Some people fondly believe that chess-playing computers work by internally trying out all possible combinations of chess moves. They find this belief comforting when a computer beats them, but their belief is utterly false. There are far too many possible chess moves: the search-space is billions of times too large to allow blind stumbling to succeed. The art of writing a good chess program is thinking of efficient short cuts through the search-space. Cumulative selection, whether artificial selection as in the computer model or natural selection out there in the real world, is an efficient searching procedure, and its consequences look very like creative intelligence. That, after all, is what William Paley’s Argument from Design was all about. Technically, all that we are doing, when we play the computer biomorph game, is finding animals that, in a mathematical sense, are waiting to be found. What it feels like is a process of artistic creation. Searching a small space, with only a few entities in it, doesn’t ordinarily feel like a creative process. A child’s game of hunt the thimble doesn’t feel creative. Turning things over at random and hoping to stumble on the sought object usually works when the space to be searched is small. As the search-space gets larger, more and more sophisticated searching procedures become necessary. Effective searching procedures become, when the search-space is sufficiently large, indistinguishable from true creativity.
The computer biomorph models make these points well, and they constitute an instructive bridge between human creative processes, such as planning a winning strategy at chess, and the evolutionary creativity of natural selection, the blind watchmaker. To see this, we must develop the idea of Biomorph Land as a mathematical ‘space’, an endless but orderly vista of morphological variety, but one in which every creature is sitting in its correct place, waiting to be discovered. The 17 creatures of Figure 5 are arranged in no special order on the page. But in Biomorph Land itself each occupies its own unique position, determined by its genetic formula, surrounded by its own particular neighbours. All the creatures in Biomorph Land have a definite spatial relationship one to another. What does that mean? What meaning can we attach to spatial position?
The space we are talking about is genetic space. Each animal has its own position in genetic space. Near neighbours in genetic space are animals that differ from one another by only a single mutation. In Figure 3, the basic tree in the centre is surrounded by 8 of its 18 immediate neighbours in genetic space. The 18 neighbours of an animal are the 18 different kinds of children that it can give rise to, and the 18 different kinds of parent from which it could have come, given the rules of our computer model. At one remove, each animal has 324 (18 × 18, ignoring back-mutations for simplicity) neighbours, the set of its possible grandchildren, grandparents, aunts or nieces. At one remove again, each animal has 5,832 (18 × 18 × 18) neighbours, the set of possible great grandchildren, great grandparents, first cousins, etc.
What is the point of thinking in terms of genetic space? Where does it get us? The answer is that it provides us with a way to understand evolution as a gradual, cumulative process. In any one generation, according to the rules of the computer model, it is possible to move only a single step through genetic space. In 29 generations it isn’t possible to move farther than 29 steps, in genetic space, away from the starting ancestor. Every evolutionary history consists of a particular pathway, or trajectory, through genetic space. For instance, the evolutionary history recorded in Figure 4 is a particular winding trajectory through genetic space, connecting a dot to an insect, and passing through 28 intermediate stages. It is this that I mean when I talk metaphorically about ‘wandering’ through Biomorph Land.
I wanted to try to represent this genetic space in the form of a picture. The trouble is, pictures are two-dimensional. The genetic space in which the biomorphs sit is not two-dimensional space. It isn’t even three-dimensional space. It is nine-dimensional space! (The important thing to remember about mathematics is not to be frightened. It isn’t as difficult as the mathematical priesthood sometimes pretends. Whenever I feel intimidated, I always remember Silvanus Thompson’s dictum in Calculus Made Easy: ‘What one fool can do, another can’.) If only we could draw in nine dimensions we could make each dimension correspond to one of the nine genes. The position of a particular animal, say the scorpion or the bat or the insect, is fixed in genetic space by the numerical value of its nine genes. Evolutionary change consists of a step by step walk through ninedimensional space. The amount of genetic difference between one animal and another, and hence the time taken to evolve, and the difficulty of evolving from one to the other, is measured as the distance in nine-dimensional space from one to the other.
Alas, we can’t draw in nine dimensions. I sought a way of fudging it, of drawing a two-dimensional picture that conveyed something of what it feels like to move from point to point in the ninedimensional genetic space of Biomorph Land. There are various possible ways in which this could be done, and I chose one that I call the triangle trick. Look at Figure 6. At the three corners of the triangle are three arbitrarily chosen biomorphs. The one at the top is the basic tree, the one on the left is one of ‘my’ insects, and the one on the right has no name but I thought it looked pretty. Like all biomorphs, each of these three has its own genetic formula, which determines its unique position in nine-dimensional genetic space.
The triangle lies on a flat two-dimensional ‘plane’ that cuts through the nine-dimensional hypervolume (what one fool can do, another can). The plane is like a flat piece of glass stuck through a jelly. On the glass is drawn the triangle, and also some of the biomorphs whose genetic formulae entitle them to sit on that particular flat plane. What is it that entitles them? This is where the three biomorphs at the corners of the triangle come in. They are called the anchor biomorphs.
