This blog is a summary of the chapter ‘The Jump to Universality’ from David Deutsch’s book, The Beginning of Infinity. All errors of interpretation are mine alone. I (obviously!) recommend reading the book.
First, here is a pithy summary of universality in a tweet (disregard his typo).
You can also listen to Brett Hall explain universality (his podcast Tokcast goes through both the Fabric of Reality and The Beginning of Infinity, and is absolutely superb).
Okay, now here is my elongated summary.
Early writing systems used stylised pictures – pictograms – to represent words or concepts. But no system accomplished having a pictogram for every word, thus failing at achieving universality.
It is likely that scribes, in ancient times, may have used new rules, not new pictograms because installing a lot of “intellectual software” – that this picture means this meaning becomes increasingly difficult. In short, it doesn’t scale very well. It may not have had universal reach but it certainly had some reach.
And reach always has an explanation. Why (does reach always have an explanation): because it had a regularity - all the words in being: any given language is built out of only a few dozen ‘elementary sounds’.
Since a rule works by exploiting regularities in the language, it implicitly encodes those regularities, and so contains more knowledge than the list. An ‘alphabet’ writing system achieves universality because it covers not only every word but every possible word in its language.
Thus, it allows the system to be used to coin new words in an easy and decentralised way. Opportunities for universality may have existed through human history. It was likely not achieved because ancient innovations – those who created pictograms or the Phoenician (it is suspected) alphabet were only concerned with specific problems they were confronting – to write particular words – and to do that, one of them invented a rule that happened to be universal. This, Deutsch observes, is a recurring theme in the early history of many fields that universality, when it was achieved, was not the primary objective, if it was an objective all.
Hierarchies There is a hierarchy of responsibility. Numbers are greater than numerals. Why? Because numbers are more scalable. Numerals, however, are programmable. Consider the roman numeral system. First, they added rules such as:
Placing symbols side by side – which means adding them together
Symbols must be written in order of decreasing value, from left to right; and
Adjacent symbols must be replaced by the symbol for their combined value, whenever possible.
These enabled new functions to be performed, such as arithmetic. In short, it allowed numbers to be manipulated. Importantly, it is the system that performed the arithmetic; the human brain, while enacting them, were simply executing the program. And it is the program that instructs its computer what to do, not vice-versa. And so, in a way, the Roman-numeral system used us to do arithmetic. It was only by causing people to do this that the Roman-numeral system survived. The ability to manipulate numbers independently of tallying our counting opened up applications such as calculating prices, wages, taxes, interest rates and so on. It is this reach that enabled Romans to scale their accounting system and build a successful civilisation, and then empire. But the Roman system was still bounded – it still had a highest value symbol, and hence would not be universal for doing arithmetic without tallying.
By contrast, today’s number system – which originated in India – has ten symbols, the digits 0 to 9, and its universality is due to a rule that the value of a digit depends on its position in the number. Around 1900 BCE the Babylonian system invented a near universal system. It had 59 digits. However, it had some shortcomings: one issue it was that it had no way of representing zeros; it used spaces instead of zeros. The second issue was that it had no decimal point. This suggests that universality was not the system’s main objective, and that it was not greatly valued, even when it was achieved.
In the 3rd Century BCE, Archimedes used a Greek system, which was similar to the Roman one but with a highest-valued symbol M for 10,000 (one myriad). The system was extended by adding other digits on top of one myriad, as a multiplication method such as Mkծ, where k was the symbol for twenty and ծ was the symbol for 4. To reach universality, all that was required was to generate multi-tier numerals such that M would mean twenty-four myriad myriad, and therefore no limit on numbers would apply. Archimedes’ system used powers of a myriad myriad. But one limiting factor was that he required the exponent to be an existing number; that is to say, it could not easily exceed a myriad myriad, thus petering out at the number we call 10 to the power of 800,000,000. Had Archimedes been willing to allow his rules to be applied without arbitrary limits, he could have invented a much better system just by removing the arbitrary limits from the existing Greek system. One reason Archimedes – who twice extended the Greek system – may not have extended the numeral system indefinitely, was that he, and ancient Greek culture may not have the concept of an abstract (more on this in a future blog post) number at all; that is, in their view, numbers could only refer to objects. In short, he may not have wanted the system to jump to universality.
We may speculate that to appreciate universality at the time of its discovery, you must either value abstract knowledge for its own sake or expect it to yield unforeseeable benefits. And that would’ve been unnatural in societies that experienced change. It is only with the enlightenment that such norms change. The central idea being that progress is both desirable attainable, and by extension, universality. In parallel, universal moral innovations such as justice, legitimacy took hold. And universality was being sought, in its own right.
A significant move to universality occurred when movable-type printing was invented. It consisted of individual pieces of metal, each embossed with one letter of the alphabet. This was a jump because one could easily merely arrange the type into words and sentences. One does not have to know, in order to manufacture type, what the documents will say: it is universal.
