose a set of possible states for each tile (e.g. on/off, black/white, 1/2/3/4, etc.) there are a huge number of possible "settings" for the space as a whole -- practically an infinite number.
Now if you pick a "rule" that all tiles obey which determine under what conditions a tile will change from one state to another every "clock tick", you have a "cellular automota". Each tile acts as an independent "cell", which is "automated" by the universal rule. Traditionally, a rule can consider only the current state of the cell in question, and the states of the cells immediately surrounding it (i.e., touching it).
Conway, back in the late 70's, discovered that if you use a checkerboard tiling, two cell states ("live"/"dead"), and a very simple rule:
then the results are very "lifelike" in behavior. And pretty much impossible to predict. It's also possible to construct "machines" that consist of many tiles working together to form stable patterns and/or actions.
Death If an occupied cell has 0, 1, 4, 5, 6, 7, or 8 occupied neighbors, the organism dies
(0, 1 neighbors: of loneliness; 4 thru 8: of overcrowding).Survival If an occupied cell has two or three neighbors, the organism survives to the next generation. Birth If an unoccupied cell has three occupied neighbors, it becomes occupied.
For a marvelous visual demonstration of Conway's rule in action, go to http://www.radicaleye.com/lifepage/java.html
Be sure to play with the library of interesting patterns at the same site, there are some truly amazing ones. And use the zoom out button ("<") to ensure that you're seeing the behavior of the entire pattern and not just one isolated section of it.
Wolfram's book seems to be a treatise claiming that our universe itself might actually be the result of such mechanisms -- perhaps at the tiniest level every "piece" of space is a little "computer" which uses the same simple rule as every other piece of space, and the laws of physics and everything else is a result of the high-level complexity that can result from such simple components and rules.
All well and good, but there should be a very heavy emphasis on the "may".
Sure, cellular automata can result in unexpected complexity at the larger level. But that hardly means that it is necessarily the underlying explanation for complexity in the universe.
This book strikes me as the work of a self-absorbed thinker who has become too attached to one "aha" idea and makes the mistake of trying to prematurely apply it everywhere.
If Wolfram had actually come up with a tiling, and a "rule", that truly would result in the physical behavior of the universe as we observe it, then he'd really likely be on to something. But instead, it seems he's simply shouting, "cellular automota can produce complex behavior, Eureka, it's The Big Answer!".
Otherwise bright individuals have made this same mistake with prior complex fields, such as the attempt to apply chaos theory to explain anything we don't currently fully understand, and especially the way that the bizarre behavior of quantum theory has been hammered into an explanation for such diverse things as "ESP", reincarnation, and human consciousness.
On the latter example, the otherwise sensible and brilliant Roger Penrose wrote an embarrassingly sophomoric book ("The Emperpor's New Mind") which attempted to argue that human consciousness must have its origins in the quantum world. Stripped of its verbiage, the book boiled down to nothing more than a late night college bull session: "Like, wow, man, quantum behavior is trippy, trippy enough to be like the expansiveness of the mind, man..."
Wolfram's point, that cellular automota *can* give rise to behavior which acts like laws of physics, is hardly the same as showing that cellular automota *is* behind the physics of our universe.
As soon as he turns up an actual set of C.A. rules that truly result in the physics of our known world, it'll be an accomplishment. Right now he's just at the "wow, man..." stage.
Having met the guy, worked with people who worked for him, and knowing about the true story of where Mathematica came from, all I can say is truer words have never been written.
He should get credit for being a far greater self-salesman than pretty much any other mathematical physicist around. If that's the kind of credit you're seeking....
He has a way to go before he unseats Hawking in that department.
Don't let the terminology fool you. There's nothing twilight-zone-ish about imaginary numbers, it's just an accident of word choice.
When people first started thinking about numbers, they thought in simplistic terms of whole numbers: 1, 2, 3, 4...
These are now called "natural" numbers, simply because they're naturally the first ones that come to mind when primitive peoples (or children) first start enumerating things.
Eventually people began realizing that they needed a number which referred to no things at all, and thus the zero was born. At the time it was a pretty revolutionary concept.
After a while, people started requiring numbers that could go below zero, as when accounting for money owed versus money earned. This was the rise of negative numbers.
The positive/zero/negative numbers altogether were called "integers", from the latin word for "untouched, undivided, whole".
However, the shortcomings of whole numbers became apparent, because how do you handle fractions? If you sell someone half of a butchered cow, you have to be able to include that in your accounting. So the "rational" numbers were added. Not because they were sensible, but because they were a "ratio" of one whole number divided by another (e.g. 1/2, 3/4, 99/100, etc.).
This covered most trade transactions, but eventually mathematicians began analyzing things that couldn't be expressed as a simple fraction (ratio). There were numeric quantities that fell between the rational numbers -- when written as decimals, the digits were nonrepeating (as all the rational numbers were). Because these were numbers which were not formed by a ratio (e.g. "rational"), they were dubbed "irrational" numbers.
