The Science of Discworld II

by Terry Pratchett

  ‘Oh, no, sir. It’s impossible to trace.’

  ‘Impossible?’

  ‘Yes, sir. Every week Hex sends a message to company headquarters readjusting the total of messages sent, sir. Anyway, there’s so many I don’t think anyone checks.’

  ‘Oh? Well, that’s all right then,’ said Ridcully. ‘It never really happens, and no one can find out it’s us in any case. Can we send all our messages that way?’

  ‘Well, technically yes, sir, but I think that might be abusing the—’

  ‘We are academics, Stibbons,’ said the Dean. ‘And information should be allowed to flow freely.’

  ‘Exactly,’ said the Lecturer in Recent Runes. ‘An untrammelled flow of information is essential to a progressive society. This is the age of the semaphore, after all.’

  ‘Obviously it flows to us,’ said Ridcully.

  ‘Oh, certainly,’ said the Dean. ‘We don’t want it flowing away from us. We’re talking about flow here, not spread.’

  ‘You wanted a message sent?’ said Ponder, before the wizards got too deeply into this.

  ‘And we really don’t have to pay?’ said Ridcully.

  Ponder sighed. ‘No, sir.’

  ‘Jolly good,’ said the Archchancellor. ‘Have this one sent to the kingdom of Lancre, will you? They’ve only got one clacks tower. Got your notebook? Message begins:

  “To Mistress Esmerelda Weatherwax. How are you? I am fine. An interesting problem has arisen …”

  EIGHTEEN

  BIT FROM IT

  A SEMAPHORE IS A SIMPLE and time-honoured example of a digital communication system. It encodes letters of the alphabet using the positions of flags, lights, or something similar. In 1795 George Murray invented a version that is close to the system currently used in Discworld: a set of six shutters that could be opened or closed, thus giving 64 different ‘codes’, more than enough for the entire alphabet, numbers 0 to 10 and some ‘special’ codes. The system was further developed but ceased to be cutting-edge technology when the electric telegraph heralded the wired age. The Discworld semaphore (or ‘clacks’) has been taken much further, with mighty trunk route towers carrying bank after bank of shutters, aided by lamps after dark, and streaming messages bi-directionally across the continent. It is a pretty accurate ‘evolution’ of the technology: if we too had failed to harness steam and electricity, we might well be using something like it …

  There is enough capacity on that system even to handle pictures – seriously. Convert the picture to a 64 × 64 grid of little squares that can be black, white or four shades of grey, and then read the grid from left to right and top to bottom like a book. It’s just a matter of information, a few clever clerks to work out some compression algorithms, and a man with a shallow box holding 4,096 wooden blocks, their six sides being, yes, black, white and four shades of grey. It’ll take them a while to reassemble the pictures, but clerks are cheap.

  Digital messages are the backbone of the Information Age, which is the name we currently give to the one we’re living in, in the belief that we know a lot more than anyone else, ever. Discworld is comparably proud of being in the Semaphore Age, the Age of the Clacks. But what, exactly, is information?

  When you send a message, you are normally expected to pay for it – because if you don’t, then whoever is doing the work of transmitting that message for you will object. It is this feature of messages that has got Ridcully worried, since he is wedded to the idea that academics travel free.

  Cost is one way to measure things, but it depends on complicated market forces. What, for example, if there’s a sale on? The scientific concept of ‘information’ is a measure of how much message you’re sending. In human affairs, it seems to be a fairly universal principle that for any given medium, longer messages cost more than short ones. At the back of the human mind, then, lurks a deep-seated belief that messages can be quantified: they have a size. The size of a message tells you ‘how much information’ it contains.

  Is ‘information’ the same as ‘story’? No. A story does convey information, but that’s probably the least interesting thing about stories. Most information doesn’t constitute a story. Think of a telephone directory: lots of information, strong cast, but a bit weak on narrative. What counts in a story is its meaning. And that’s a very different concept from information.

  We are proud that we live in the Information Age. We do, and that’s the trouble. If we ever get to the Meaning Age, we’ll finally understand where we went wrong.

  Information is not a thing, but a concept. However, the human tendency to reify concepts into things has led many scientists to treat information as if it is genuinely real. And some physicists are starting to wonder whether the universe, too, might be made from information.

  How did this viewpoint come about, and how sensible is it?

  Humanity acquired the ability to quantify information in 1948, when the mathematician-turned-engineer Claude Shannon found a way to define how much information is contained in a message – he preferred the term signal – sent from a transmitter to a receiver using some kind of code. By a signal, Shannon meant a series of binary digits (‘bits’, 0 and 1) of the kind that is ubiquitous in modern computers and communication devices, and in Murray’s semaphore. By a code, he meant a specific procedure that transforms an original signal into another one. The simplest code is the trivial ‘leave it alone’; more sophisticated codes can be used to detect or even correct transmission errors. In the engineering applications, codes are a central issue, but for our purposes here we can ignore them and assume the message is sent ‘in plain’.

