8/7/2019 20756

1/6

COMPUTING SCIENCE

THEWORLDACCORDING TOWOLFRAM

Brian Hayes

A reprint from

American Scientistthe magazine of Sigma Xi, the Scientific Research Society

Volum e 90, Num ber 4

JulyAugust, 2002

pages 308312

This reprint is provided for personal and noncommercial use. For any other use, please send a

request to Permissions, American Scientist, P.O. Box 13975, Research Triangle Park, NC, 27709, U.S.A.,

or by electronic mail to perms@am sci.org. 2002 Brian Hayes.

8/7/2019 20756

2/6

Arthur C. Clarke wrote: Any sufficientlyadvan ced technology is indistinguish-able from ma gic. In the sam e vein, per-

haps any sufficiently ad vanced science risks be-ing mistaken for the raving of a crackpot.

This uncomfortable thought is prompted by abook that has landed on my desk with a five-poun d thu d. The title promises A New Kind of Sci-ence, and inside are claims no less extravagant. I

have discovered vastly more than I ever thoughtpossible, the author s preface boasts, and in factwhat I have done now touches almost every exist-ing area of science, and quite a bit besides. In1,200 pages, this one volumethe work of onepersonundertakes to explain the structure ofspace and time at the deepest level, clears up themysteries of entropy and randomness, and elbowsaside mathematics and statistics as central tools ofscientific analysis. Along the way, the book also

corrects the errors of Darwinism, shows wherechaos theory went wrong, explains the forms ofseashells, tree leaves and snow flakes, and puts hu-man free will on a proper philosophical footing.

The author acknowledges that these grandambitions may be met w ith a measure of skepti-cism. If I myself were just to pick up this booktoday without having spent the past twentyyears thinking about its contents, I have littledoubt that I too would not believe many of thethings it says. But he urges patience and perse-verance. To assimilate the new ideas w ill requirean investment of years comparable to learningan area like physics. Those mired in the old kindof science may resist at first, but, In time I expectthat the ideas in this book will come to pervadenot only science and technology but also many

areas of general thinking. And w ith this its meth-

ods will eventually become a standard part ofedu cationmu ch as mathem atics is today.

Such grand iose visions are often a telltale signof the crank, and there are other reasons to bewary h ere. The author has been w orking in seclu-sion and secrecy for 10 years, and during thattime has subm itted none of his results to peer re-view. The publisher of the volum e is the author sown comp any, so that even now there was been

no opportunity for independent editorial judg-ment. The circum stances suggest a person cut offfrom the social context of science.

And yet the author ofA New Kind of Science is

not an outsider, and he is not a crank or a crack-pot. H e is Stephen Wolfram, p hysicist, comp uterscientist and entrepreneur. Wolfram published hisfirst paper in p article physics at age 15 and earneda Ph.D. at 20. He was a precocious professor at

Caltech, then moved to the Institute for Ad-vanced Stud y and later the University of Illinoisbefore leaving the academic world to create thesoftware called Mathematica. His ideas deserve areading even if they are presented in a mannerthat raises both eyebrows and hackles.

Before proceeding further I should disclose myown occasional interactions with Stephen Wol-fram. As a magazine editor I once invited him towrite an article; some years later I wrote a reviewof Mathematica (and accepted a complimentarycopy of the software, as well as a T-shirt); on an-

other occasion I wrote for The Mathematica Jour-nal, which was sponsored by Wolfram Research,

Inc. More recently, as a condition of being al-lowed to see A New Kind of Science in advance ofpu blication, I signed a non disclosure agreement,which has now expired. And a few weeks agoWolfram and I met to talk about the book.

Playing by the Rules

Science is usually viewed as an indu ctive art, whichstarts with observations or experiments and pro-ceeds toward laws of nature. Wolfram stands thisprocess on its head, suggesting that we start by list-ing all possible laws of nature and then see whichof them might correspond to observed events.

Is such a scheme feasible? If we set out to com-

pile a catalogue of all conceivable natural laws,where would we begin, and how would weknow when to stop? Wolfram answers by intro-du cing abstract systems small enough and sim-ple enough that we can enum erate all the rules orprograms that might possibly govern their be-havior. In these systems you dont have to govery far down the list before the rules start mak-ing toy worlds with interesting properties.

