By now, thanks to Newsweek, Time Magazine, the New York Times, and such, we have all heard about Stephen Wolfram and his new kind of book. It is a new kind of book in that a work of major scientific significance first finds its way through a book rather than via scientific journals, to the public rather than to peers. It is a new kind of book in that its author, a brilliant physicist with impeccable credentials, was already a millionaire by virtue of his contribution to computation and educational technology (Mathematica) when the book was launched. It is a new kind of book in being a work on science with no references to other works in a bibliography. Like A Brief History of Time by the other Stephen, this book has also become a best-seller, this too has received immense attention from the popular press. Like the other book that sold more than a million copies, this too will be bought and cherished by a greater number than will read and assimilate it from cover to cover. I am quite sure translators are working round the clock to bring out the book in other idioms also.
No one can deny that Wolfram has made a crucial discovery in the context of computer programming. One of his major discoveries is that starting from a few simple rules, one can generate an amazing variety of complexity. It is well known that there is a fascinating interconnection between a Fibonacci sequence and the symmetry in pea-pods or sea-shells. Wolfram’s recognition is a far richer generalization in the sophisticated language of computer programming of a very similar idea. This interesting discovery is repeated several times throughout his book, because this indeed is Wolfram’s eureka! He has successfully applied this principle in countless instances to show how everything from snowflakes to living organisms, from visual perception to freewill and more, emerge at the qualitative level from a set of fairly simply rules.
Another important thesis of the book hinges on the Principle of Computational Equivalence (p. 5): The sophistication in the behavior of a system is a reflection of the sophistication of the computation that is involved in it. The complexity we observe is in fact the result of the computational rules followed by the system. So, if we find simple behavior, the corresponding computation was simple, and likewise for complex behavior. Complex computation does not mean that the underlying rules are not simple.
This principle implies that it is just not possible to make predictions about fairly complex systems using the usual mathematics. In fact, “almost any system whose behavior is not obviously simple performs computations that are in the end exactly equivalent in their sophistication.” (p. 741) Thus, we are able to predict the orbit of a comet because the corresponding computation is simple, and we cannot predict how a species will evolve because the computation here is complex..
Thus Wolfram’s investigations have thrown a whole new light on the phenomenon of complexity. He has uncovered the larger implications of the power and potential of cellular automata in our efforts to unscramble the roots of complexity: a theme that is going to be one of the major foci of scientific interest in the new century. And for this, his new methodology is certainly interesting, impressive, fascinating, and deserving of loud applause and admiring recognition. It will certainly instigate serious further investigation by the scientific community. There is every reason to believe that Wolfram’s approach will lead to many more significant and meaningful results. Indeed, his book could have been more appropriately entitled: A New Approach to Complexity, for this is what the book is all about.
Anyone who is familiar with Wolfram’s previous contributions to fundamental physics will know that this is the work of an prodigious mind. The author has published richly on particle physics, cosmology, computational theory, cellular automata and more.
With all that, not everyone may fully agree with some of the assertions in the book which begins with an outline of the Foundations for a New Kind of Science. Wolfram reveals that it took him “the better part of twenty years to build the intellectual structure that is needed” (p.1) to complete his work. In this introductory chapter, he makes some general observations about the scientific enterprise. It is difficult to accept some of these without qualifications. For example, according to Wolfram (p.1), “in the past throughout the exact sciences it has usually been assumed that these rules (which systems follow) must be ones based on traditional mathematics.” But there is more to science than the exact sciences. He himself includes biology and much more. For example, in the theory of evolution, of tectonic plates, of rock formation, in archaeology, etc., one is not really dependent on traditional mathematics for their validity.
He notes that (p. 2) “our everyday experience of building things tends to give us the intuition that creating complexity is somehow difficult, and requires rules or plans that are themselves difficult.” This is not necessarily always so. From a few simple definitions and postulates, all the complexity of Euclidean geometry arises; from a few sounds and their combinations all the myriad words and rhythms of languages emerge; from a few simple strokes can result in a magnificent painting; and from just eight octaves countless complex melodies have been composed.
Further on, Wolfram makes the incomplete statement (p. 4) “When mathematics was introduced into science it provided for the first time an abstract framework in which scientific conclusions could be drawn without direct reference to physical reality.” I am sure Wolfram doesn’t mean what this statement says. For, while it is true that one can draw predictive conclusions from the mathematization of a physical situation, this can never be done without incorporating some aspects physical reality into the equations.