  Shannon’s information measure puts a number to the extent to which our uncertainty about the bits that make up a signal is reduced by what we receive. In the simplest case, where the message is a string of 0s and 1s and every choice is equally likely, the amount of information in a message is entirely straightforward: it is the total number of binary digits. Each digit that we receive reduces our uncertainty about that particular digit (is it 0 or 1?) to certainty (‘it’s a 1’, say) but tells us nothing about the others, so we have received one bit of information. Do this a thousand times and we have received a thousand bits of information. Easy.

  The point of view here is that of a communications engineer, and the unstated assumption is that we are interested in the bit-by-bit content of the signal, not in its meaning. So the message 111111111111111 contains 15 bits of information, and so does the message 111001101101011. But Shannon’s concept of information is not the only possible one. More recently, Gregory Chaitin has pointed out that you can quantify the extent to which a signal contains patterns. The way to do this is to focus not on the size of the message, but on the size of a computer program, or algorithm, that can generate it. For instance, the first of the above messages can be created by the algorithm ‘every digit is a 1’. But there is no simple way to describe the second message, other than to write it down bit by bit. So these two messages have the same Shannon information content, but from Chaitin’s point of view the second contains far more ‘algorithmic information’ than the first.

  Another way to say this is that Chaitin’s concept focuses on the extent to which the message is ‘compressible’. If a short program can generate a long message, then we can transmit the program instead of the message and save time and money. Such a program ‘compresses’ the message. When your computer takes a big graphics file – a photograph, say – and turns it into a much smaller file in JPEG format, it has used a standard algorithm to compress the information in the original file. This is possible because photographs contain numerous patterns: lots of repetitions of blue pixels for the sky, for instance. The more incompressible a signal is, the more information in Chaitin’s sense it contains. And the way to compress a signal is to describe the patterns that make it up. This implies that incompressible signals are random, have no pattern, yet contain the most information. In one way this is reasonable: when each successive bit is maximally unpredictable,
you learn more from knowing what it is. If the signal reads 111111111111111 then there is no great surprise if the next bit turns out to be 1; but if the signal reads 111001101101011 (which we obtained by tossing a coin 15 times) then there is no obvious guess for the next bit.

  Both measures of information are useful in the design of electronic technology. Shannon information governs the time it takes to transmit a signal somewhere else; Chaitin information tells you whether there’s a clever way to compress the signal first, and transmit something smaller. At least, it would do if you could calculate it, but one of the features of Chaitin’s theory is that it is impossible to calculate the amount of algorithmic information in a message – and he can prove it. The wizards would approve of this twist.

  ‘Information’ is therefore a useful concept, but it is curious that ‘To be or not to be’ contains the same Shannon information as, and less Chaitin information than, ‘xyQGRlfryu&d%sk0wc’. The reason for this disparity is that information is not the same thing as meaning. That’s fascinating. What really matters to people is the meaning of a message, not its bit-count, but mathematicians have been unable to quantify meaning. So far.

  And that brings us back to stories, which are messages that convey meaning. The moral is that we should not confuse a story with ‘information’. The elves gave humanity stories, but they didn’t give them any information. In fact, the stories people came up with included things like werewolves, which don’t even exist on Roundworld. No information there – at least, apart from what it might tell you about the human imagination.

  Most people, scientists in particular, are happiest with a concept when they can put a number to it. Anything else, they feel, is too vague to be useful. ‘Information’ is a number, so that comfortable feeling of precision slips in without anyone noticing that it might be spurious. Two sciences that have gone a long way down this slippery path are biology and physics.

  The discovery of the ‘linear’ molecular structure of DNA has given evolutionary biology a seductive metaphor for the complexity of organisms and how they evolve, namely: the genome of an organism represents the information that is required to construct it. The origin of this metaphor is Francis Crick and James Watson’s epic discovery that an organism’s DNA consists of ‘code words’ in the four molecular ‘letters’ A C T G, which, you’ll recall, are the initials of the four possible ‘bases’. This description led to the inevitable metaphor that the genome contains information about the corresponding organism. Indeed, the genome is widely described as ‘containing the information needed to produce’ an organism.

  The easy target here is the word ‘the’. There are innumerable reasons why a developing organism’s DNA does not determine the organism. These non-genomic influences on development are collectively known as ‘epigenetics’, and they range from subtle chemical tagging of DNA to the investment of parental care. The hard target is ‘information’. Certainly, the genome includes information in some sense: currently an enormous international effort is being devoted to listing that information for the human genome, and also for other organisms such as rice, yeast, and the nematode worm Caenorhabditis elegans. But notice how easily we slip into cavalier attitudes, for here the word ‘information’ refers to the human mind as receiver, not to the developing organism. The Human Genome Project informs us, not organisms.

  This flawed metaphor leads to the equally flawed conclusion that the genome explains the complexity of an organism in terms of the amount of information in its DNA code. Humans are complicated because they have a long genome that carries a lot of information; nematodes are less complicated because their genome is shorter. However, this seductive idea can’t be true. For example, the Shannon information content of the human genome is smaller by several orders of magnitude than the quantity of information needed to describe the wiring of the neurons in the human brain. How can we be more complex than the information that describes us? And some amoebas have much longer genomes than ours, which takes us down several pegs as well as casting even more doubt on DNA as information.