Wolfram first formulated these ideas in studiesof cellular automatageometric arrays of mini-malist comp uting elements, called cells. Each cell

308 American Scientist, Volume 90

COMPUTING SCIENCE

THEWORLDACCORDING TOWOLFRAM

Brian H ayes

Brian Hayes is Senior Writer forAmerican Scientist. Address:

211 Dacian A venue, Durham, NC 27701; bhayes@amsci.org

8/7/2019 20756

3/6

in an automaton has a finite number of possiblestates, and it spends its time continually recom-puting this state. To do so it looks at its own pre-sent state and at the states of a few neighboringcells, then app lies a fixed rule to determine its nextstate. As soon as all the cells have u pd ated theirstates in this way, the process starts over. All thecells use the sam e rule and operate in synchrony.

In the early 1980s Wolfram began a systematicsurvey of one-dimensional cellular automata,where the cells are arranged in a line or a ring.

The operation of a one-dimensional autom aton isparticularly easy to visualize: If you d raw th e line

of cells horizontally and show successive statesof the system going down the p age, the result is atwo-dimensional spacetime diagram that revealsthe entire evolution of the system.

Consider a one-dimensional automaton wherethe cells have just two possible states (blackandwhite) and w here each cell interacts only with itsnearest neighbors. There are eight possible con-figurations of a cell and its two neighbors, andfor each of these configurations the next state canbe either blackor white. Hence there are 28 possi-ble rules for the evolution of the system. These

256 rules are all the available cand idates for lawsof nature in this tiny un iverse.

Can anything interesting ever hap pen in aworld of such meager resources? Wolfram identi-fies four broad categories of behavior, which are il-lustrated in Figure 1. First are simple repetitivepattern s, such as a ll white cells or all black cells, oralternation in stripes and checkerboards. The sec-ond category includes self-similar, fractal patterns,whose characteristic feature Wolfram describes asnesting. In the third category are ru les that gen-

erate apparent randomness; although the outputis not a totally patternless spatter of black and

white cells, the sequence of states of a single cellcan have the statistical properties of a random se-ries. Finally, a few ru les yield localized structuresthat maintain their coherence while movingthrough the cellular space. These persistent struc-tures are reminiscent of particles being created andannihilated in a qu antum field theory.

It was the random p atterns and the persistentstructures th at caugh t Wolframs attention.Where does all that complexity come from? Thenaive assumption might have been that simplerules would yield only simple behavior, and the

2002 JulyAugust

Figure 1. Four kinds of behavior seen in one-dimensional cellular automata were classified by Stephen Wolfram, who also devised the

scheme for numbering the rules. Each panel is a spacetime diagram, with space extending horizontally and time proceeding downward, so

that successive configurations of the system are drawn one below the other. In each case the initial configuration is a line of white cells with

either one or two black cells in the center. All of the systems shown are among the simplest nontrivial cellular automata, wi th just two s tates

per cell (black and white) and a neighborhood of three cells. Rule 250 yields simple alternation of black and white cells. Rule 90 produces a

nested, recursive pattern. Parts of the pattern generated by Rule 30 show evidence of randomness, and Rule 110 gives rise to objects that

move across the space and interact with one another. (The illustration of Rule 110 is shown at half the scale of the others.) Both the illustra-

tions that accompany this article are taken from A New Kind of Science and are reproduced courtesy of Stephen Wolfram, LLC.

Rule 250

Rule 30

Rule 90

Rule 110

8/7/2019 20756

4/6

complexity of the pattern would grow steadilyalong with the complexity of the und erlyingrules. But Wolfram observed a d ifferent relation:There is a threshold of complexity. If the ru les arekept simple enough to stay below th e threshold,the outcome is always rather dull. (For example,a cellular automaton in which each cells nextstate depends only on that same cells presentstate is inevitably repetitious.) But once the ru les

get just a little more intricate, all four categoriesof behavior appear. Whats more, once the sys-tem has crossed the threshold, further em bellish-ments of the rules have little effect. New detailsmay appear, but all the patterns can still be as-signed to th e same four basic categories.

