Sebastian Hayes

Stephen Wolfram is  known to many computer  users as the millionaire designer/inventor of the computer software package Mathematica.  However, as a pioneer of a ‘new kind of science’ he has  been markedly less successful : Freeman Dyson reputedly gave the above book a one-word review, “Worthless”, and I only heard about it six years after its first publication because of an interview with the great man (who lives as a recluse in Concord, Massachusetts) reported in the New Scientist.

Wolfram’s main claim to fame is his discovery that “very simple rules can give rise to considerable complexity”. At first sight, this doesn’t sound earth-shattering but, then, neither did the 2^nd Law of Thermo-dynamics which in its original formulation by Clausius was simply  the innocuous sounding observation that “Heat does not spontaneously move from a colder to a warmer body”.

Wolfram has made a twenty-five year long study of cellular automata and, partly because of his own software, has been able to investigate their hidden properties more extensively than anyone else.

What is a cellular automaton?  It presupposes

1. A grid which can be extended indefinitely;
2. An initial ‘seed’, usually a single cell (square of grid) coloured black with empty cells to left and right all along the first row;
3. A rule which specifies the colour of every cell in each new row according to the colour of the cells in the row above.

In the simplest type of cellular automata there are only two permitted colours, black and white, and a rule attributes the colour of a new cell according to the colour of the three cells directly above it.

The rule can be presented visually, for example Rule 1 will transform the triplet in one row into, say, the triplet in the row below, Rule 2 will transform the triplet into another triplet, say,  and so on.  (The limitations of the symbols I can access on this site do not enable me to go into too much detail unfortunately.)

If we start with a single black cell and the rest of the row empty, i.e. …. …..and apply one set of rules which makes every triplet in the next row with the exception of which gives the same in the line below we end up with the ‘Maya temple’ pattern

…. ……
…. ……
…. ……

where every new line has two more black squares at the extremities compared to the row above.

Now, since there are two possibilities, and , for each of the trios numbered 1 to 8 in the top line of the Rule, there are altogether 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸ = 256 different Rules, each giving a different pattern of black and white cells.

Wolfram labels the first one 0 since, apart from the initial seed  which is present in every first row, the relevant rule produces a completely blank expanse after the initial row. The last rule No. 255 does the opposite and makes everything below the first row black.

In between there are various intermediary patterns, all different.

The patterns (which I am unable to give in detail on this computer) fall into three main categories:  1. patterns of fixed size; 2. growing repetitive patterns; 3. complicated, non-repetitive patterns.

Of the more complex patterns, quite a few are ‘nested’ : the Sierpinski Triangle, for example, is generated by Rule 90. Wolfram observes that this well-known pattern “is exactly Pascal’s Triangle of binomial coefficients reduced modulo 2, [where] black cells correspond to odd binomial coefficients” (Wolfram, p. 870) — something I did not know.

That such simple procedures can generate fractal shapes is in itself surprising but even more surprising is that one or two out of the 256 generate patterns that end up by becoming completely random though containing occasional localized pockets of more ordered behaviour. Many other examples illustrate the same phenomenon.

So far, so good. Wolfram seems to have won the first round on points by showing that “however certain one might be that simple programs could never do more than produce simple behaviour, the pictures of the past few pages [of this book] should forever disabuse one of that notion” (Wolfram, p. 39). Moreover, there seems, in the majority of cases, to be no way of predicting the type of behaviour that a particular rule will produce.

Playing around with various more restricted, or more extensive, types of cellular automata Wolfram reaches a very significant conclusion :

“Looking at many examples [of mobile automata], a certain theme emerges : complex behaviour almost never occurs except when large numbers of cells are active at the same time, Indeed, there is, it seems, a significant correlation between overall activity and the likelihood of complex behaviour.”                                                                                                                                                                       (Wolfram, p. 76)

Why is this significant? Because, precisely, the behaviour of living creatures involves the co-operation of astronomical numbers of individual cells, and we are at every turn confronted by “the extreme simplicity of the principle [of the DNA], on the other the endless complexity of the outcome” as one biologist puts it.

