ARE THE REAL NUMBERS
|position in the sequence. (||3
|is more than||2
|and less than||5
numbers have customary meaning of the word number.
Complex or imaginary numbers do not have that meaning, because we cannot place them with respect to order. They are algebraic entities. They obey the laws of computation, and they can be solutions to an equation.
Irrational numbers can also have that customary meaning. Apart from unique irrationals such as π and e, the names and definitions of the irrationals come from the categories of functions: roots, sines, arcsines, logarithms, and so on.
But we cannot determine the order of irrational numbers from their
|names. Is more than or less tan||5π
|? 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.|
It should not be unwarranted to say that if a theory of irrationals does not satisfy those conditions, then it lacks a firm foundation.
Every irrational number, then -- by which we mean every one that exists -- will 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. That distance will be the x-coördinate of the point P.
Now 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 point on the x-axis -- every distance from 0 -- there should correspond a number.
Can the concept of a continuum be applied to numbers? Or is that merely an idea, a figment of the imagination? For if the word number is to retain its customary meaning, then such a continuum is impossible. Numbers -- the values of x -- have names. But we cannot name every element in a continuum. Names are discrete. A continuum of names is an absurdity.
There is no arithmetical continuum.
(That simple argument is called 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, motion -- does 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.
Yet it is now claimed that the real numbers are a continuum; that they are like the points on a line, and corresponding to every point there is a real number. To claim that, the word number had to be given a completely different meaning. 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 name or symbol cannot obey laws of computation. And they cannot be solutions to an equation. They are called "numbers" only to be able to say that to every point on the x-axis there corresponds a "number." They are the ones 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. What looks like this --
.24059165378. . .
-- will satisfy the definition of real numbers.
To actually construct a decimal expansion, of course, there must be an algorithm. But if there are to be algorithms for computing a continuum of real numbers, then there must be a continuum of algorithms -- which again is absurd. Algorithms are discrete. If it is not possible, then, to compute each next digit of what appears as
.24059165378. . . ,
then it does not signify a limit. What is more, we cannot place it with respect to order. It is nothing but a sequence of made-up digits followed by three dots. And since it has no name, it cannot enter into computation. It is not possible to name the sum of such "infinite decimals," we cannot name their difference, their product, or their quotient. Such a symbol is not the symbol of a number.
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.
In other words, algorithms are discrete. And since mathematics is done more and more on computers these days, then for all practical purposes that's the end of it. No algorithm. No irrational number.
Is a line really composed of points?
This whole problem of a continuum of numbers is premised on the assumption, the concept, that a line is composed of points. Is that necessary? Or is "point" simply the word we use to call attention to a specific place, such as the boundary of an interval or where two lines meet? Points are individuals, and we indicate them one at a time. That is what we do, and that is all we need to mean when we say that that point exists.
(We may say there are an "infinite" number of points on a line, which is a brief way of saying that there is no limit to the number of points we could indicate.)
Points -- like pitches on a violin string -- exist potentially. The pitch of a string does not exist until it is sounded: a violin string is not composed of pitches. And a straight line, such as the x-axis, is not composed of points.
But say it were. Now the most common and important application of calculus is to motion, where the independent variable is time t. Then if the abstract x-axis is composed of points, its application to time must also be composed of points. That is, time -- the t-axis -- will be composed of points, or, we would say, instants. Is that a valid presumption?
First, like any continuous quantity, time is not inherently composed of intervals., yet we can conveniently decompose time into any intervals,
any units of measure -- hours, minutes, seconds -- we please, however small.
Time will then be composed of those intervals, which will then have common boundaries, to which we give the name "instants."
Those instants will be the values of t. "Let t = 10 seconds." But time cannot be composed of those instants, because instants are not intervals. They are not units of time. They cannot be further divided as required by the definition of a continuum.
If time had components and they were instants, then at every instant there would be no change of time, no change of position, which means: at every instant there is no motion. That is the arrow paradox of Zeno. But because time continues and has no inherent components -- it is not composed of instants -- that paradox is not valid.
Since the t-axis then is not composed of points, then neither can the x-axis -- the so-called real line -- of which time is but an application.
To accept that an infinite number of points of zero length can add up to a positive length, not only calls for belief more typical of the demands of religion; it approves division by 0.
For the ambiguous semantics of the word point, see Lesson 1.
Why the obsession with an infinity of points and a continuum of numbers? Again, it 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 our experience requires that. When we let a variable approach a limit or do a calculation, we name a number. That is all anyone has ever done or will ever need to. A limit is a number that we name.
By expressions such as "all" values -- or "all" anything -- then, it is sufficient to mean all that we name or produce. We have the clearest model for that in Euclid's Elements, in which statements with the word "all" or "every" -- such as "All right angles are equal" -- refer to all that exist; that is, all we have actually drawn.
When a number exists
We have the decimal system of numeration. That is what enables us to name any number. That is our instrument of construction.
A number -- that individual -- will exist, we assert, at the moment it is named. It will exist in the most actual sense of the word. Naming is how we produce a number, how we bring a verbal or written symbol for that idea into this world. For if we cannot bring an idea into this world, then it is nothing but an idea, which is to say, a fantasy.
("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.)
Opposing that is the idea of an actual infinity -- an infinite list of names and numerals -- every one of which is imagined to exist now. "The natural numbers." "The real numbers." One might ask where such collections exist (In some transcendent world?); but let that go. Those are concepts abstracted from individuals, and they exist only as names of kinds of numbers. They are instances of a doctrine that has no consequences and is unnecessary. Calculus has need only of numbers we name and bring into this world.
It is pointless to do with more what can be done with less.
At first, the infinite was a philosophical and religious concept. As for the use of that word in calculus, it is completely different from its definition in the theory of infinite sets. Here is how the "Prince 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, to be of any value, then 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 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. The theory of real numbers belongs to 19th century modernism, a movement which sought "freedom" from the values of the past and 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 © 2014 Lawrence Spector