ARE THE REAL NUMBERS
|position in the sequence. (||3
|is more than||2
|and less than||5
Rational numbers have the customary meaning of the word number.
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.
An arithmetical continuum?
The concept of a continuum comes from geometry. A line is the classic example. It is a continuum of length.
The job of arithmetic when confronted 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 number. Since there is no limit to the smallness of the distances between two points, there should be no limit to the smallness of the differences beween two real numbers.
Does that make sense? Or is it a figment of the imagination? For if the word number is to retain its customary meaning, implying both a name and a symbol, then a continuum of numbers is impossible. We cannot name every element in a continuum. Names are discrete. A continuum of namesthe differences between them being arbitrarily smallis an absurdity.
There is no arithmetical continuum.
(That simple argument is the semantic rejection.)
Time, distance, motion 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 continuum time, 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, which is sufficient. 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 that they do, the word number had to be given a completely different meaninghaving nothing to do with measuring. What distinguishes that meaning is that, in addition to the customary 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 or a unique 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. Again that 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.
THIS WHOLE PROBLEM of a continuum of numbers began with the assumption, the concept, that a linethe x-axisis composed of points. But does calculus really require that? A "point"the idea of position onlyis simply the name we give to the extremity of a distance from 0. We indicate points and their coördinates one at a time. That is what we do. And having done that, that is all we need to mean when we say that that point exists.
It is obvious that, in no additive sense, is a line composed of points. To accept that an infinite number of points of zero length will add up to a positive length, calls for credulity more typical of the demands of religion. And it approves division by 0.
Again, the obsession with an infinity of points and a continuum of numbers seemed to be demanded by coördinate geometry. To "every" point on the x-axis there should 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 do a calculation, we name numbers.
We name a number c that corresponds to one point on the x-axis. We name a number L that corresponds to one point on the graph of f(x). (We're supposed to name numbers ε and δ; but we don't.) Calculus deals with numbers by means of their names. A continuum of numbers is never an issue.
It is pointless to do with more what can be done with less.
To summarize: For numbers to be useful in calculus and science, the word number must have its customary meaning. As for a "real number," the original definition is perfectly clear and sufficient. It is what we call any rational or irrational number. Any definition that defines them so that they form a continuum, completely departs from that meaning, and has nothing to do with measuring nor was it ever intended to. 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 © 2016 Lawrence Spector