The image of DNA-space is very useful for geneticists who are considering possible changes to DNA sequences, such as ‘point mutations’ where one code letter is changed, say as the result of a copying error. Or an incoming high-energy cosmic ray. Viruses, in particular, mutate so rapidly that it makes little sense to talk of a viral species as a fixed thing. Instead, biologists talk of quasi-species, and visualise these as clusters of related sequences in DNA-space. The clusters slosh around as time passes, but they stay together as one cluster, which allows the virus to retain its identity.
In the whole of human history, the total number of people has been no more than ten billion, a mere 11-digit number. This is an incredibly tiny fraction of all those possibilities. So actual human beings have explored the tiniest portion of DNA-space, just as actual books have explored the tiniest portion of L-space. Of course, the interesting questions are not as straightforward as that. Most sequences of letters do not make up a sensible book; most DNA sequences do not correspond to a viable organism, let alone a human being.
And now we come to the crunch for phase spaces. In physics, it is reasonable to assume that the sensible phase space can be ‘pre-stated’ before tackling questions about the corresponding system. We can imagine rearranging the bodies of the solar system into any configuration in that imaginary phase space. We lack the engineering capacity to do that, but we have no difficulty imagining it done, and we see no physical reason to remove any particular configuration from consideration.
When it comes to DNA-space, however, the important questions are not about the whole of that vast space of all possible sequences. Nearly all of those sequences correspond to no organism whatsoever, not even a dead one. What we really need to consider is ‘viable-DNA-space’, the space of all DNA sequences that could be realised within some viable organism. This is some immensely complicated but very thin part of DNA-space, and we don’t know what it is. We have no idea how to look at a hypothetical DNA sequence and decide whether it can occur in a viable organism.
The same problem arises in connection with L-space, but there’s a twist. A literate human can look at a sequence of letters and spaces and decide whether it constitutes a story; they know how to ‘read’ the code and work out its meaning, if it’s in a language they understand. They can even make a stab at deciding whether it’s a good story or a bad one. However, we do not know how to transfer this ability to a computer. The rules that our minds use, to decide whether what we’re reading is a story, are implicit in the networks of nerve cells in our brains. Nobody has yet been able to make those rules explicit. We don’t know how to characterise the ‘readable books’ subset of L-space.
For DNA, the problem is compounded because there isn’t some kind of fixed rule that ‘translates’ a DNA code into an organism. Biologists used to think there would be, and had high hopes of learning the ‘language’ involved. Then the DNA for a genuine (potential) organism would be a code sequence that told a coherent story of biological development, and all other DNA sequences would be gibberish. In effect, the biologists expected to be able to look at the DNA sequence of a tiger and see the bit that specified the stripes, the bit that specified the claws, and so on.
This was a bit optimistic. The current state of the art is that we can see the bit of DNA that specifies the protein from which claws are made, or the bits that make the orange, black and white pigments of the fur that show up as stripes, but that’s about as far as our understanding of DNA narrative goes. It is now becoming clear that many non-genetic factors go into the growth of an organism, too, so even in principle there may not be a ‘language’ that translates DNA into living creatures. For example, tiger DNA turns into a baby tiger only in the presence of an egg, supplied by a mother tiger. The same DNA, in the presence of a mongoose egg, would not make a tiger at all.
Now, it could be that this is just a technical problem: that for each DNA code there is a unique kind of mother-organism that turns it into a living creature, so that the form of that creature is still implicit in the code. But theoretically, at least, the same DNA code could make two totally different organisms. We give an example in The Collapse of Chaos, where the developing organism first ‘looks’ to see what kind of mother it is in, and then develops in different ways depending on what it sees.
Complexity guru Stuart Kauffman has taken this difficulty a stage further. He points out that while in physics we can expect to pre-state the phase space of a system, the same is never true in biology. Biological systems are more creative than physical ones: the organisation of matter within living creatures is of a different qualitative nature from the organisation we find in inorganic matter. In particular organisms can evolve, and when they do that they often become more complicated. The fish-like ancestor of humans was less complicated than we are today, for example. (We’ve not specified a measure of complexity here, but that statement will be reasonable for most sensible measures of complexity, so let’s not worry about definitions.) Evolution does not necessarily increase complexity, but it’s at its most puzzling when it does.
Kauffman contrasts two systems. One is the traditional thermodynamic model in physics, of N gas molecules (modelled as hard spheres) bouncing around inside their 6N-dimensional phase space. Here we know the phase space in advance, we can specify the dynamic precisely, and we can deduce general laws. Among them is the Second Law of Thermodynamics, which states that with overwhelming probability the system will become more disordered as time passes, and the molecules will distribute themselves uniformly throughout their container.
The second system is the ‘biosphere’, an evolving ecology. Here, it is not at all clear which phase space to use. Potential choices are either much too big, or much too limited. Suppose for a moment that the old biologists’ dream of a DNA language for organisms was true. Then we might hope to employ DNA-space as our phase space.
