The earliest writing systems used stylized pictures – ‘pictograms’ – to represent words or concepts. So a symbol like ‘’ might stand for ‘sun’, and ‘’ for ‘tree’. But no system ever came close to having a pictogram for every word in its spoken language. Why not?
Originally, there was no intention to do so. Writing was for specialized applications such as inventories and tax records. Later, new applications would require larger vocabularies, but by then scribes would increasingly have found it easier to add new rules to their writing system rather than new pictograms. For example, in some systems, if a word sounded like two or more other words in sequence, it could be represented by the pictograms for those words. If English were written in pictograms, that would allow us to write the word ‘treason’ as ‘’. This would not represent the sound of the word precisely (nor does its actual spelling, for that matter), but it would approximate it well enough for any reader who spoke the language and was aware of the rule.
Following that innovation, there would have been less incentive to coin new pictograms – say ‘’ for ‘treason’. Coining one would always have been tedious, not so much because designing memorable pictograms is hard – though it is – but because, before one could use it, one would somehow have to inform all intended readers of its meaning. That is hard to do: if it had been easy, there would have been much less need for writing in the first place. In cases where the rule could be applied instead, it was more efficient: any scribe could write ‘’ and be understood even by a reader who had never seen the word written before.
However, the rule could not be applied in all cases: it could not represent any new single-syllable words, nor many other words. It seems clumsy and inadequate compared to modern writing systems. Yet there was already something significant about it which no purely pictographic system could achieve: it brought words into the writing system that no one had explicitly added. That means that it had reach. And reach always has an explanation. Just as in science a simple formula may summarize a mass of facts, so a simple, easily remembered rule can bring many additional words into a writing system, but only if it reflects an underlying regularity. The regularity in this case is that all the words in any given language are built out of only a few dozen ‘elementary sounds’, with each language using a different set chosen from the enormous range of sounds that the human voice can produce. Why? I shall come to that below.
As the rules of a writing system were improved, a significant threshold could be crossed: the system could become universal for that language – capable of representing every word in it. For example, consider the following variant of the rule that I have just described: instead of building words out of other words, build them out of the initial sounds of other words. So, if English were written in pictograms, the new rule would allow ‘treason’ to be spelled with the pictograms for ‘Tent’, ‘Rock’, ‘EAgle’, ‘Zebra’, ‘Nose’. That tiny change in the rules would make the system universal. It is thought that the earliest alphabets evolved from rules like that.
Universality achieved through rules has a different character from that of a completed list (such as the hypothetical complete set of pictograms). One difference is that the rules can be much simpler than the list. The individual symbols can be simpler too, because there are fewer of them. But there is more to it than that. 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, for instance, contains knowledge of what words sound like. That allows it to be used by a foreigner to learn to speak the language, while pictograms could at most be used to learn to write it. Rules can also accommodate inflections such as prefixes and suffixes without adding complexity to the writing system, thus allowing written texts to encode more of the grammar of sentences. Also, a writing system based on an alphabet can cover not only every word but every possible word in its language, so that words that have yet to be coined already have a place in it. Then, instead of each new word temporarily breaking the system, the system can itself be used to coin new words, in an easy and decentralized way.
Or, at least, it could have been. It would be nice to think that the unknown scribe who created the first alphabet knew that he was making one of the greatest discoveries of all time. But he may not have. If he did, he certainly failed to pass his enthusiasm on to many others. For, in the event, the power of universality that I have just described was rarely used in ancient times, even when it was available. Although pictographic writing systems were invented in many societies, and universal alphabets did sometimes evolve from them in the way I have just described, the ‘obvious’ next step – namely to use the alphabet universally and to drop the pictograms – was almost never taken. Alphabets were confined to special purposes such as writing rare words or transliterating foreign names. Some historians believe that the idea of an alphabet-based writing system was conceived only once in human history – by some unknown predecessors of the Phoenicians, who then spread it throughout the Mediterranean – so that every alphabet-based writing system that has ever existed is either descended from or inspired by that Phoenician one. But even the Phoenician system had no vowels, which diminished some of the advantages I have mentioned. The Greeks added vowels.
It is sometimes suggested that scribes deliberately limited the use of alphabets for fear that their livelihoods would be threatened by a system that was too easy to learn. But perhaps that is forcing too modern an interpretation on them. I suspect that neither the opportunities nor the pitfalls of universality ever occurred to anyone until much later in history. Those ancient innovators only ever cared about the specific problems they were confronting – to write particular words – and, in order to do that, one of them invented a rule that happened to be universal. Such an attitude may seem implausibly parochial. But things were parochial in those days.
And indeed it seems to be 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 at all. A small change in a system to meet a parochial purpose just happened to make the system universal as well. This is the jump to universality.
