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? Supposedly[16] 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-dimension phase space of thermodynamics, for example, is too big. It includes non-physical states. By a quirk of mathematics, the laws of motion of 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 jus 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.
5. 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 the 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, slood[17] 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 village and simply scaled up.
The wizards hadn't been hard to find. Hex located them easily, 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 staff 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 the swept his feet from under him on the back-swing. A cheer went up 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.
'You've detected elves?' said Ridcully, forcing enough wizards apart to give them seats.
'The place is lousy with glamour, sir.' said Ponder, sitting down.
'You're telling me,' said Ridcully. He glanced along the table. 'Oh, yes. We've found a friend.
Dee, this is Mister Stibbons. Remember we told you about him?'
It was then that Ponder realised there were a couple of non-wizards in the party. It was quite hard to spot one, though, since for all practical purposes he fitted in well. He even had the right kind of beard.
'Er ... the noddlepate?' said Dee.
'No, that's Rincewind,' said Ridcully. 'Ponder is the clever one. And this ...' he turned to the Librarian, and words failed even him, 'Is ... a ... friend of theirs.'
'From Spanish,' said Rincewind, who didn't know what noddlepate meant but had formed a pretty good idea.
'Dee here is a sort of local wizard,' said Ridcully, in the loud voice he thought was a confidential whisper. 'Sharp as a tack, mind like a razor, but spends all his spare time trying to do magic!'
'Which doesn't work here,' said Ponder.
'Right! But everyone believes it does, despite everything. Amazing! That's what elves can do to a place.' Ridcully leaned forward, conspiratorially. 'They came straight through our world and straight on into this one and we got caught up in the ... what's it you call it when it's all swirly and chilly as hell?'
'Trans-dimensional flux, sir,' said Ponder.
'Right. We'd have been totally lost if our friend Dee here hadn't been working a magic circle at the time.'
There was silence from Rincewind and Ponder. Then Rincewind said: 'You said magic doesn't work here.'
'As with this crystal sphere,' said a voice from Rincewind's pocket, 'this world is quite capable of maintaining a passive receptor. Rincewind removed the scrying stone from his pocket. 'But that is mine,' said Dee, staring at it.
'Sorry,' said Rincewind. 'We just sort of found it and sort of picked it sort of up.'
'But it speaks!' gasped Dee. 'An ethereal voice!'
'No, it's just from another world that is much bigger than this one and can't be seen,' said Ridcully. 'There's nothing mysterious about it at all.'
With trembling fingers, Dee took the sphere from Rincewind and held it in front of his eyes.
'Speak!' he commanded.
'Permission denied,' said the crystal. 'You do not have the rights to do this.'
'Where did you tell him you came from?' Rincewind whispered Ridcully, as Dee tried to polish the ball with the sleeve of his robe.
'I just said we'd dropped in from another sphere,' said Ridcully. 'After all, this universe is full of spheres. He seemed to be quite happy about that. I didn't mention the Discworld at all, in case it confused him.
Rincewind looked at Dee's shaking hands and the manic glint in his eye.
'I just want to be clear,' he said slowly. 'You appeared in a magic circle, you told him you're from another sphere, he'd just spoken to crystal ball, you've explained to him that magic doesn't work and you don't want to confuse him?'
'Make him any more confused than he is already, you mean,' said the Dean. 'Confusion is the natural state of mind here, believe us. Do you know they think numbers are magical? Doing sums can get you into real trouble in these parts.'
'Well, some numbers are magi—' Ponder began.
'Not here they're not,' said the Archchancellor. 'Here I am, out in the open air, no magical protection and I'm going to say the number that comes after seven. Here it comes: eight. There.
Nothing happened. Eight! Eighteen! Two fat ladies in very tight corsetry, eighty-eight! Oh someone pull Rincewind out from under the table, will you?'
While the Professor of Cruel and Unusual Geography was having some of it brushed off his robes, Ridcully continued: 'It's a mad world. No narrativium. People makin' up history as they go along. Brilliant men spendin' their time wondering how many angels can dance on the head of a pin—'
'Sixteen,' said Ponder.
'Yes, we know that because we can go and look, but here it's just another silly question,' said Ridcully. 'It'd make you cry. The history of this place goes backwards half the time. It's a mess.
A parody of a world.'
'We made it,' said the Lecturer in Recent Runes.
'We didn't make it this badly,' said the Dean. 'We've seen the history books here. There were some great civilisations thousands of years ago. There was a place like Ephebe that was really beginning to find things out. The wrong things, mostly, but at least they were making an effort.
Even had a decent pantheon of gods. All gone now. Our chum here and his friends think everything worth knowing has been discovered and forgotten and, frankly, they're not totally wrong.'
'What can we do about it?' said Ponder.
'You can talk to Hex on that thing?'
Yes, sir.'
'Then Hex can do the magic back at UU and we'll find out what the elves did,' said Ridcully.
'Er,' Rincewind began, 'do we have the right to interfere?' They all stared at him.
'I mean, we never did it before,' he went on. 'Remember all those other creatures that evolved here? The intelligent lizards? The intelligent crabs? Those dog things? They all got completely wiped out by ice ages and falling rocks and we never did anything to stop it.'[18]
They went on staring.
'I mean, elves are just another problem, aren't they?' said Rincewind. Maybe ... maybe they're just another form of big rock? Maybe ... maybe they always turn up when intelligence gets going? And the species is either clever enough to survive them or it ends up buried in the bedrock like all the others? I mean, perhaps it's a kind of, of a test? I mean…
It dawned on Rincewind that he was not carrying the meeting, wizards were glaring at him.
'Are you suggesting that someone somewhere is awarding marks, Rincewind?' said Ponder.
'Well, obviously there is no—'
'Good. Shut up,' said Ridcully. 'Now, lads, let's get back to Mort Lake and get started.'
'Mort Lake?' said Rincewind. 'But tha
t's in Ankh-Morpork!' 'There's one here, too,' said the Lecturer in Recent Runes, beaming.
'Amazing, isn't it? We never guessed. This world is a cheap parody of our own. As Above, So Below and all that.'
'But without magic,' said Ridcully. 'And with no narrativium. It doesn't know where it's going.'
'But we do, sir,' said Ponder, who had been scribbling in notebook.
'Do we?'
'Yes, sir. Remember? In about a thousand years' time it's going to be hit by a really big rock. I keep looking at the numbers, sir, an that's what it means.'
'But I thought we found there'd been a race that built huge structures to get off the place?' 'That's right, sir.'
'Can a new species turn up in a thousand years?'
'I don't think so, sir.'
'You mean these are the ones that leave?'
'It seems like it, sir,' said Ponder.
The wizards looked at the people in the courtyard. Of course, the presence of beer always greases the rungs of the evolutionary ladder, but even so ...
At a nearby table, one man threw up on another one. There was general applause.
T think,' said Ridcully, summing up the general mood, 'that we are going to be here for some time.'
6. THE LENS-GRINDER'S PHILOSOPHY
John Dee, who lived from 1527 to 1608, was a court astrologer to Mary Tudor. At one point he was imprisoned for being a magician, but in 1555 they let him out again, presumably for not being one. Then he became an astrologer for Queen Elizabeth I. He devoted much of his life to the occult, both alchemy and astrology. On the other hand, he was also the author of the first English translation of Euclid's Elements, the renowned treatise on geometry. Actually, if you believe the printed word, the book is attributed to Sir Henry Billingsley, but it was common knowledge that Dee did all the work, and he even wrote a long and erudite preface. Which may be why it was common knowledge that Dee did all the work.
The Science of Discworld II - The Globe tsod-2 Page 6