  Underlying the widespread belief that DNA complexity explains organism complexity (even though it clearly doesn’t) are two assumptions, two scientific stories that we tell ourselves. The first story is DNA as Blueprint, in which the genome is represented not just as an important source of control and guidance over biological development, but as the information needed to determine an organism. The second is DNA as Message, the ‘Book of Life’ metaphor.

  Both stories oversimplify a beautifully complex interactive system. DNA as Blueprint says that the genome is a molecular ‘map’ of an organism. DNA as Message says that an organism can pass that map to the next generation by ‘sending’ the appropriate information.

  Both of these are wrong, although they’re quite good science fiction – or, at least, interestingly bad science fiction with good special effects.

  If there is a ‘receiver’ for the DNA ‘message’ it is not the next generation of the organism, which does not even exist at the time the ‘message’ is being ‘sent’, but the ribosome, which is the molecular machine that turns DNA sequences (in a protein-coding gene) into protein. The ribosome is an essential part of the coding system; it functions as an ‘adapter’, changing the sequence information along the DNA into an amino acid sequence in proteins. Every cell contains many ribosomes: we say ‘the’ because they are all identical. The metaphor of DNA as information has become almost universal, yet virtually nobody has suggested that the ribosome must be a vast repository of information. The structure of the ribosome is now known in high detail, and there is no sign of obvious ‘information-bearing’ structure like that in DNA. The ribosome seems to be a fixed ‘machine’. So where has the information gone? Nowhere. That’s the wrong question.

  The root of these misunderstandings lies in a lack of attention to context. Science is very strong on content, but it has a habit of ignoring ‘external’ constraints on the systems being studied. Context is an important but neglected feature of information. It is so easy to focus on the combinatorial clarity of the message and to ignore the messy, complicated processes carried out by the receiver when it decodes the message. Context is crucial to the interpretation of messages: to their meaning. In his book The User Illusion Tor Nørretranders introduced the term exformation to capture the role of the context, and Douglas Hofstadter made the same general point in Gödel, Escher, Bach. Observe how, in the next chapter, the otherwise incomprehensible message ‘THEOSTRY’ becomes obvious when context is taken into account.

  Instead of thinking about a DNA ‘blueprint’ encoding an organism, it’s easier to think of a CD encoding music. Biological development is like a CD that contains instructions for building a new CD-player. You can’t ‘read’ those instructions without already having one. If meaning does not depend upon context, then the code on the CD should have an invariant meaning, one that is independent of the player. Does it, though?

  Compare two extremes: a ‘standard’ player that maps the digital code on the CD to music in the manner intended by the design engineers, and a jukebox. With a normal jukebox, the only message that you send is some money and a button-push; yet in the context of the jukebox these are interpreted as a specific several minutes’ worth of music. In principle, any numerical code can ‘mean’ any piece of music you wish; it just depends on how the jukebox is set up, that is, on the exformation associated with the jukebox’s design. Now consider a jukebox that reacts to a CD not by playing the tune that’s encoded on it, as a series of bits, but by interpreting that code as a number, and then playing some other CD to which that number has been assigned. For instance, suppose that a recording of Beethoven’s Fifth Symphony starts, in digital form, with 11001. That’s the number 25 in binary. So the jukebox reads the CD as ‘25’, and looks for CD number 25, which we’ll assume is a recording of Charlie Parker playing jazz. On the other hand, elsewhere in the jukebox is CD number 973, which actually is Beethoven’s Fifth Symphony. Then a CD
of Beethoven’s Fifth can be ‘read’ in two totally different ways: as a ‘pointer’ to Charlie Parker, or as Beethoven’s Fifth Symphony itself (triggered by whichever CDs start with 973 in binary). Two contexts, two interpretations, two meanings, two results.

  Whether something is a message depends upon context, too: sender and receiver must agree upon a protocol for turning meanings into symbols and back again. Without this protocol a semaphore is just a few bits of wood that flap about. Tree branches are bits of wood that flap about, too, but no one ever tries to decode the message being transmitted by a tree. Tree rings – the growth rings that appear when you saw through the trunk, one ring per year – are a different matter. We have learned to ‘decode’ their ‘message’, about climate in the year 1066 and the like. A thick ring indicates a good year with lots of growth on the tree, probably warm and wet; a thin ring indicates a poor year, probably cold and dry. But the sequence of tree rings only became a message, only conveyed information, when we figured out the rules that link climate to tree growth. The tree didn’t send its message to us.

  In biological development the protocol that gives meaning to the DNA message is the laws of physics and chemistry. That is where the exformation resides. However, it is unlikely that exformation can be quantified. An organism’s complexity is not determined by the number of bases in its DNA sequence, but by the complexity of the actions initiated by those bases within the context of biological development. That is, by the meaning of the DNA ‘message’ when it is received by a finely tuned, up-and-running biochemical machine. This is where we gain an edge over those amoebas. Starting with an embryo that develops little flaps, and making a baby with those exquisite little hands, involves a series of processes that produce skeleton, muscles, skin, and so on. Each stage depends on the current state of the others, and all of them depend on contextual physical, biological, chemical and cultural processes.