Just as writing dates back to the dawn of civilization, so do numerals. Mathematicians nowadays distinguish between numbers, which are abstract entities, and numerals, which are physical symbols that represent numbers; but numerals were discovered first. They evolved from ‘tally marks’ ( . . .) or tokens such as stones, which had been used since prehistoric times to keep track of discrete entities such as animals or days. If one made a mark for each goat released from a pen, and later crossed one out for each goat that returned, then one would have retrieved all the goats when one had crossed out all the marks.
That is a universal system of tallying. But, like levels of emergence, there is a hierarchy of universality. The next level above tallying is counting, which involves numerals. When tallying goats one is merely thinking ‘another, and another, and another’; but when counting them one is thinking ‘forty, forty-one, forty-two . . . ’
It is only with hindsight that we can regard tally marks as a system of numerals, known as the ‘unary’ system. As such, it is an impractical system. For instance, even the simplest operations on numbers represented by tally marks, such as comparing them, doing arithmetic, and even just copying them, involves repeating the entire tallying process. If you had forty goats, and sold twenty, and had tally-mark records of both those numbers, you would still have to perform twenty individual deletion operations to bring your record up to date. Similarly, checking whether two fairly close numerals were the same would involve tallying them against each other. So people began to improve the system. The earliest improvement may have been simply to group the tally marks – for instance, writing instead of . This made arithmetic and comparison easier, since one could tally whole groups and see at a glance that is different from Later, such groups were themselves represented by shorthand symbols: the anci
ent Roman system used symbols like , and to represent one, five, ten, fifty, one hundred, five hundred, and one thousand. (So they were not quite the same as the ‘Roman numerals’ we use today.)
So this was another story of incremental improvements intended to solve specific, parochial problems. And, again, it seems that no one aspired to anything more. Even though adding simple rules could make the system much more powerful, and even though the Romans did occasionally add some such rules, they did this without ever aiming for, or achieving, universality. For some centuries, the rules of their system were:
– Placing symbols side by side means adding them together. (This rule was inherited from the tally-mark system.)
– 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.
(The subtractive rule in today’s ‘Roman numerals’, where IV represents four, was introduced later.) The second and third rules ensure that each number has only one representation, which makes comparison much easier. Without them, XIXIXIXIXIX and VXVXVXVXV would both be valid numerals, and one could not tell at a glance that they represent the same number.
By exploiting the universal laws of addition, those rules gave the system some important reach beyond tallying – such as the ability to perform arithmetic. For example, consider the numbers seven (VII) and eight (VIII). The rules say that placing them side by side – VIIVIII – is the same as adding them. Then they tell us to rearrange the symbols in order of decreasing value: VVIIIII. Then they tell us to replace the two V’s by X, and the five I’s by V. The result is XV, which is the representation of fifteen. Something new has happened here, which is more than just a matter of shorthand: an abstract truth has been discovered, and proved, about seven, eight and fifteen without anyone having counted or tallied anything. Numbers have been manipulated in their own right, via their numerals.
I mean it literally when I say that it was the system of numerals that performed arithmetic. The human users of the system did of course physically enact those transformations. But to do that, they first had to encode the system’s rules somewhere in their brains, and then they had to execute them as a computer executes its program. And it is the program that instructs its computer what to do, not vice versa. Hence the process that we call ‘using Roman numerals to do arithmetic’ also consists of the Roman-numeral system using us to do arithmetic.
It was only by causing people to do this that the Roman-numeral system survived – that is to say, caused itself to be copied from generation to generation of Romans: they found it useful, so they passed it on to their offspring. As I have said, knowledge is information which, when it is physically embodied in a suitable environment, tends to cause itself to remain so.
To speak of the Roman-numeral system as controlling us in order to get itself replicated and preserved may sound like relegating humans to the status of slaves. But that would be a misconception. People consist of abstract information, including the distinctive ideas, theories, intentions, feelings and other states of mind that characterize an ‘I’. To object to being ‘controlled’ by Roman numerals when we find them helpful is like protesting at being controlled by one’s own intentions. By that argument, it is slavery to escape from slavery. But in fact when I obey the program that constitutes me (or when I obey the laws of physics), ‘obey’ means something different from what a slave does. The two meanings explain events at different levels of emergence.
Contrary to what is sometimes said, there were also fairly efficient ways of multiplying and dividing Roman numerals. So a ship with XX crates, each containing jars in a V-by-VII grid, could be known to hold CC jars altogether without anyone having performed the lengthy count that was implicit in that numeral. And one could tell at a glance that CC was less than CCI. Thus, manipulating numbers independently of tallying or counting opened up applications such as calculating prices, wages, taxes, interest rates and so on. It was also a conceptual advance that opened the door to future progress. However, in regard to these more sophisticated applications, the system was not universal. Since there was no higher-valued symbol than (one thousand), the numerals from two thousand onwards all began with a string of ’s, which therefore became nothing more than tally marks for thousands. The more of them there were in a numeral, the more one would have to fall back on tallying (examining many instances of the symbol one by one) in order to do arithmetic.