To provid e evidence in support of this last as-sertion, Wolfram surveys an exuberant variety of

systemssome familiar, some novel, all interest-ing in their own right. He looks at cellular au-tomata with more than two states per cell, in-cluding systems where the range of possiblestates is a continuum. He considers cellular au-tomata in two and three dimensions. He studies

automata where the cells are updated one at atime rather than simultaneously. Leaving behindthe them e of cellular au tomata, he stu dies Turingmachines and grammar-driven systems w heresubstitution rules allow a string of symbols togrow and change. He looks at differential equa-tions and at iterated numerical functions. Heeven examines the sequence of prime numbers.In each case his conclusion is the same: Simplesets of rules can generate complex outputs, butpiling further complications onto the rules leads

to little add itional comp lexity in the ou tcome.

A Cellular Universe

Its all very well to dabble in disembodied bitsand streams of digits, but w hat has all this to dowith science in ou r w orld of atoms and energy?Wolframs answer is that the sam e kinds of sim-

ple rules or programs are found at work every-where in the u niverse.

One case where the mapp ing between the tworealms is qu ite direct comes from biology, wherecertain mollusk shells are decorated with pat-terns similar to those produced by some cellularautomata. And the resemblance is surely not acoincidence: The shell patterns are d eposited by arow of pigment-secreting cells (that is, biological

cells) that appear to act much like a one-dimen-sional cellular automaton, with neighboring cellscommunicating through the exchange of chemi-cal signals. These resemblances have been notedbefore, but Wolfram argues for an unusuallystrong version of the id ea, claiming that all possi-ble cellular autom aton ru les in a certain class areobserved on mollusk shells.

Elsewhere in biology, Wolfram applies similarmethod s of analysis to other pigment patterns, to

the arrangem ent of stems and branches in plants,and to the shapes of leaves. In physics he treatsthe growth of snowflakes and other crystals, thefracturing of solids and the onset of turbu lence in

fluids. There is even a brief discussion of eco-nomics, suggesting that the kind of randomnessobserved in some cellular automata could ac-count for p rice fluctuations in stock markets.

In another chapter ofA New Kind of Science

Wolfram presents his version of the thesis that theuniverse as a whole is something like a cellularautomaton. The model looks below the level ofeveryday experience and even beyond th e eventsstud ied in high-energy physics, where the worldseems to be made up of electrons and quarks andother elementary particles, moving through a

continuu m of space. In Wolframs view of the uni-verse there is no continuum, and particles are a

mere epiphenomenon; indeed, motion and geom-etry are also little more than illusions. In this cos-mology, space and time are both assumed to bediscrete, just as they are in a cellular autom aton,

310 American Scientist, Volume 90

Figure 2. Pigment patterns on snail shells might be products of a biological system working something like a one-

dimensional cellular automaton. Wolfram argues that various mollusks illustrate all possible patterns generated by a

specific class of automata. The shells show n are identified as the banded marble cone (left) and the textile cone (right).

8/7/2019 20756

5/6

and the only things that move are signals passingfrom cell to cell.

The most obvious way of implementing thisidea wou ld be to par tition space into tiny cubicalvolumes, creating a cellular automaton on a three-dimensional grid. Wolfram looks with disfavoron this simplest solution because it imp oses a par-ticular geometry on space and a lso requires somekind of master clock to synchronize the up dating

of all the cells throughout the grid . His alternativeis a mod el where the cells are nod es of a free-formnetwork that has a well-defined topology but nospecific geometry. In other word s, the connectionsbetween nodes are all determined beforehand,but the spatial coordinates of the nodes are leftunsp ecified. Concepts such as shape an d p ositionhave no meaning at this level: The geometry of

space emerges from the model rather than beingbuilt into it. Specifically, our world is perceivedas being three-dimensional because each node ofthe network has three incoming links from othernodes and three outgoing links, creating the sameconnectivity as a three-dimensional lattice. An-

other feature of the model is that updating thecells requires no synchronizing master clock; up -dating events propagate along the links of the net-work itself. Making each link a one-way conduitdefines a direction for time and causality; thewhole structure is called a causal net.

This theory of everything is one of the bookswilder flights of fancy; there is no immediateprospect of testing it by experiment. But the sameis true of all other attempts to explain the struc-ture of the u niverse at this level of detail. In any

case, Wolfram is not coy in his manner of propos-ing the mod el. When I asked him h ow seriouslyhe intends it to be taken, he said he would be

quite surprised if something very much like itdoesnt turn out to be right.

What to Make of It

Wolfram warns that developing an intuition forhis new kind of science will take months, evenfor the most talented and open-minded people.For me I suppose it may take years. But even if Iam p remature in passing judgment, I want to givea preliminary assessment of a few major themes.

