Appendix 2
ARE THE REAL NUMBERS
REALLY NUMBERS?
The customary meaning of the word number
Rational and irrational numbers
WHEN CONFRONTED WITH WHAT IS CONTINUOUS—distance, time, motion—the job of arithmetic is to come up with the name of a number to be its measure. The term real number was coined by René Descartes in 1637. It was to distinguish it from an imaginary or complex number, and it was comprised of the rational and irrational numbers.
In the 19th century, certain mathematicians thought that a more rigorous definition was required.
In coördinate geometry we measure length as the distance from 0 along the x-axis. That distance will be the x-coördinate of a point P. Now since length is continuous—there is no limit to the smallness of the difference between two lengths—it was thought that the values of x should reflect that by being a continuum of numbers. Can the concept of a continuum be applied to numbers? And is it even necessary?
The customary meaning of the word number
We deal with numbers completely through their names. Consider the natural numbers. Each one has a defined name, a symbol, and a knowable position in the sequence. ("5" comes after "4" and before "6.") Those same properties must belong to whatever we call a number. They clearly belong to each rational number, they must also belong to each irrational number, for they are the numbers we need for measuring. Calculus is a theory of measuring. A limit is a number that we name.
It is the name, symbol, and sequential position of
what we call a number
that allow us to count, calculate, and measure.
That is the customary and expected meaning—use—of the word number.
Rational and irrational numbers
It is clear that every rational number has that meaning. Each one has a defined name, a symbol, and a knowable position in the sequence. (
is more than 
and less than 
.)
An irrational number can also have that meaning. But we cannot determine the order of irrational numbers from their names. Is 
more than or less tan
? The only way to decide is to compare their rational approximations. And that will depend on the existence of a method, an algorithm, to actually produce one.
Specifically, we must be able to decide whether an irrational number is less than or greater than any rational number we name. For it is the rational numbers whose order we know. Is the irrational number less than or greater than 2.71828103594612074?
We say, then, that the sentence "This is an irrational number" means:
| 1) | This number has a name and a symbol, it can therefore obey rules of computation; and |
| 2) | we can decide whether it is less than or greater than any rational number we name. |
Every irrational number, then—by which we mean every one that exists—will have the customary meaning of the word number.
A continuum of numbers?
When numbers have that meaning, they cannot constitute a continuum. For that would mean that there is no limit to the smallness of the difference between them. But each such number has a name—and the difference between names is not arbitrarily small.
To create the present theory of real numbers, the word continuum and the word number had to be given a completely different meaning. As we pointed out in Lesson 1, mathematicians of the 19th century abandoned the classical concept of a continuum, and they began with abstract elements they called "points" (see here), to which they ascribed a primary logical existence. They defined a "continuum," and specifically a "line," as a "set" of those "points." Those abstract points then replaced the geometrical points on the x-axis, which was then called the real line.
The theory continued with geometrical language, however. "Between" any two such points it was postulated that there is always a third. In other words, there is no limit to the smallness of the "distance" between them. To each of those points there was then said to correspond a real "number." Since those points constituted a continuum, so did the real numbers.
The first thing to note is that the word number does not have its customary meaning. There is now no limit to the smallness of the differences between them—there are therefore "numbers" with no names. In fact, the reals will be teeming with nameless "numbers." Otherwise, they could not fill out a continuum.
Such numbers clearly were not intended to be useful. Something without a name and a unique symbol cannot obey rules of computation. And they cannot be a solution to an equation. They are the "numbers" that truly deserve to be called imaginary.
We should state that the theory of real numbers was never concerned with measuring; that is, with calculus or science. The theory was an abstract, logical creation. It belongs to 19th century modernism, a movement which, along with art and music, sought "freedom" from the values of the past and what was easily understood.
A symbol was required to represent the general real number. That symbol became an infinite decimal.
Now there is a method, an algorithm, that allows us to construct as many decimal digits as we please for π:
π = 3.141592653589793. . .
The symbol on the right with three dots (ellipsis) is called an infinite decimal. It signifies this sequence of rational numbers:
3.1, 3.14, 3.141, 3.1415, . . .
π is the limit of that sequence.
Abstracting from that, it was asserted that every real number is symbolized by an infinite decimal.
.24059165378. . .
A decimal is a way of representing a number. And numbers have names. The name of this decimal --
.2405
-- is "Two thousand four hundred five ten-thousandths."
An infinite decimal however has no name. It is not that we will never
.24059165378. . .
finish naming it. We cannot even begin.
To actually construct a decimal expansion, there must be an algorithm, a rule for computing each next digit. But if there are to be algorithms for computing a continuum of numbers, then there must be a continuum of algorithms—the differences between them being arbitrarily small. But that is absurd. Algorithms are discrete.
In fact, the English mathematician and father of artificial intelligence Alan Turing proved the following:
To compute the decimal expansion of a real number, it is possible to create an algorithm for only a countable number of them.
If it is not possible then to compute each next digit of what might appear as
.24059165378. . . ,
then it is nothing but a made up sequence of digits followed by three dots. It does not signify a limit. What is more, in spite of what is postulated we could not know where to place it with respect to order. That alone tells us that the general real number does not deserve the word number.
*
This whole idea of a continuum of numbers followed from the definition of the "real line" as being composed of those abstract "points." But calculus is a theory of measurement and requires geometrical points, which are the extremities of a length. A geometrical point is the word we use to indicate a position. The extremity of a distance from 0 is called a point, as is the boundary of an interval. We indicate points and their coördinates one at a time. ("Let a boundary of this interval be x = 5.") When we let a variable approach a limit or evaluate a function, we name a number. That is all anyone has ever done or will ever need to. Calculus has no need for a continuum of numbers. It is not necessary. Calculus requires numbers with names.
When does a number exist?
When, in mathematics, may we say that something exists? Any thing exists when it is the effect of a cause. A straight line will exist when we draw it. A point will exist as an extremity of the line. And a number, we say, will exist when we have named it. It will exist at the moment we say, write, read, or hear its name.
Like any thought, a number will exist when we experience it.
("Do you mean to say that the number 100 does not exist until I name it?" That is correct, and you have just named it.)
What is it that exists prior to naming a number? It is the decimal system of positional numeration. Together with the names of functions, that is what enables us to name a number.
"Cube root of 238,096.608,404,009,650,000,412."
That is the first time that number has ever existed.
*
Distance, time, motion: They are continuous. The numbers we need to measure what is continuous—to name their relationship to a unit of measure—are not. That is the tension between geometry and arithmetic, a tension realized by Pythagoras with his discovery of what we call the irrational—and he called "without a name" (alogos). That tension was brought to a head with the introduction of coördinate geometry, which has been the dominant methodology since the 17th century, and which of course we take for granted. Geometry is concerned with continuous objects, while the domain of arithmetic is numbers and their discrete names.
It should be no wonder, then, that neither a teacher nor a text can give an example of a variable approaching a limit continuously, but only as a sequence of discrete rational numbers. Why not? Because no such thing exists.
To summarize: For numbers to be useful in calculus and science, they must have names; the word number must have its customary meaning. As for a "real " number, the original definition is perfectly clear and sufficient. It is a number that is not imaginary; it is any rational or irrational number. Any definition that defines them so that they form a continuum, has nothing to do with measuring. Nor was it ever intended to.
And so, "Are the real numbers really numbers?" Inasmuch as the word number does not have its customary meaning, they most definitely are not.
Appendix 3: Is a line really composed of points?
Copyright © 2022 Lawrence Spector