Again, this doesn't mean that they're nonsensical, it just means that they aren't formed by a ratio of integers. The circumference of a circle (pi) is one such number out of many.
"Irrational" numbers were the first unfortunately named term, because they give the impression that the numbers are insane or something. But it's just an accident of terminology.
The entire continuum of numbers on the number line (integers, rationals, and irrationals) now filled the entire number line. There was not a spot on the number line which didn't fall into one of the above three categories (detail: the integer numbers are actually a subset of the rational numbers). The number line was complete. Because it contained every number that could be expressed to describe an actual single quantity, the set of all such numbers was rather grandiosely named the "real" number line, and the numbers were called "real" numbers.
Eventually, though, math required something more.
The square root of negative numbers, for example, didn't fit anywhere on the "real" number line. Nor did a lot of other mathematical operations which cropped up under perfectly ordinary calculations. It was discovered that if you just presumed the existence of the square root of a negative number, and used it in calculations, suddenly a of calculations became quite straightforward.
Because these new numbers didn't fall anywhere on the number line that had been labelled "real", some witty mathematician (Rene Descartes decided to humorously call them "imaginary" numbers (actually, Descartes coined the term in order to belittle the concept, but the name stuck).
But there was nothing actually imaginary about them. "Complex" numbers (formed by a combination of a "real" number quantity and an "imaginary" number quantity) are used in countless computations that have perfectly real-world applications. For example, electrical engineers use them all the time when doing AC power calculations -- the "real" component represents the instantaneous voltage, and the "imaginary" component represents the power phase. Einstein's relativity equations describe the "real" world beautifully when the three spatial dimensions are represented as "real" quantities and time as an "imaginary" quantity. Many navigation equations become dead simple when east/west is measured as a "real" number and north/south as an "imaginary" component -- and yet no one would argue that the north/south direction is not actually real. And so on.
In fact, the "reality" of imaginary numbers can be seen when you realize that the most intuitive way to understand complex (e.g., real+imaginary) numbers is as points on a *plane*, instead of on a *line*. "Real" numbers can only denote positions on a *line* (think of them as a value along the "X" coordinate). The "imaginary" part can be thought of as the value along the "Y" coordinate, which necessarily doesn't not fall on the one-dimensional "real" number line.
In short, "real" numbers are a limited form of math: one-dimensional math. "complex" numbers (which include an "imaginary" component) are TWO-dimensional math, which is much more direct and straightforward for many types of REAL-world calculations.
There's nothing "imaginary" about imaginary numbers. It's just an accident of nomenclature.
Of course, I have a suggestion! He could suddenly be overcome with a speech depriving malady.
Being mute would:
a) relieve the rest of us of having to listen to him and
b) one up Hawking by making even machine assisted speech impossible.
Then, everyone happy happy happy! Except for Steven H.
No, he didn't.
What he did prove was that a sufficiently complex formal system can be either consistent, or complete, but not both at the same time.
This only "proves" that computers can't do what people can do if you're silly enough to claim that human minds are both complete (e.g. can logically solve all possible problems) *and* consistent (e.g. never self-contradictory).
If you truly believe that we are, I've got several bridges to sell you.
Since we *aren't* complete, and *aren't* consistent, there's no impediment for a computer to equal our less than perfect performance, and Godel's theorem does not apply.
Godel proved that there are things we know are true, but cannot be arrived at given a starting set of rules like a computer program.
Right -- nor can they be arrived at via logical deduction by humans, either, since logical deduction is itself a formal system.
We "know" them by taking something as a given because we choose to *believe* it to be true, not because we can prove that it is. A computer can do this as well -- accepting a proposition as a premise is easy for a computer.
Ernest Nagel in his 1958 book "Godel's Proof" correctly concludes this, "Godel's conclusions bear on the question whether a calculating machine can be constructed that would match the human brain in mathematical intelligence...the brain appears to embody a structure of rules of operation which is far more powerful than the structure of currently conceived artifical machines. There is no immediate prospect of replacing the human mind by robots."
What was correct about the "immediate prospect" of computer ability in *1958* is hardly the case now.
Godel's proof is a killer logical problem for atheists like Hofstader who want to reduce man to nothing more than a sophisticated machine.
You didn't actually *read* his book, did you? He does nothing of the sort.
It's mathematically proven that our thinking brains are outside the system.
No, it's only "mathematically proven" that our brains aren't consistent -- i.e., we don't operate on Spock-like pure logic. But then we knew that already. Humans do much of their work via intuition, leaps of faith, rules-of-thumb, believing what we wish were true, and so on.
And computer programs can be just as fuzzy-headed as humans when they're set up right. Computer programs can operate on presumption, illogical trial-and-error, etc.
Hofstader trys to get around Godel's proof by suggesting that machines can change their own logical set of rules they function by. The end result is Hofstader spiriling down to nowhere in his book "Godel, Escher, Bach".
You clearly didn't read "Godel, Escher, Bach", or if you did you didn't understand it -- nor are you really clear on what Godel did and did not say. Try again.
Three words: State table lookup.
No equations needed.
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