Remember that the whole idea of ‘distance’ in genetic ‘space’ is that genetically similar biomorphs are near neighbours, genetically different biomorphs are distant neighbours. On this particular plane, the distances are all calculated with reference to the three anchor biomorphs. For any given point on the sheet of glass, whether inside the triangle or outside it, the appropriate genetic formula for that point is calculated as a ‘weighted average’ of the genetic formulae of the three anchor biomorphs. You will already have guessed how the weighting is done. It is done by the distances on the page, more precisely the nearnesses, from the point in question to the three anchor biomorphs. So, the nearer you are to the insect on the plane, the more insect-like are the local biomorphs. As you move along the glass towards the tree, the ‘insects’ gradually become less insect-like and more tree-like. If you walk into the centre of the triangle the animals that you find there, for instance the spider with a Jewish seven-branched candelabra on its head, will be various ‘genetic compromises’ between the three anchor biomorphs.
Figure 6
But this account gives altogether too much prominence to the three anchor biomorphs. Admittedly the computer did use them to calculate the appropriate genetic formula for every point on the picture. But actually any three anchor points on the plane would have done the trick just as well, and would have given identical results. For this reason, in Figure 7 I haven’t actually drawn the triangle. Figure 7 is exactly the same kind of picture as Figure 6. It just shows a different plane. The same insect is one of the three anchor points, this time the right-hand one. The other anchor points, in this case, are the spitfire and the bee-flower, both as seen in Figure 5. On this plane, too, you will notice that neighbouring biomorphs resemble each other more than distant biomorphs. The spitfire, for instance, is part of a squadron of similar aircraft, flying in formation. Because the insect is on both sheets of glass, you can think of the two sheets as passing, at an angle, through each other. Relative to Figure 6, the plane of Figure 7 is said to be ‘rotated about’ the insect.
Figure 7
The elimination of the triangle is an improvement to our method, because it was a distraction. It gave undue prominence to three particular points
in the plane. We still have one further improvement to make. In Figures 6 and 7, spatial distance represents genetic distance, but the scaling is all distorted. One inch upwards is not necessarily equivalent to one inch across. To remedy this, we must choose our three anchor biomorphs carefully, so that their genetic distances, one from the other, are all the same. Figure 8 does just this. Again the triangle is not actually drawn. The three anchors are the scorpion from Figure 5, the insect again (we have yet another ‘rotation about’ the insect), and the rather nondescript biomorph at the top. These three biomorphs are all 30 mutations distant from each other. This means that it is equally easy to evolve from any one to any other one. In all three cases, a minimum of 30 genetic steps must be taken. The little blips along the lower margin of Figure 8 represent units of distance measured in genes. You can think of it as a genetic ruler. The ruler doesn’t only work in the horizontal direction. You can tilt it in any direction, and measure the genetic distance, and hence the minimum evolution time, between any point on the plane and any other (annoyingly, that is not quite true on the page, because the computer’s printer distorts proportions, but this effect is too trivial to make a fuss about, although it does mean that you will get slightly the wrong answer if you simply count blips on the scale).
These two-dimensional planes cutting through nine-dimensional genetic space give some feeling for what it means to walk through Biomorph Land. To improve that feeling, you have to remember that evolution is not restricted to one flat plane. On a true evolutionary walk you could ‘drop through’, at any time, to another plane, for instance from the plane of Figure 6 to the plane of Figure 7 (in the vicinity of the insect, where the two planes come close to each other).
Figure 8
I said that the ‘genetic ruler’ of Figure 8 enables us to calculate the minimum time it would take to evolve from one point to another. So it does, given the restrictions of the original model, but the emphasis is on the word minimum. Since the insect and the scorpion are 30 genetic units distant from one another, it takes only 30 generations to evolve from one to the other if you never take a wrong turning; if, that is, you know exactly what genetic formula you are heading towards, and how to steer towards it. In real-life evolution there is nothing that corresponds to steering towards some distant genetic target.
Let’s now use the biomorphs to return to the point made by the monkeys typing Hamlet, the importance of gradual, step-by-step change in evolution, as opposed to pure chance. Begin by relabelling the graticules along the bottom of Figure 8, but in different units. Instead of measuring distance as ‘number of genes that have to change in evolution’, we are going to measure distance as ‘odds of happening to jump the distance, by sheer luck, in a single hop’. To think about this, we now have to relax one of the restrictions that I built into the computer game: we shall end by seeing why I built that restriction in in the first place. The restriction was that children were only ‘allowed’ to be one mutation distant from their parents. In other words, only one gene was allowed to mutate at a time, and that gene was allowed to change its ‘value’ only by +1 or –1. By relaxing the restriction, we are now allowing any number of genes to mutate simultaneously, and they can add any number, positive or negative, to their current value. Actually, that is too great a relaxation, since it allows genetic values to range from minus infinity to plus infinity. The point is adequately made if we restrict gene values to single figures, that is if we allow them to range from –9 to +9.