The core notionthat simple rules or pro-grams can yield complex behavioris surelyboth true an d important. Whether it constitutes

a new kind of science remains to be seen.

A closely related point is Wolframs insistencethat programs or algorithms are the best way ofexpressing ideas in th e sciences. In particular, ex-plicit rules of evolution are preferable to equa-tions, which merely state the constraints that asystem mu st satisfy withou t necessarily showinghow to satisfy them. Perhaps hes right; my ownexperience is that I und erstand best wh at I canprogram . But Wolfram wou ld expand this obser-vation into a broad indictment of the mathe-matical framework trad itionally used in the exactsciences, w hich is reckless overkill. Wolfram ob-

viously needs that framework (and uses it ex-pertly) in his own work.

The concepts of random ness and complexity

are central to the argum ent of this book, and yetWolfram is curiously lax about defining them.He doesnt address the issue directly until 550pages into his narrative, after many referencesto intrinsic generation of randomness in cel-lular autom ata and other simple systems. If you

are accustomed to thinking of randomness asan inherent p roperty of a patternsomethingthat can be traced back to the way it was creat-edintrinsic generation makes no sense inthis context, because the m echanism that creat-ed the pattern is totally deterministic. It turnsout that Wolfram defines random ness and com-plexity in terms of how patterns are perceivedrather than how they are created. Roughlyspeaking, if it looks random , it is random. Fair

enough, but it remains unclear just what is be-ing generated intrinsically.

With the exception of the cosmic causal net,the examples that illustrate applications of Wol-

frams ideas are strangely bland. Snowflakes, flu-id tu rbulence, branching in plants, pigment pat-terns in animalsthese are all rather shopwornspecimens, which have long been explained bymodels of the same general type. (Alan Turinggave a computational account of leopard spotsand zebra stripes 50 years ago.) If the new kindof science is to have mu ch generality, it will needto show its worth in other areas. In d evelopmen-tal biology, for example, can we write a simpleprogram that explains the complex structure ofCaenorhabditis elegans? Anatomists have tracedthe paths of all 959 somatic cells in this worm,

but expressing the und erlying algorithm in terms

of a few simp le rules looks like a challenge.Wolframs commen ts on evolutionary biology

are perhaps the lamest passages of the entirebook. Noting that some traits of some organismsseem to explore the entire space of available vari-ations, he concludes that Darwinian selectioncant be acting on those traits. It is my suspi-cion, he writes, that at least many of the visu-ally most striking differencesassociated forexample with texture and pigmentation pat-ternsin the end have almost nothing to do withnatural selection. And instead what I believe isthat such differences are in essence just reflec-tions of completely random changes in underly-

ing genetic programs. He writes as if he wereunaware that a debate between neutralists and

selectionists had ever entered biology.All of the programm able systems explored in A

New Kind of Science have a d istinctive trait in com-mon: They have a densely occupied space of pro-grams. In a cellular automaton, for example, anyrule relating a neighborhood configuration to anext state is a valid program . Other programmablesystemssuch as desktop computers and theDNA-reading apparatus of the living cellaremuch choosier about w hat they will recognize as a

2002 JulyAugust

8/7/2019 20756

6/6

valid program. If you try feeding them randomstrings of bits or random sequences of nucleotides,youre in for a frustrating experience. Its not obvi-ous to me how the p aradigm of complex behaviorfrom simple rules can be extended to such systems.

Clarity and Modesty

So mu ch for the content ofA New Kind of Science.The book also has a person al and historical con-

text that requires comment.In a note titled clarity and mod esty, Wolfram

explains that he had to sacrifice the latter to attainthe former. But immod esty is only half the issuehere. The problem is not just the rosy spotlightthat Wolfram shines u pon himself at center stage;its also the utter darkness that enshrouds all theother actors in this drama. The main text ofANew Kind of Science (850 pages) nam es no namesat all; the only w ork attribu ted to a sp ecific indi-vidual is Wolframs. The notes at the end of thebook (another 350 pages in smaller typ e) do m en-tion nam es of people, but br iefly, grud gingly andoften dismissively. (Its remarkable how many

discoverers failed to appreciate the significanceof their own work.) And many of the historicalnotes manage to p resent an anonymized h istory,written in the passive voice: By the end of the1950s it had been noted that.... Over the courseof the 1960s constructions were found ....

