Numbers: Their Tales, Types, and Treasures.
Chapter 11: Numbers and Philosophy
11.11.THE PROBLEM WITH REALLY HUGE NUMBERS
Mathematicians firmly believe that the statement
8,864,759,012 + 7,938,474,326 = 16,803,223,338
is correct in the same sense as 5 + 3 = 8, although it appears to be fairly impossible to verify that claim simply by counting.
On the other hand, numbers as large as these appear in economics, and we even have a name for the result: sixteen billion eight hundred three million two hundred twenty-three thousand three hundred thirty-eight. The Gross Domestic Product of the United States was worth 16.8 trillion US dollars in 2013, and this number is still a thousand times larger than the result above. Obviously, numbers like these are handled without any problems in our culture.
The sequence of number words is built according to a system that at least in principle, has no end. It does have practical limits, however, because we tend to run out of names and symbols for extremely large numbers. According to the common system that is used in the United States, we have names for
a billion…1,000,000,000
a trillion…1.000,000,000,000
a quadrillion…1,000,000,000,000,000
a quintillion…1,000,000,000,000,000,000
and so on, with a new name for every three zeros added to the number. The prefixes bi-, tri-, quadri-, and so on are derived from Latin number words. With this system, we will eventually arrive at a centillion, which would be a 1 followed by 303 zeros. But these numbers play no role in our life and are typically not named at all. Besides, any naming scheme is only a temporal solution, and one can easily construct an example that is outside the range of names, and then the problem of finding appropriate number words would arise again. How would one name a number like a 1 followed by a quintillion zeros?
The scientific notation uses powers of 10 to describe large numbers. For example, a billion would be 109, a trillion 1012, and so on. Here the exponent gives the number of zeros following the leading 1 and describes how often 10 is multiplied by itself in order to produce that number. Hence, for example,
1 billion = 109 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000,000,
and a quintillion would be 1018.
This notation is by far more effective than the attempt to give all numbers an English name. In fact, we can easily write down a number like 101018 = 10(1018), which would be a 1 followed by 1018 (a quintillion) zeros. But the numbers in scientific notation are also only approximate. For most practical purposes, the following number would simply be written as 3 × 1015:
3,000,000,000,219,325 = 3.000000000219325 × 1015.
We see from this that for most numbers, in order to describe them exactly, the scientific notation would not really provide a notational abbreviation. In particular, one could generally not describe a number with a quintillion arbitrary digits with scientific notation, except in an approximate sense.
Among the very large numbers, a few have gained special popularity. In their book Mathematics and the Imagination, Edward Kasner and James Newman describe extremely large numbers and introduce the name “googol” for the number 10100. It is said that this name was invented by Kasner's nine-year-old nephew in about 1920. Later, the name was used in a slightly changed form to name the Internet search engine Google, thus indicating the huge amount of data on the World Wide Web. A googol is indeed unimaginably large, and the total number of particles in the observable universe is usually estimated to be much lower (putting aside the fact that the number of particles is a rather ill-defined quantity). Yet one can easily define, using mathematical notation, even much larger numbers, such as
10google = 10(10100) (which is sometimes called a googolplex)
or even googolplexgoogolplex, which would be a googolplex multiplied by itself a googolplex times.
What about the meaning of huge numbers like googolplex? They have absolutely no use in counting, because nobody can count that far and there are no collections in the observable physical universe that have nearly that many elements. Obviously, we have a precise algorithmic description for some of these huge numbers, like googolgoogol, which is 10100 taken to the power of 10100. But the numbers having a relatively short description are the exception. A “typically huge” number, which in the usual decimal notation would have, say, about a googolplex digits, is not just a 1 followed by the corresponding number of zeros. Rather, the digits 0, 1, 2…9 would follow each other in a fairly random manner, and in general there is no rule or notation or “compression algorithm” that could describe all these typical numbers in a shorter way.
Can we say that such a huge number, for which we have no means to properly write it, exists in any reasonable sense? What would we mean by the word existence, when there is not even a symbolic representation? There is no collection of concrete objects that this number would represent. If there is a number with about googolplex digits, but if the number of digits is unknown, it would be forever beyond the reach of humanity to determine the exact number of its digits. Hence, you could not even distinguish this number from a number that is about a million times as large, because how would you distinguish a number with about googolplex digits from a number with six more digits? You could not do the simplest arithmetic with this number, and of course you could not write it down, because there is not enough matter, space, and time in the universe for that task. Considering the impossibility of realizing such a number, or even describing it exactly, do these large numbers have any meaning? How could we claim that every huge number has a unique successor? Of course, we could just call any huge number by the variable name n, and write “n + 1” for its successor. But (except in a few cases) we do not have exact expressions to substitute for the variable name. Hence one could not describe exactly to which concrete number the letter n would refer. Exact huge numbers, in general, are not represented in this universe, not even in symbolic notation. Therefore, nothing in reality or imagination would correspond to the successor of that number n, because nothing in reality or imagination corresponds exactly to n.
The branch of philosophy of mathematics that would not accept objects or expressions that nobody can construct in any practical sense is called ultrafinitism. According to this view, not even the concept of natural numbers would be accepted without restrictions, and, of course, an ultrafinitist would refuse to talk about infinity. To most mathematicians, this view would be too extreme. Reducing mathematics to finite and not-too-large objects would restrict mathematics and its usefulness in an intolerable way. (And, by the way, how would one define “not too large”?)
Most mathematicians are not particularly worried by the fact that there are natural numbers so huge that they cannot be conceptualized exactly. Typically, when applying numbers to reality, approximate quantities are sufficient, and extremely large numbers would rarely be needed. In theory, the natural numbers are just a sequence whose structure is axiomatically described by the Peano axioms. As a mathematician, one typically does not care about the practical realizability of particular numbers. That every number has a unique successor is simply true by assumption; it needs no practical verification. Mathematicians usually think not in terms of concrete realizations but in terms of rules that are given axiomatically. Mathematics is the art of arguing with some chosen logic and some chosen axioms. As such, it is simply one of the oldest games with symbols and words.
And, moreover, the usefulness of mathematics is by no means limited to finite objects or to those that can be represented with a computer. Mathematical concepts depending on the idea of infinity, like real numbers and differential calculus, are useful models for certain aspects of physical reality.