Similarly, movable type was invented in China in the 11th century. This could be because of a lack of interest or because the Chinese writing system used thousands of pictograms which decreased the immediate advantages of such a universal printing system.
What are the characteristics or patterns inherent in universal systems? They are customisable or programmable, when compared with their manual predecessors. This enables n combinations and leads to a scalable system.
The Significance of Computation
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. At the time, these were performed by ‘computers’ – the term for armies of clerks – who 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.
The Lost Century
Babbage’s design was sound, except a few trivial mistakes and in 1991 a team led by the engineer Doron Swade at London’s Science Museum successfully implemented it [the Difference Engine], using engineering tolerances achievable in Babbage’s time.
Yes, the Difference Engine had a limited a repertoire; it’s able to exist because of a regularity among all the mathematical functions that occur in physics, and hence in navigation and engineering. They are analytical functions. This could be described as part of the self-similarity properties of nature. 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, thus proving its universality. 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 analytical functions. It also made use 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 comes close to it, in terms of its physical constitution – not its computations per se. To program it to prior out a given table, one initialises certain cogs. In doing so, Babbage realised that this programming phase could itself be automated: the settings could be prepared on punched cards like the Jacquard weaving machine and transformed mechanically into the cogs. Thus, reducing errors and increasing the repertoire of the machine.
Babbage then realised that if the machine could also punch new cards for its own later use, and could control which punched card it would read next, then something qualitatively new would happen: the jump to universality. Babbage called this machine the Analytical Machine, and his colleague, Ada Lovelace, knew that it would be capable of computing anything that human computers could, spanning algebra, play chess, compose music, process images etc. Thus, it could be thought of as a universal classic computer.
It [the universal classical computer] would be:
capable of making scientific predictions;
a universal simulator – able to predict the behaviour, to any desired accuracy of any physical object, given the relevant laws of physics.
This is part of the ‘self-similarity’ of nature whereby objects that are unlike each other share the same mathematical relationships.
Unfortunately, their enthusiasm didn’t extend beyond them. Thus, making it another might-have-been of history. Had they looked around: it could’ve been used for electrical relays (switches controlled by electric currents) which were about to be mass produced for the technological revolution of telegraphy. A re-designed 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.
So, the computer revolution could have occurred 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. Babbage and Lovelace also thought about one application of universal computers that has not been achieved to this day, namely Artificial Intelligence. 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, at the time, they denied that it could. They argued that ‘the Analytical Engine has no pretentions whatever to original thinking. 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.’ Alan Turing called this Lady Lovelace’s objection. In short, they failed to appreciate the universality of the laws of physics. Lovelace’s mistake was their mistaken premise that low-level computations steps cannot possibly add up to a higher-level ‘I’ that affects anything; thereby concluding that AI is impossible.
In sum, because of Babbage’s failure to build a universal computer or to persuade others to do so, an entire century would pass before the first one was built.
History of Modern Computing
In 1936, Turing developed the definitive theory of universal classical computers. His motivation was to use the theory abstractly to study the nature of mathematical proof.
The classic computer was built a few years later during the second World War for specific wartime applications. The British computers, named Colossus (in which Turning was involved) were used for code-breaking; the American one, ENIAC, was designed to solve the equations needed for aiming large guns; the German one was a programmable calculator built, interestingly, out of relays, just as Babbage should have done. All three had the technological features necessary to be a universal computer, but none of them was quite configured for this. ENIAC was the only one allowed to jump to universality. After the war, it was put to diverse uses for which it had never been designed i.e. weather forecasting and the hydrogen-bomb project.
In 1970, a jump in universality occurred: in the post-war era, technology became miniaturised, with ever more microscopic switches being implemented in each device. From then on, designers of any information-processing device could start with a micro-processor and then customise it – program it – to perform the specific tasks needed for that device.
Modern super-computers, washing machines and phones all have the same thing in common: they all have an identical repertoire of computations. Another common trait: 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.
From all of this, it’s worth explicitly stating one fact: all computers are digital. There is no such thing as an analogue computer. That is because of the need for error correction. Without error correction, all information-processing is bounded. Error correction is the beginning of infinity.
Why can there be no analogue computers? Simply because in an analogue system, errors accumulate until the information is no longer useful. What is needed is a system that takes for granted that errors will occur – but corrects them once they do, at the lowest level of emergence. But in analogue computation, error correction runs into the basic problem that there is no way of distinguishing an erroneous value from a correct one at sight because it is in the very nature of analogue computation that every value could be correct. That is not so in a computation that confines itself to whole numbers. So, all universal computers are digital. For example, Babbage’s computers assigned only ten different meanings to the whole continuum of angles at which a cog wheel might be oriented. Making the representation digital in that way allowed the cogs to carry out error-correct automatically after each step. Any slight drift in the orientation of the wheel away from its ten ideal positions would immediately be corrected back to the nearest one as it clicked into place.
