ARE THE REAL NUMBERS
|position in the sequence. (||3
|is more than||2
|and less than||5
Rational numbers have the customary meaning.
Irrational numbers 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 specify. For it is the rational numbers whose order we know. Is the irrational number less than or greater than 2.71828103594612074?
1.414213562373095 < < 1.414213562373096.
There is a procedure that enables us to calculate as many digits as we please of . Therefore we can place it with respect to order. That guarantees that it is a number. For if it did not have that customary meaning, it would be of no use to calculus, geometry, or science.
We say, then, that the sentence "This is an irrational number" means:
|1)||This irrational number has a name; and|
|2)||we can decide whether it is less than or greater than any rational number we specify.|
Every irrational number, thenby which we mean every one that existswill thus have the customary meaning of the word number.
A continuum of numbers?
The concept of a continuum comes from geometry. A line is the classic example. It is a continuum of length. There is no limit to the smallness of the difference between two lengths.
The job of arithmetic when faced with what is continuous is to come up with the name of a number to be its measure, relative to a unit of measure.
In coördinate geometry, we measure length as the distance from 0 along the x-axis. And since length is continuous, it was thought that the values of x should reflect that by being a continuum of numbers. That is, to every distance from 0every point on the x-axisthere should correspond a real number. Since there is no limit to the smallness of the distance between two points, there should be no limit to the smallness of the difference beween two real numbers.
But if the word number is to have its customary and therefore intelligible meaning, then it will not be possible to name every real numberand a name is what gives a number its meaning. Differences between names are not arbitrarily small. A continuum of names is an absurdity.
There is no continuum of numbers.
(That simple argument is the semantic rejection.)
Thus it will not be possible to assign a number—to measure—the length of every line. Equivalently, it will not be possible to name the endpoint of every distance from 0. Time, distance, motion: they are continuous. Numbers 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. A continuum of the numbers we need for measuring and that corresponds to an actual continuumtime, distance, motiondoes not exist.
Whether it is even necessary is another question.
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.
The term real number was coined by René Descartes in 1637. It was to distinguish it from an imaginary or complex number. Now, we can define a rational number, and they exist. An irrational number can be defined (not rational), and they will exist (). It is perfectly clear, then, when by a real number we mean any rational number or any irrational number that exists. The word number has its customary meaning. And they will not form a continuum.
To claim they do, the word number had to be given a completely different meaning. What distinguishes that meaning is that, in addition to the customary—nameable—rational and irrationals, there are now "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 its associated symbol cannot obey laws of computation. And they cannot be solutions to an equation. They are the "numbers" that truly deserve to be called imaginary.
There is a method, an algorithm, that allows us to construct as many decimal digits as we please for the irrational number π:
π = 3.141592653589793. . .
The symbol on the right is called an infinite decimal. It represents 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, and especially an irrational, could be symbolized by an infinite decimal.
To actually construct a decimal expansion, of course, there must be an algorithm, a rule. But if there are to be rules for computing a continuum of numbers, then there must be a continuum of rulesthe differences between which will be arbitrarily small. Which is absurd. Rules are discrete.
If it is not possible then to compute each next digit of what might appear as
.24059165378. . . ,
then it does not signify a limit. What is more, we cannot place it with respect to order. It is not the symbol of a number.
In fact, the English mathematician and father of computer science 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.
The obsession with a continuum of numbers seemed to be demanded by coördinate geometry. The x-axis was conceived to be composed of points, and to "every" point there must correspond a number that is its coördinate. But nothing in the actual practice of calculus requires that. When we let a variable approach a limit or when we evaluate a function, we name numbers. And we name them one at a time.
Thus if we say that the domain of f(x) = x2 is
−∞ < x < ∞,
we mean that x may take any rational or irrational value we name.
"Let x = 5." "Let x = −1." "Let x = ."
Now, what is it that enables us to name a number? Is there some infinite list of names and numerals from which we choose? "The set of natural numbers." "The set of rational numbers." "The set of real numbers." Those are names of kinds of numbers. They are ideas abstracted from individuals. They exist only as names. Rather, what enables us to name a number is the decimal system of positional numeration, together with the names of functions.
"Square root of 238,096,608,009,650,000,412."
Upon naming the number, we may then say that it exists. The number will be at the moment we directly experience it, which is the moment we say, write, read, or hear its name. Not because those names pre-exist in some transcendental world.
As for the use of the word infinite in calculus, it is completely different from its definition in the theory of infinite sets. Here is how "the foremost of mathematicians," Carl Friedrich Gauss, put it: "I protest against the use of an infinite quantity as something completed; this is never allowed in mathematics. The infinite is only a manner of speaking, the true meaning being a limit to which certain ratios can come as near as desired, while others are permitted to increase without bound."
To summarize: If numbers are to be useful, then they must have names. But if every number had a name, they could not be elements in a continuum, because names are discrete. That is the inherent contradiction in the project to create a continuum of numbers.
As for a "real number," the original definition is perfectly clear and sufficient. It is what we call any rational or irrational number, every one of which has a name. Any definition that defines them so that they form a continuum, completely departs from that original meaning and is not concerned with measuring, which requires names ("5.3872 meters"). That theory of real numbers belongs to 19th century modernism, a movement which sought "freedom" from the values of the past and from what was accessible only to the many. Those real numbers are an abstract creation; a kind of logical sport; and the most prominent current example of fantasy mathematics.
Copyright © 2020 Lawrence Spector