Just as one could upgrade the vocabulary of an ancient writing system by adding pictograms, so one could add symbols to a system of numerals to increase its range. And this was done. But the resulting system would still always have a highest-valued symbol, and hence would not be universal for doing arithmetic without tallying.
The only way to emancipate arithmetic from tallying is with rules of universal reach. As with alphabets, a small set of basic rules and symbols is sufficient. The universal system in general use today 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. For instance, the digit 2 means two when written by itself, but means two hundred in the numeral 204. Such ‘positional’ systems require ‘placeholders’, such as the digit 0 in 204, whose only function is to place the 2 into the position where it means two hundred.
This system originated in India, but it is not known when. It might have been as late as the ninth century, since before that only a few ambiguous documents seem to show it in use. At any rate, its tremendous potential in science, mathematics, engineering and trade was not widely realized. At approximately that time it was embraced by Arab scholars, yet was not generally used in the Arab world until a thousand years later. This curious lack of enthusiasm for universality was repeated in medieval Europe: a few scholars adopted Indian numerals from the Arabs in the tenth century (resulting in the misnomer ‘Arabic numerals’), but again these numerals did not come into everyday use for centuries.
As early as 1900 BCE the ancient Babylonians had invented what was in effect a universal system of numerals, but they too may not have cared about its universality – nor even been aware of it. It was a positional system, but very cumbersome compared with the Indian one. It had 59 ‘digits’, each of which was itself written as a numeral in a Roman-numeral-like system. So using it for arithmetic with numbers occurring in everyday life was actually more complicated than using Roman numerals. It also had no symbol for zero, so it used spaces as placeholders. It had no way of representing trailing zeros, and no equivalent of the decimal point (as if, in our system, the numbers 200, 20, 2, 0.2 and so on were all written as 2, and were distinguished only by context). All this suggests that universality was not the system’s main design objective, and that it was not greatly valued when it was achieved.
Perhaps an insight into this recurring oddity is provided by a remarkable episode in the third century BCE involving the ancient Greek scientist and mathematician Archimedes. His research in astronomy and pure mathematics led him to a need to do arithmetic with some rather large numbers, so he had to invent his own system of numerals. His starting point was a Greek system with which he was familiar, similar to the Roman one but with a highest-valued symbol M for 10,000 (one myriad). The range of the system had already been extended with the rule that digits written above an M would be multiplied by a myriad. For instance, the symbol for twenty was κ and the symbol for four was δ, so they could write twenty-four myriad (240,000) as .
If only they had allowed that rule to generate multi-tier numerals, so that would mean twenty-four myriad myriad, the system would have been universal. But apparently they never did. Even more surprisingly, nor did Archimedes. His system used a different idea, similar to modern ‘scientific notation’ (in which, say, two million is written 2×106), except that instead of powers of ten it used powers of a myriad myriad. But, again, he then required the exponent (the power to which the myriad myriad was raised) to be an existing Gr
eek numeral – that is to say, it could not easily exceed a myriad myriad or so. Hence this construction petered out after the number that we call 10800,000,000. If only he had not imposed that additional rule, he would have had a universal system, albeit an unnecessarily awkward one.
Even today, only mathematicians ever need numbers above 10800,000,000, and only rarely at that. But that cannot be why Archimedes imposed the restriction, for he did not stop there. Exploring the concept of numbers further, he set up yet another extension, this time amounting to an even more unwieldy system with base 10800,000,000. Yet, once again, he allowed this number to be raised only to powers not exceeding 800,000,000, thus imposing an arbitrary limit somewhere in excess of 106.4×1017.
Why? Today it seems very perverse of Archimedes to have placed limits on which symbols could be used at which positions in his numerals. There is no mathematical justification for them. But, if Archimedes had been willing to allow his rules to be applied without arbitrary limits, he could have invented a much better universal system just by removing the arbitrary limits from the existing Greek system. A few years later the mathematician Apollonius invented yet another system of numerals which fell short of universality for the same reason. It is as though everyone in the ancient world was avoiding universality on purpose.
The mathematician Pierre Simon Laplace (1749–1827) wrote, of the Indian system, ‘We shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.’ But was this really something that escaped them, or something that they chose to steer clear of? Archimedes must have been aware that his method of extending a number system – which he used twice in succession – could be continued indefinitely. But perhaps he doubted that the resulting numerals would refer to anything about which one could validly reason. Indeed, one motivation for that whole project was to contradict the idea – which was a truism at the time – that the grains of sand on a beach could literally not be numbered. So he used his system to calculate the number of grains of sand that would be needed to fill the entire celestial sphere. This suggests that he, and ancient Greek culture in general, may not have had the concept of an abstract number at all, so that, for them, numerals could refer only to objects – if only objects of the imagination. In that case universality would have been a difficult property to grasp, let alone to aspire to. Or maybe he merely felt that he had to avoid aspiring to infinite reach in order to make a convincing case. At any rate, although from our perspective Archimedes’ system repeatedly ‘tried’ to jump to universality, he apparently did not want it to.
The Beginning of Infinity Page 16