Here is an even more speculative possibility. The largest benefits of any universality, beyond whatever parochial problem it is intended to solve, come from its being useful for further innovation. And innovation is unpredictable. So, to appreciate universality at the time of its discovery, one must either value abstract knowledge for its own sake or expect it to yield unforeseeable benefits. In a society that rarely experienced change, both those attitudes would be quite unnatural. But that was reversed with the Enlightenment, whose quintessential idea is, as I have said, that progress is both desirable and attainable. And so, therefore, is universality.
Be that as it may, with the Enlightenment, parochialism and all arbitrary exceptions and limitations began to be regarded as inherently problematic – and not only in science. Why should the law treat an aristocrat differently from a commoner? A slave from a master? A woman from a man? Enlightenment philosophers such as Locke set out to free political institutions from arbitrary rules and assumptions. Others tried to derive moral maxims from universal moral explanations rather than merely to postulate them dogmatically. Thus universal explanatory theories of justice, legitimacy and morality began to take their place alongside universal theories of matter and motion. In all those cases, universality was being sought deliberately, as a desirable feature in its own right – even a necessary feature for an idea to be true – and not just as a means of solving a parochial problem.
A jump to universality that played an important role in the early history of the Enlightenment was the invention of movable-type printing. Movable type consisted of individual pieces of metal, each embossed with one letter of the alphabet. Earlier forms of printing had merely streamlined writing in the same way that Roman numerals streamlined tallying: each page was engraved on a printing plate and thus all the symbols on it could be copied in a single action. But, given a supply of movable type with several instances of each letter, one does no further metalwork. One merely arranges the type into words and sentences. One does not have to know, in order to manufacture type, what the documents that it will eventually print are going to say: it is universal.
Even so, movable type did not make much difference when it was invented in China in the eleventh century, perhaps because of the usual lack of interest in universality, or perhaps because the Chinese writing system used thousands of pictograms, which diminished the immediate advantages of a universal printing system. But when it was reinvented by the printer Johannes Gutenberg in Europe in the fifteenth century, using alphabetic type, it initiated an avalanche of further progress.
Here we see a transition that is typical of the jump to universality: before the jump, one has to make specialized objects for each document to be printed; after the jump, one customizes (or specializes, or programs) a universal object – in this case a printing press with movable type. Similarly, in 1801 Joseph Marie Jacquard invented a general-purpose silk-weaving machine now known as the Jacquard loom. Instead of having to control manually each row of stitches in each individual bolt of patterned silk, one could program an arbitrary pattern on punched cards which would instruct the machine to weave that pattern any number of times.
The most momentous such technology is that of computers, on which an increasing proportion of all technology now depends, and which also has deep theoretical and philosophical significance. The jump to computational universality should have happened in the 1820s, when the mathematician Charles Babbage designed a device that he called the Difference Engine – a mechanical calculator which represented decimal digits by cogs, each of which could click into one of ten positions. His original purpose was parochial: to automate the production of tables of mathematical functions such as logarithms and cosines, which were heavily used in navigation and engineering. At the time, they were compiled by armies of clerks known as ‘computers’ (which is the origin of the word), and were notoriously error-prone. The Difference Engine would make fewer errors, because the rules of arithmetic would be built into its hardware. To make it print out a table of a given function, one would program it only once with the definition of the function in terms of simple operations. In contrast, human ‘computers’ had to use (or be used by) both the definition and the general rules of arithmetic thousands of times per table, each time being an opportunity for human error.
Unfortunately, despite pouring a fortune of his own money and that of the British government into the project, Babbage was such a poor organizer that he never succeeded in building a Difference Engine. But his design was sound (apart from a few trivial mistakes), and in 1991 a team led by the engineer Doron Swade at London’s Science Museum successfully implemented it, using engineering tolerances achievable in Babbage’s time.
By the standards of today’s computers and even calculators, the Difference Engine had an extremely limited repertoire. But the reason it could exist at all is that there is a regularity among all the mathematical functions that occur in physics, and hence in navigation and engineering. These are known as analytic functions, and in 1710 the mathematician Brook Taylor had discovered that they can all be approximated arbitrarily well using only repeated additions and multiplications – the operations that the Difference Engine performs. (Special cases had been known before that, but the jump to universality was proved by Taylor.) Thus, to solve the parochial problem of computing the handful of functions that needed to be tabulated, Babbage created a calculator that was universal for calculating analytic functions. It also made use of the universality of movable type, in its typewriter-like printer, without which the process of printing the tables could not have been fully automated.