The limitation that the information being processed must be digital does not take away from the universality of digital computers – or the laws of physics. The laws of physics are such that the behaviour of any physical object – and that includes any computer – can be simulated with any desired accuracy by a universal digital computer. It is a just a matter of approximating continuously variable quantities by a sufficiently fine grid of discrete ones. Because of the necessity of error correction, all jumps to universality occur in digital systems. Another major fact to note: all jumps to universality occurred on Earth, under the auspices of human beings, except one: it happened during the early evolution of life (I will elaborate on this below).
Evolution and Universality
Genes in present-day organisms replicate themselves by a very complicated and very indirect chemical route. In most species they act as templates for forming stretches of a similar molecule, RNA. Those then act as programs which direct the synthesis of the body’s constituent chemicals, especially enzymes, which are catalysts. A catalyst is a kind of constructor – promoting a change amongst other chemicals whilst remaining unchanged itself. Those catalysts in turn control all the chemical production and regulatory functions in an organism, and hence define the organism itself, crucially including a process that makes a copy of the DNA – the replicator.
Evolution produced replicators that caused themselves to be replicated ever faster and more reliably. The most successful replicators may have been RNA molecules. They may have had catalytic properties of their own, depending on the precise sequence of their constituent molecules (or bases, which are similar to those of DNA). As a result, the replication process became ever less like straightforward catalysis and ever more like programming – in a language, or genetic code, that used it [bases] as its alphabet.
Genes are replicators that can be interpreted as instructions in a genetic code. Genomes are groups of genes that are dependent on each other for replication. The process of copying a gene is called a living organism. Thus, the genetic code is also a language for specifying organisms. At some point, the system switched to replicators made of DNA, which more stable than RNA and therefore more suitable for storing large amounts of information.
Initially, the genetic code and the mechanism that interpreted it were both evolving along with everything else in the organisms. But there came a time when the code stopped evolving yet the organisms continued to do so. At the moment, the system was coding for nothing more complex than primitive, single-celled creatures.
Virtually all subsequent organisms on Earth have not only been based on DNA replicators but have used exactly the same alphabet of bases, grouped into three-base ‘words’, with only small variations in the meanings of those ‘words’.
As a language for specific organisms, the genetic code has displayed phenomenal reach. It evolved only to specify organisms with no nervous systems, no ability to move or exert forces, no internal organs and no sense organs, whose lifestyle consisted of little more than synthesising their own structural constituents and then dividing into two. And yet the same language today specifies the hardware and software for countless multi-cellular behaviours that had no close analogue in those organisms, such as running and flying and breathing and mating and recognising predators and prey, and nano-technology such as immune systems, and even, a brain that is capable of explaining quasars, designing other organisms from scratch, and wondering why it exists.
During the entire revolution of the genetic code, it was displaying far less reach. It may be that each successive variant of it was used to specify only a few species that were very similar to each other. One hypothesis Deutsch proposes is that it must have been a frequent occurrence that a species embodying new knowledge was specified in a new variant of the genetic code. But then the evolution stopped, at a point when it had already attained enormous reach. This was a jump to universality.
Interestingly it was not used to make anything other than bacteria for over a billion years after the system had reach universality. It is not known why this is the case. Is the genetic code a universal constructor? Possibly. It can also be used to program organisms to perform constructions outside their bodies: birds build nests; beavers build dams for example.
In 1994 the computer scientist and molecular biologist Leonard Adleman designed and built a computer composed of DNA together with some simple enzymes and demonstrated that it was capable of performing some sophisticated computations. And it was clear that a universal classical computer could be made in a similar way. Hence, we know that wherever that, whatever that other universality of the DNA system was, the universality of computation had also been inherent in it for billions of years – without being used until Adleman did.
The mysterious universality of DNA as a constructor may have been the first universality to exist. Of all the different forms of universality, the most significant physically, is the characteristic universality of people, merely that they are universal explainers, which makes them universal constructors as well. It is the only kind of universality capable of transcending its parochial origins: universal computers cannot really be universal unless there are people present to provide energy and maintenance – indefinitely. And the same is true of other technologies. Even life on Earth will eventually be distinguished unless people decide otherwise. Only people can rely on themselves into the unbounded future.
To sum up:
All knowledge growth is by incremental improvement, but in many fields there comes a point when one of the incremental improvements in a system of technology or knowledge causes a sudden increase in reach, making it a universal system in the relevant domain.
Since the Enlightenment, humans have been seeking universal jumps and explanations, and valued them for their own sake as well as for their utility.
Because error-correction is essential in processes of potentially unlimited length, the jump to universality only ever happens in digital systems.