So, within these wide limits, we are theoretically allowing mutation, at a stroke, in a single generation, to change any combination of the nine genes. Moreover, the value of each gene can change any amount, so long as it doesn’t stray into double figures. What does this mean? It means that, theoretically, evolution can jump, in a single generation, from any point in Biomorph Land to any other. Not just any point on one plane, but any point in the entire ninedimensional hypervolume. If, for instance, you should want to jump in one fell swoop from the insect to the fox in Figure 5, here is the recipe. Add the following numbers to the values of Genes 1 to 9, respectively: –2,2,2,–2,2,0,–4,–1,1. But since we are talking about random jumps, all points in Biomorph Land are equally likely as destinations for one of these jumps. So, the odds against jumping to any particular destination, say the fox, by sheer luck, are easy to calculate. They are simply the total number of biomorphs in the space. As you can see, we are embarking on another of those astronomical calculations. There are nine genes, and each of them can take any of 19 values. So the total number of biomorphs that we could jump to in a single step is 19 times itself 9 times over: 19 to the power 9. This works out as about half a trillion biomorphs. Paltry compared with Asimov’s ‘haemoglobin number’, but still what I would call a large number. If you started from the insect, and jumped like a demented flea half a trillion times, you could expect to arrive at the fox once.
What is all this telling us about real evolution? Once again, it is ramming home the importance of gradual, step-by-step change. There have been evolutionists who have denied that gradualism of this kind is necessary in evolution. Our biomorph calculation shows us exactly one reason why gradual, step-by-step change is important. When I say that you can expect evolution to jump from the insect to one of its immediate neighbours, but not to jump from the insect directly to the fox or the scorpion, what I exactly mean is the following. If genuinely random jumps really occurred, then a jump from insect to scorpion would be perfectly possible. Indeed it would be just as probable as a jump from insect to one of its immediate neighbours. But it would also be just as probable as a jump to any other biomorph in the land. And there’s the rub. For the number of biomorphs in the land is half a trillion, and if no one of them is any more probable as a destination than any other, the odds of jumping to any particular one are small enough to ignore.
Notice that it doesn’t help us to assume that there is a powerful nonrandom ‘selection pressure’. It wouldn’t matter if you’d been promised a king’s ransom if you achieved a lucky jump to the scorpion. The odds against your doing so are still half a trillion to one. But if, instead of jumping you walked, one step at a time, and were given one small coin as a reward every time you happened to take a step in the right direction, you would reach the scorpion in a very short time. Not necessarily in the fastest possible time of 30 generations, but very fast, nevertheless. Jumping could theoretically get you the prize faster — in a single hop. But because of the astronomical odds against success, a series of small steps, each one building on the accumulated success of previous steps, is the only feasible way.
The tone of my previous paragraphs is open to a misunderstanding which I must dispel. It sounds, once again, as though evolution deals in distant targets, homing in on things like scorpions. As we have seen, it never does. But if we think of our target as anything that would improve survival chances, the argument still works. If an animal is a parent, it must be good enough to survive at least to adulthood. It is possible that a mutant child of that parent might be even better at surviving. But if a child mutates in a big way, so that it has moved a long distance away from its parent in genetic space, what are the odds of its being better than its parent? The answer is that the odds against are very large indeed. And the reason is the one we have just seen with our biomorph model. If the mutational jump we are considering is a very large one, the number of possible destinations of that jump is astronomically large. And because, as we saw in Chapter 1, the number of different ways of being dead is so much greater than the number of different ways of being alive, the chances are very high that a big random jump in genetic space will end in death. Even a small random jump in genetic space is pretty likely to end in death. But the smaller the jump the less likely death is, and the more likely is it that the jump will result in improvement. We shall return to this theme in a later chapter.
That is as far as I want to go in drawing morals from Biomorph Land. I hope that you didn’t find it too abstract. There is an
other mathematical space filled, not with nine-gened biomorphs but with flesh and blood animals made of billions of cells, each containing tens of thousands of genes. This is not biomorph space but real genetic space. The actual animals that have ever lived on Earth are a tiny subset of the theoretical animals that could exist. These real animals are the products of a very small number of evolutionary trajectories through genetic space. The vast majority of theoretical trajectories through animal space give rise to impossible monsters. Real animals are dotted around here and there among the hypothetical monsters, each perched in its own unique place in genetic hyperspace. Each real animal is surrounded by a little cluster of neighbours, most of whom have never existed, but a few of whom are its ancestors, its descendants and its cousins.
Sitting somewhere in this huge mathematical space are humans and hyenas, amoebas and aardvarks, flatworms and squids, dodos and dinosaurs. In theory, if we were skilled enough at genetic engineering, we could move from any point in animal space to any other point. From any starting point we could move through the maze in such a way as to recreate the dodo, the tyrannosaur and trilobites. If only we knew which genes to tinker with, which bits of chromosome to duplicate, invert or delete. I doubt if we shall ever know enough to do it, but these dear dead creatures are lurking there forever in their private corners of that huge genetic hypervolume, waiting to be found if we but had the knowledge to navigate the right course through the maze. We might even be able to evolve an exact reconstruction of a dodo by selectively breeding pigeons, though we’d have to live a million years in order to complete the experiment. But when we are prevented from making a journey in reality, the imagination is not a bad substitute. For those, like me, who are not mathematicians, the computer can be a powerful friend to the imagination. Like mathematics, it doesn’t only stretch the imagination. It also disciplines and controls it.