Despite — in some ways even because of — the discovery of the DNA and the subsequent genome project, the phenomenon we call ‘life’ remains as mysterious as ever. It is to the highest degree paradoxical that cellular automata, since they are basically just patterns on a computer screen produced by  rules that human beings such as Wolfram specify, have found their commonest and most successful applications in the modelling of living systems. For, whatever else they may be, cellular automata are not the product of random mutation plus natural selection which, as all good Darwinists know, are the two forces responsible for most of what we see around us that moves.

As deliberately designed ‘pure’ products of human intelligence, cellular automata would seem to have more in common with certain abstract mathematical systems than with physics or biology. And indeed a good deal of the adverse, not to say violent, reaction to Wolfram’s claims comes from the mathematical, rather than the scientific, community. Why is this?  Basically, because, although mathematics itself is not in crisis — pure mathematics is more flourishing than ever — the relevance of higher mathematics to what goes on in the real world has become a highly troublesome issue that orthodox mathematicians would rather not confront, as they suspect that the answer will not be favourable to them.

Broadly speaking, I fear that the dreadful truth is that  “Nature does not do mathematics” any more than New Labour ‘does God’, to quote Alisteir Campbell’s unforgettable remark. Euclidian geometry has a certain relevance to situations where close packing in regular arrays is significant, and thus to crystallography, but you will look in vain for the standard geometric shapes, the circle, triangle and even the dead straight line in the natural world around you. The extraordinarily varied, irregular and complicated forms of plants and animals could not be a more striking contrast to the simple ideal shapes Euclid studies with such absorption and that Plato immortalized as his ‘Ideas’. Mandelbrot’s fractals do occasionally look rather more like natural objects, but no plant is self-similar and the sea-horses of the Mandelbrot Set have nothing to do with real sea-horses.

‘Modern’ mathematics — I mean from Descartes and Newton onwards — deals essentially in algebraic formulae which in the vast majority of applications require the completely unrealistic assumption of ‘continuity’. The equation of a curve y = f(x) is absolute : it delimits the curve everywhere and for all time (barring certain so-called singularities). As a French physicist I once knew  put it, “It is ‘our fault’ if we cannot see all the features of the curve at a glance, they are essentially all there in the formula”. This is quite different from definition by recursion where a mathematical entity is built up step by step from an initial ‘seed’ and, although most mathematical functions can be defined recursively, mathematicians have a marked preference for the analytic way of doing things.

As for continuity, we now know that exchanges of energy, which account for practically all physical and chemical behaviour, are not continuous, but must obey quantum laws. Nonetheless, Calculus methods are still employed in, for example, population studies and molecular thermo-dynamics where we know for a fact that the independent variable can never be smaller than a single molecule or a single human being. It is doubtless because of the long shadow mathematics has thrown over physics, that, even today, it is automatically assumed that Space and Time are ‘continuous’ even though this is by no means self-evident and has always struck me as being a dead weight that physicists insist on carrying around with them (essentially because they have all been trained in the same mathematical school).

As it happens, cellular automata score on both these points. They are built up step by step, row by row, and the Rule is essentially a method for getting from one state of a system to the next, not a formula which is ‘true for all time’. This goes some way to explaining the success of applications of cellular automata to living systems, also to systems involving a very large number of individual elements, e.g.  fluid mechanics. The key question is : which way of proceeding is Nature’s way? In the days when all scientists believed in a supremely intelligent Creator God, as Newton and Boyle and Leibnitz did, the analytic approach clearly had the advantage, hence the very idea of Nature ‘obeying laws’. As for systems with large numbers of entities, there was, prior to the invention of computers, no alternative to Calculus but now it is much less necessary to solve differential equations and concoct analytic formulae since it is often possible simply to slog it out numerically.

There is, I think, a consensus now that organisms, including ourselves, are not masterminded by a transcendent intelligent Being. But nor do they in general ‘know what they are doing’ mathematically and scientifically speaking. Humble unicellular organisms perform miracles of chemical engineering that even our current technology cannot rival. A cheetah stalking its prey is ignorant of equations of motion and children easily learn to ride a bicycle without knowing anything about angular momentum or gyroscopic stability. For all this, ‘instinct’ (whatever that is) plus trial and error suffice, and, according to one of the two most successful scientific theories of all time, trial and error (random mutation) combined with endurance testing (’natural selection’) suffice to explain most of what we see around us in the organic world.