Babbage originally had no conception of computational universality. Nevertheless, the Difference Engine already comes remarkably close to it – not in its repertoire of computations, but in its physical constitution. To program it to print out a given table, one initializes certain cogs. Babbage eventually realized that this programming phase could itself be automated: the settings could be prepared on punched cards like Jacquard’s, and transferred mechanically into the cogs. This would not only remove the main remaining source of error, but also increase the machine’s repertoire. Babbage then realized that if the machine could also punch new cards for its own later use, and could control which punched card it would read next (say, by choosing from a stack of them, depending on the position of its cogs), then something qualitatively new would happen: the jump to universality.
Babbage called this improved machine the Analytical Engine. He and his colleague the mathematician Ada, Countess of Lovelace, knew that it would be capable of computing anything that human ‘computers’ could, and that this included more than just arithmetic: it could do algebra, play chess, compose music, process images and so on. It would be what is today called a universal classical computer. (I shall explain the significance of the proviso ‘classical’ in Chapter 11, when I discuss quantum computers, which operate at a still higher level of universality.)
Neither they nor anyone else for over a century afterwards imagined today’s most common uses of computation, such as the internet, word processing, database searching, and games. But another important application that they did foresee was making scientific predictions. The Analytical Engine would be a universal simulator – able to predict the behaviour, to any desired accuracy, of any physical object, given the relevant laws of physics. This is the universality that I mentioned in Chapter 3, through which physical objects that are unlike each other and dominated by different laws of physics (such as brains and quasars) can exhibit the same mathematical relationships.
Babbage and Lovelace were Enlightenment people, and so they understood that the universality of the Analytical Engine would make it an epoch-making technology. Even so, despite great efforts, they failed to pass their enthusiasm on to more than a handful of others, who in turn failed to pass it to anyone. And so the Analytical Engine became one of the tragic might-have-beens of history. If only they had looked around for other implementations, they might have realize
d that the perfect one was already waiting for them: electrical relays (switches controlled by electric currents). These had been one of the first applications of fundamental research into electromagnetism, and they were about to be mass produced for the technological revolution of telegraphy. A redesigned Analytical Engine, using on/off electrical currents to represent binary digits and relays to do the computation, would have been faster than Babbage’s and also cheaper and easier to construct. (Binary numbers were already well known. The mathematician and philosopher Gottfried Wilhelm Leibniz had even suggested using them for mechanical calculation in the seventeenth century.) So the computer revolution would have happened a century earlier than it did. Because of the technologies of telegraphy and printing that were being developed concurrently, an internet revolution might well have followed. The science-fiction authors William Gibson and Bruce Sterling, in their novel The Difference Engine, have given an exciting account of what that might have been like. The journalist Tom Standage, in his book The Victorian Internet, maintains that the early telegraph system, even without computers, did create an internet-like phenomenon among the operators, with ‘hackers, on-line romances and weddings, chat-rooms, flame wars . . . and so on’.
Babbage and Lovelace also thought about one application of universal computers that has not been achieved to this day, namely so-called artificial intelligence (AI). Since human brains are physical objects obeying the laws of physics, and since the Analytical Engine is a universal simulator, it could be programmed to think, in every sense that humans can (albeit very slowly and requiring an impractically vast number of punched cards). Nevertheless, Babbage and Lovelace denied that it could. Lovelace argued that ‘The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths.’
The mathematician and computer pioneer Alan Turing later called this mistake ‘Lady Lovelace’s objection’. It was not computational universality that Lovelace failed to appreciate, but the universality of the laws of physics. Science at the time had almost no knowledge of the physics of the brain. Also, Darwin’s theory of evolution had not yet been published, and supernatural accounts of the nature of human beings were still prevalent. Today there is less mitigation for the minority of scientists and philosophers who still believe that AI is unattainable. For instance, the philosopher John Searle has placed the AI project in the following historical perspective: for centuries, some people have tried to explain the mind in mechanical terms, using similes and metaphors based on the most complex machines of the day. First the brain was supposed to be like an immensely complicated set of gears and levers. Then it was hydraulic pipes, then steam engines, then telephone exchanges – and, now that computers are our most impressive technology, brains are said to be computers. But this is still no more than a metaphor, says Searle, and there is no more reason to expect the brain to be a computer than a steam engine.
But there is. A steam engine is not a universal simulator. But a computer is, so expecting it to be able to do whatever neurons can is not a metaphor: it is a known and proven property of the laws of physics as best we know them. (And, as it happens, hydraulic pipes could also be made into a universal classical computer, and so could gears and levers, as Babbage showed.)