However, as we’ve just seen, only a tiny, intricate subset of that space would really be of interest – but we can’t work out which subset. When you add to that the probable non-existence of any such language, the whole approach falls apart. On the other hand, if the phase space is too small, entirely reasonable changes might take the organisms outside it altogether. For example, tiger-space might be defined in terms of the number of stripes on the big cat’s body. But if one day a big cat evolves that has spots instead of stripes, there’s no place for it in the tiger phase space. Sure, it’s not a tiger … but its mother was. We can’t sensibly exclude this kind of innovation if we want to understand real biology.
As organisms evolve, they change. Sometimes evolution can be seen as the opening-up of a region of phase space that was sitting there waiting, but was not occupied by organisms. If the colours and patterns on an insect change a bit, all that we’re seeing is the exploration of new regions of a fairly well-defined ‘insect-space’. But when an entirely new trick, wings, appears, even the phase space seems to have changed.
It is very difficult to capture the phenomenon of innovation in a mathematical model. Mathematicians like to pre-state the space of possibilities, but the whole point about innovation is that it opens up new possibilities that were previously not envisaged. So Kauffman suggests that a key feature of the biosphere is the inability to pre-state a phase space for it.
At risk of muddying the waters, it is worth observing that even in physics, pre-stating the phase space is not as straightforward as it might appear. What happens to the phase space of the solar system if we allow bodies to break up, or merge? Supposedly5 the Moon was splashed off the Earth when it collided with a body about the size of Mars. Before that event, there was no Moon-coordinate in the phase space of the solar system; afterwards, there was. So the phase space expanded when the Moon came into being. The phase spaces of physics always assume a fixed context. In physics, you can usually get away with that assumption. In biology, you can’t.
There’s a second problem in physics, too. That 6N-dimensional phase space of thermodynamics, fo
r example, is too big. It includes non-physical states. By a quirk of mathematics, the laws of motion for elastic spheres do not prescribe what happens when three or more collide simultaneously. So we must excise from that nice, simple 6N-dimensional space all configurations that experience a triple collision somewhere in their past or future. We know four things about these configurations. They are very rare. They can occur. They form an extremely complicated cloud of points in phase space. And it is impossible, in any practical sense, to determine whether a given configuration should or should not be excised. If these unphysical states were a bit more common, then the thermodynamic phase space would be just as hard to pre-state as that for the biosphere. However, they are a vanishingly small proportion of the whole, so we can just about get away with ignoring them.
Nonetheless, it is possible to go some way towards pre-stating a phase space for the biosphere. While we cannot pre-state a space of all possible organisms, we can look at any given organism and at least in principle say what the potential immediate changes are. That is, we can describe the space of the adjacent possible, the local phase space. Innovation then becomes the process of expanding into the adjacent possible. This is a reasonable and fairly conventional idea. But, more controversially, Kauffman suggests the exciting possibility that there may be general laws that govern this kind of expansion, laws that have exactly the opposite effect to the famous Second Law of Thermodynamics. The Second Law in effect states that thermodynamic systems become simpler as time passes; all of the interesting structure gets ‘smeared out’ and disappears. In contrast, Kauffman’s suggestion is that the biosphere expands into the space of the adjacent possible at the maximum rate that it can, subject to hanging together as a biological system. Innovation in biology happens as rapidly as possible.
More generally, Kauffman extends this idea to any system composed of ‘autonomous agents’. An autonomous agent is a generalised life-form, defined by two properties: it can reproduce, and it can carry out at least one thermodynamic work cycle. A work cycle occurs when a system does work and returns to its original state, ready to do the same again. That is, the system takes energy from its environment and transforms it into work, and does so in such a manner that at the end of the cycle it returns to its initial state.
A human being is an autonomous agent, and so is a tiger. A flame is not: flames reproduce by spreading to inflammable material nearby, but they do not carry out a work cycle. They turn chemical energy into fire, but once something has been burnt, it can’t be burnt a second time.
This theory of autonomous agents is explicitly set in the context of phase spaces. Without such a concept, it cannot even be described. And in this theory we see the first possibility of obtaining a general understanding of the principles whereby, and wherefore, organisms complicate themselves. We are starting to pin down just what it is about lifeforms that makes them behave so differently from the boring prescription of the Second Law of Thermodynamics. We paint a picture of the universe as a source of ever-increasing complexity and organisation, instead of the exact opposite. We find out why we live in an interesting universe, instead of a dull one.
1 There’s a Special Theory as well, but no one bothers with it much because it’s self-evidently a load of marsh-gas. [This footnote is a footnote in the original quotation. So this is a metafootnote.]
2 The bean-counters don’t even know how to count beans sensibly. Are we surprised?
3 A tour of any airport bookshop will show that this is reasonable.
4 But Joycean scholars would be furious if we excluded Finnegan’s Wake, which reads exactly like that.
5 See The Science of Discworld, ‘A giant leap for moonkind’.
FIVE
REMARKABLY LIKE ANKH-MORPORK
‘HOW CAN YOU COMMUNICATE LIKE THIS?’ panted Ponder, as they jogged along beside a broad river.