The book has no bibliography; the only refer-ences listed are Wolframs own pu blications.

Here is a precis of Wolframs history of cellularautom ata. He discovered them in 1981 (althoughhe had made important precursor experiments

earlier, as a teenager). Some time later h e learnedthat John von Neumann had had the idea 30years earlier. But von Neumann missed making

the crucial discovery that simple rules could p ro-duce complex behavior, and so did others whotoyed w ith the systems in the intervening years.By the late 1970s, research on systems equ iva-lent to cellular automata had largely peteredout. But then the publication of Wolframs pa-pers redefined and reinvigorated th e field, draw -ing in many followersalthough most of themwent off on the wrong tangent or wasted theirtime on trivial details.

It didn t hap pen that w ay. To begin with, inter-est in cellular automata did notpeter out in the1970s; it was thriving. The field continued to

grow in th e 1980s, when Wolframs participation

doubtless helped, but m ore important w as workon the p hysics of computation and reversible cel-lular automata. Wolfram took no p art in that. Themain actors were Edward Fredkin, Charles Ben-nett, Tommaso Toffoli and Norman Margoluswho were also, as it happ ens, the ones who ex-plained to Wolfram in 1981 the nature an dhistorical context of his own work.

Wolframs telling of the mollusk-shell story isanother notable example of delusional history.The shells have been known since antiquity, hesays, but almost no efforts to und erstand the ori-

gins of such patterns seem ever to have beenmade. He then mentions one such effort, byC. H. Wadd ington and Russell Cowe, but says it

was too narrow. He ignores entirely a cellular au-tomaton devised in 1973 by G. T. Herman andW. H. Liu to model one of the shell patterns. Ac-cording to Wolfram, only his 1982 discoverybrought the matter to w ide attention and led oth-ers to take it up. Among those others was Hans

Meinhardt, but Wolfram disparages his modelsas being too elaborate. In fact, Meinhard t came tothe problem independently of Wolfram; whatbrought the shell to his attention was seeing it onhis dinner plate in an Italian restaurant. And hismodels are elaborate because they account for thespecific features observed on actual shells, ratherthan simply d eclaring that all shells look like onecellular autom aton or another.

At the end of a long lunch with Wolfram, our

conversation turned to these matters of historyand attribution, then dr ifted on to a related topic.I asked w hy he had not given his new kind of sci-ence a nam e. In the book, he refers constantly to

the new k ind of science I describe in this book,which gets cumbersome after the first 50 repeti-tions. He said h ed struggled to come up with asuitable name, but nothing quite fit the bill. AtWolfram Research, people spoke of NKS, hesaid, or sometimes Wolfram science.

Are there any sciences named for people? Iwondered aloud.

Well, theres Newtonian p hysics, he replied.It was not the first time the names Wolfram

and Newton have been mentioned in the samebreath, and I suppose it might be taken as furtherevidence of an ego bursting all bounds. But I see

it in another light: He is simply too modest to

name the field Wolframian science. That part isleft to u s.

Bibliography

Fredkin, Edward. 1992. A new cosmogony.http:/ / www.digitalphilosophy.org/ new_cosmogony.htm

Herman, G. T., and W. H. Liu. 1973. The daughter of Celia,the French flag and th e firing squ ad: Progress report ona cellular linear iterative-array simulator. Simulation21:3341.

Meinhardt, Han s. 1995. The Algorithmic Beauty of Sea Shells.With contributions and images by PrzemyslawPrusinkiewicz and Deborah R. Fowler. New York:Springer-Verlag.

Toffoli, Tommaso, and Norman Margolus. 1990. Invertible

cellular au tomata: A review. Physica D 45:229-253Turing, Alan M. 1952. The chemical basis of morphogene-

sis. Philosophical Transactions of the Royal Society, Series B237:3772.

von N eum ann , J. 1966.Theory of Self-Reproducing Automata.Edited and completed by A. Burks. Champaign, Ill.:University of Illinois Press.

Waddington, C. H., and Russell J. Cowe. 1969. Computersimulation of a molluscan pigmentation pattern . Journalof Theoretical Biology 25:219225.

Wolfram , Stephen . 1986. Theory and Applications of CellularAutomata. Singapore: World Scientific.

Wolfram, Stephen. 2002. A New Kind of Science. Champaign,Ill.: Wolfram Med ia, Inc.

312 American Scientist, Volume 90