Ironically, Lady Lovelace’s objection has almost the same logic as Douglas Hofstadter’s argument for reductionism (Chapter 5) – yet Hofstadter is one of today’s foremost proponents of the possibility of AI. That is because both of them share the mistaken premise that low-level computational steps cannot possibly add up to a higher-level ‘I’ that affects anything. The difference between them is that they chose opposite horns of the dilemma that that poses: Lovelace chose the false conclusion that AI is impossible, while Hofstadter chose the false conclusion that no such ‘I’ can exist.
Because of Babbage’s failure either to build a universal computer or to persuade others to do so, an entire century would pass before the first one was built. During that time, what happened was more like the ancient history of universality: although calculating machines similar to the Difference Engine were being built by others even before Babbage had given up, the Analytical Engine was almost entirely ignored even by mathematicians.
In 1936 Turing developed the definitive theory of universal classical computers. His motivation was not to build such a computer, but only to use the theory abstractly to study the nature of mathematical proof. And when the first universal computers were built, a few years later, it was, again, not out of any special intention to implement universality. They were built in Britain and the United States during the Second World War for specific wartime applications. The British computers, named Colossus (in which Turing was involved), were used for code-breaking; the American one, ENIAC, was designed to solve the equations needed for aiming large guns. The technology used in both was electronic vacuum tubes, which acted like relays but about a hundred times as fast. At the same time, in Germany, the engineer Konrad Zuse was building a programmable calculator out of relays – just as Babbage should have done. All three of these devices had the technological features necessary to be a universal computer, but none of them was quite configured for this. In the event, the Colossus machines never did anything but code-breaking, and most were dismantled after the war. Zuse’s machine was destroyed by Allied bombing. But ENIAC was allowed to jump to universality: after the war it was put to diverse uses for which it had never been designed, such as weather forecasting and the hydrogen-bomb project.
The history of electronic technology since the Second World War has been dominated by miniaturization, with ever more microscopic switches being implemented in each new device. These improvements led to a jump to universality in about 1970, when several companies independently produced a microprocessor, a universal classical computer on a single silicon chip. From then on, designers of any information-processing device could start with a microprocessor and then customize it – program it – to perform the specific tasks needed for that device. Today, your washing machine is almost certainly controlled by a computer that could be programmed to do astrophysics or word processing instead, if it were given suitable input–output devices and enough memory to hold the necessary data.
It is a remarkable fact that, in that sense (that is to say, ignoring issues of speed, memory capacity and input–output devices), the human ‘computers’ of old, the steam-powered Analytical Engine with its literal bells and whistles, the room-sized vacuum-tube computers of the Second World War, and present-day supercomputers all have an identical repertoire of computations.
Another thing that they have in common is that they are all digital: they operate on information in the form of discrete values of physical variables, such as electronic switches being on or off, or cogs being at one of ten positions. The alternative, ‘analogue’, computers, such as slide rules, which represent information as continuous physical variables, were once ubiquitous but are hardly ever used today. That is because a modern digital computer can be programmed to imitate any of them, and to outperform them in almost any application. The jump to universality in digital computers has left analogue computation behind. That was inevitable, because there is no such thing as a universal analogue computer.
That is because of the need for error correction: during lengthy computations, the accumulation of errors due to things like imperfectly constructed components, thermal fluctuations, and random outside influences makes analogue computers wander off the intended computational path. This may sound like a minor or parochial consideration. But it is quite the opposite. Without error-correction all information processing, and hence all knowledge-creation, is necessarily bounded. Error-correction is the beginning of infinity.
For example, tallying is universal only if it is digital. Imagine that some ancient goatherds had tried to tally the total length of their flock instead of the number. As each goat left the en
closure, they could reel out some string of the same length as the goat. Later, when the goats returned, they could reel that length back in. When the whole length had been reeled back in, that would mean that all the goats had returned. But in practice the outcome would always be at least a little long or short, because of the accumulation of measurement errors. For any given accuracy of measurement, there would be a maximum number of goats that could be reliably tallied by this ‘analogue tallying’ system. The same would be true of all arithmetic performed with those ‘tallies’. Whenever the strings representing several flocks were added together, or a string was cut in two to record the splitting of a flock, and whenever a string was ‘copied’ by making another of the same length, there would be errors. One could mitigate their effect by performing each operation many times, and then keeping only the outcome of median length. But the operations of comparing or duplicating lengths can themselves be performed only with finite accuracy, and so could not reduce the rate of error accumulation per step below that level of accuracy. That would impose a maximum number of consecutive operations that could be performed before the result became useless for a given purpose – which is why analogue computation can never be universal.
The Beginning of Infinity Page 17