‘Since the physics of Roundworld are subordinate to the physics of the real world, I can use anything considered to be a communication device,’ said the voice of Hex, slightly muffled in Rincewind’s pocket. ‘The owner of this device believes it to be one such. Also, I can deduce much information from this world’s footprint in L-space. And the Archchancellor was right. There is much Elvish influence here.’
‘You can extract information from Roundworld books?’ said Ponder.
‘Yes. The phase space of books that relate to this world contains ten to the power of 1,100 to the power of n volumes,’ said Hex.
‘That’s enough books to fill the univ— hold on, what is n?’
‘The number of all possible universes.’
‘Then that’s enough books to fill all possible universes! Well … as close as makes no difference, anyway.’
‘Correct. That is why there is never enough bookshelf space. However, because of the subordinate temporal matrix of this world, I can use virtual computing,’ said Hex. ‘Once you know what the answer is, the process of calculation can be seriously reduced. Once the correct answer is found, the fruitless channels of inquiry cease to exist. Besides, if you deduct all the books that are about golf, cats, slood1 and cookery the number is really quite manageable.’
‘Oook,’ said the Librarian.
‘He says he’s not going to have a shave,’ said Rincewind.
‘It is essential,’ said Hex. ‘We are getting strange glances from people in the fields. We do not wish to attract a mob. He must be shaved, and given a robe and hat.’
Rincewind was doubtful. ‘I don’t think that’ll fool anyone,’ he said.
‘My readings tell me that it will if you say he’s Spanish.’
‘What’s Spanish?’
‘Spain is a country some five hundred miles from this one.’
‘And people there look like him?’
‘No. But people here would be quite prepared to believe so. This is a credulous age. The elves have done a lot of damage. The greatest minds spend half their time busying themselves with the study of magic, astrology, alchemy and communion with spirits.’
‘Well? Sounds just like life at home,’ said Rincewind.
‘Yes,’ said Hex. ‘But there is no narrativium in this world. No magic. None of those things work.’
‘Then why don’t they just stop trying it?’ said Ponder.
‘My inference is that they believe it should work if only they get it right.’
‘Poor devils,’ said Rincewind.
‘They believe in those, too.’
‘There’s more houses ahead,’ said Ponder. ‘We’re coming to a city. Er … and we’ve got the Luggage with us. Hex, we haven’t just got an orangutan with us, we’ve got a box on legs!’
‘Yes. We must leave it in some bushes while we find a voluminous dress and a wig,’ said Hex calmly. ‘Fortunately, this is the right period.’
‘A dress won’t work, believe me!’
‘It will if the Librarian sits on the Luggage,’ said Hex. ‘That will bring him up to the right height and the dress will provide adequate cover for the Luggage.’
‘Now hang on a moment,’ said Rincewind. ‘You saying that people here will believe an ape in a dress and a wig is a woman?’
‘They will if you say she’s Spanish.’
Rincewind took another look at the Librarian.
‘Those elves really must have done a lot of damage,’ he said.
The city was remarkably like Ankh-Morpork, although smaller and, unbelievably, smellier. One reason for that was the large number of animals in the streets. It was as if the place had been designed as a village and simply scaled up.
The wizards hadn’t been hard to find. Hex located them easily, but in any case the noise could be heard in the next street. There was a tavern, with a courtyard, and in the courtyard a crowd of alcohol, which contained people, was watching a man trying to beat Archchancellor Ridcully with a very long and heavy staff.
He wasn’t succeeding. Ridcully, who was stripped to the waist, was fighting back very effectively, putting his wizarding s
taff to the unusual task of hitting someone. He was a lot better at it than his opponent. Most wizards would die rather than take exercise, and did, but Ridcully had the rude health of a bear and only marginally better interpersonal skills. Despite his quite considerable if erratic erudition, at heart he was a man who’d rather smack someone around the ear than develop a complicated argument.
As the rescue party arrived, he hit the man across the head and then swept his feet from under him on the back-swing. A cheer went up as the man went down.
Ridcully helped his stunned adversary to his feet and propelled him to a bench, where the man’s friends poured beer over him. Then he nodded to Rincewind and company.
‘Got here, then,’ he said. ‘Bring the stuff, did you? Who’s the Spanish lady?’
‘That’s the Librarian,’ said Rincewind. There wasn’t a great deal visible between the ruff and the red wig except an impression of extreme annoyance.
‘Is it?’ said Ridcully. ‘Oh, yes. Sorry. Been here too long. This place gets to you. Good thinking, puttin’ him in disguise. Hex suggested that, I expect.’
‘We came as quick as we could, sir,’ said Ponder. ‘How long have you been here?’
‘Couple of weeks,’ said Ridcully. ‘Not a bad place. Come and meet everyone.’
The rest of the wizards were sitting around a table. They were dressed in their normal wizarding outfits which, Rincewind had noticed, fitted in pretty well with the costumes in this town. But each man had equipped himself with a ruff, just to be on the safe side.
They nodded cheerfully at the newcomers. A forest of empty mugs in front of them went some way to explaining the cheer.
The Science of Discworld II Page 6
