The Beginning of Infinity: Explanations That Transform the World (23 page)

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 2×10
⁶
), 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 Greek 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 10
^800,000,000
. 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 10
^800,000,000
, 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 10
^800,000,000
. 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 10
^6.4×10
17
.

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.

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.