Wolfram claims that (p.8) “it has become almost universally assumed that any serious physical theory must be based on mathematical equations.” He should perhaps say mathematical concepts rather than equations. There are not too many equations in many sections of crystallography, geology, and organic chemistry. More importantly, the mathematical framework is a necessary, but not a sufficient condition for a physical theory. We need concepts and data of observation which must be tied into that framework: elements that simply cannot be extracted from programming rules alone.
Almost 350 pages of the almost 1200 pages of the book are devoted to General Notes. These contain brief explanatory and historical statements on the numerous technical terms used in the course of the book, from Abelian groups to zeta functions, with lambda calculus, quantum computers, the three body problem, and a good deal more thrown in between. Arranged alphabetically, this could be used as a good glossary of modern scientific jargon, somewhat like what one finds in the book, Who’s afraid of Schroedinger’s cat? Yet, I am not sure how many of his lay readers will decipher all the complex and technical ideas that he has so succinctly packed into dense little paragraphs.
Wolfram talks about the “popularity” of systems based on numbers, rather than their effectiveness, and goes on to say that the standard mathematical physics has been ineffective to describe complex systems (p. 115). To some, this might seem an unfair assessment of the record of applied mathematical physics to a variety of fairly complex systems, using computational techniques in many instances, but preserving the same conceptual framework.
Wolfram also expresses the not-uncommon view which divides worldviews into Eastern and Western thinking, claiming rationality for the West and relegating mysticism to the East. It is a matter of historical record that in both traditions there are rich instances of rationality as well as mysticism. The two ways of looking at the world, if one wishes to make such a dichotomy, are the following: One grants only purely rationalistic-materialistic-empirical elements in science; the other incorporates spiritual-mystical-transcendental elements in our apprehension of Reality. This is not an Eastern-Western contrast, since there have been and still are eloquent spokespersons for both views in both Western and Eastern civilizations. Wolfram is quick to distance himself from Buddhist and Taoist modes, and he points out that though he rejects reductionism his book is based on the rational tradition (p. 1196).
Wolfram informs us that he has made far more discoveries than he had expected (p. 22), and is confident his approach will lead to many new discoveries. He certainly seems to have opened a new door to comprehend complexity. But to suggest that all the science of the past four centuries was on the wrong track, and will soon be replaced by sets of programming rules sounds a bit much. Clearly, the New Kind of Science could not have led to the discovery of, for example, helium, UV radiation, bacteria, chromosomes, or any of the thousands of other revelations which traditional science has made, including the construction of computers on which Wolfram’s rules can be imposed. It is not at all clear how he can derive the space-time physics of relativity without invoking the speed of light which no computer programming can reveal.
His needlessly grandiose claims dilute the merit of an otherwise fruitful and provocative work. He would probably say that anyone who regards his discoveries as just another bunch of fruits from interesting research is missing the whole point: The mission of A New Kind of Science is to make standard science chart an altogether new course. Only the future will tell if this is a prophet’s call or aught else. Wolfram admits (p. 849) to an uncommon lack of humility in book, but feels that if he had displayed more modesty, “the cost would be a drastic reduction in clarity.” One wonders.
An attack on reductionism is not something new. In fact, it is one of the important ingredients in the post-modernist brew. Yet, it may be premature to compose requiems for reductionist research. While critics of the classical scientific mode are writing books and presenting papers which have been proclaiming that reductionism is dead, thousands of dedicated scientists are still working away in laboratories, observatories, universities and research centers, dissecting, weighing, splitting protons, measuring, analyzing, computing, hypothesizing, and engaging in other unabashedly reductionist activities.
What’s more, they continue to produce interesting results in the process. The question is not whether reductionism works, but where it does. From the fact the sweetness of a succulent fruit cannot be explained in terms of the seed’s chemical composition, it does not follow that chemists have been on the wrong track for two hundred and fifty years.
To repeat: Wolfram’s book offers new and exciting insights on the origins of complexity, and impresses the reader by its revelation of sheer simplicity at the foundation of some complex phenomena in computer language terms. But not all may be inclined to see eye-to-eye with the totalizing claims of his book. As a new chapter in science, or as an important insight into a variety of observed effects, this book will certainly be welcome. As a new kind of science, it may be suspect. There sure will be more advances in the years to come in the theory of cellular automata.
But one may seriously doubt that, even with the clarion call for A New Kind of Science, beakers and Bessel functions, microscopes and mass-spectrographs will disappear from the arena of serious science in the foreseeable future. NKP is not the TOE it claims to be.
[This was first published ten years ago.]
June 14, 2002