The Beginning of Infinity: Explanations That Transform the World (22 page)

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 ancient 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.