The great thing about cellular automata is that they are not just simple, they are, some of them at any rate, absolutely simplistic : a child of three could carry out one of Wolfram’s 256 rules and build up a pattern with coloured blocks, though she would very rapidly get bored with the activity. And yet some of these simple automata  exhibit very great complexity. Although some animals seem to possess a rudimentary sense of number, I cannot conceive of mammals and plants knowing anything at all about calculus. However, I can just about conceive of an organism, or even a genus, directing itself to carry out over and over again the basic rules governing the growth of a cellular automaton, and letting the programme run to see what comes up, while natural selection can be depended on to weed out the absolutely hopeless results. It is in this sense that Wolfram has a point.

Just possibly, something akin to the procedures that drive cellular automata  could take the inorganic into the organic, i.e. produce life. At this very moment, a Swiss Professor, Henry Markham, is heading a multi-million IBM backed project to develop a true ‘artificial mind’, not just a chess-playing computer programme but something that has consciousness and a sense of personal identity. “Markham believes,” a correspondent writes, “that consciousness is probably something that ‘emerges’ given a sufficient degree of organized complexity” (Daily Mail, Jan 4 2010). This is precisely Wolfram’s contention, indeed Wolfram goes rather further in that he seems to suggest that something we might reasonably call ‘life’ is automatically going to emerge from certain types of system whether we like it or not. (Some scientists such as Dr Blackmore have seriously suggested that new types of ‘life’ are being spawned already by the Internet.)

Because of the conclusions Wolfram derives from his observation of, or, better, experimentation with, cellular automata, the contemporary analogy between the human brain and a computer, which has become something of a tired cliché, takes on new life. For Wolfram believes that there are plenty of systems based on  precisely formulated simple rules which nonetheless, perhaps after a considerable lapse of time, exhibit highly complex ‘interesting’ behaviour which is entirely unpredictable (and he gives examples of this). He suspects that the brain is one such system and proposes  this as the solution to the age-old problem of Free Will vs. Necessity.

“Even though all the components of our brains presumably follow definite laws, I strongly suspect that their overall behaviour corresponds to an irreducible computation whose outcome can never in effect be found by reasonable laws.

(…) As a whole our brains still manage to behave with a certain apparent freedom.

Traditional science has made it very difficult to understand how this can possibly happen…. [But] in fact there can be vastly more to the behaviour of a system than one could ever foresee just by looking at its underlying rules. And fundamentally this is a consequence of the phenomenon of computational irreducibility.

For if a system is computationally irreducible this means that in effect there is a tangible separation between the underlying rules for the system and its overall behaviour…… And it is in this separation, I believe, that the basic origin of the apparent freedom we see in all sorts of systems lies.”                                                                 Wolfram, pp. 750-751

This claim constitutes a decisive break with the basic presuppositions of scientific thinking during the last five hundred years, inasmuch as Wolfram denies that the universe, or us, are predictable even in theory — though Gödel, Quantum Indeterminacy and Chaos Theory have prepared the way for this grand conclusion. Oddly, Wolfram faces two ways at once : he denies on the one hand that there is anything ‘special’ about human beings, while at the same time, by identifying  them as ‘computationally irreducible systems’ he provides them with ‘free will’ and capacity for development in unpredictable ways.

Wolfram — and for that matter Dawkins — will have to work a bit  harder if they want to convince me that man-made patterns  on a computer screen are really analogous to organisms that are subject to the constraints of actual, as opposed to virtual, space, but he has certainly convinced me that some features of his automata throw light on the dark mysteries of growth and form.

When moving on to cosmology — Wolfram is nothing if not ambitious — he abandons the idea of a grid progressively filling up with coloured cells. He  models Space/Time as “a giant network of nodes…..with a fixed  number of connections” and departs noticeably from the near universal  assumption of Space/Time continuity by giving an estimate of the size of a basic Space/Time ‘causal link’, namely  “an elementary distance of 10^-35 metres and an elementary time interval of around 10^-43 seconds” (Wolfram p. 520).

A New Kind of Science is a very long book (1,200 large pages) but it is simply and fluently written and, in the main text, contains virtually no mathematical formulae — though there is plenty of advanced mathematics and computer speak in the Notes (300 pages long) for those who might  otherwise be tempted to immediately dismiss this very ambitious work that claims to prepare the ground for the coming scientific paradigm. 

Sebastian Hayes