Appendix
THE MATHEMATICAL EXISTENCE
of
NUMBERS
The existence of irrational numbers
Is a line really composed of points?
THE WORLD IS FULL of things to count. Photons, electrons, atoms, molecules, stars, people. Each one is composed of the same indivisible units, which is what we mean by a natural number.
But until we have created a sequence of names, there is not that man-made activity called mathematics. And upon naming the natural numbers -- without which we could not count -- we can then proceed to the problem of measuring. The following makes clear what a number is.
It is the name and symbol of what we call a number that allows us to count, measure, and calculate. "Five people." "Five axioms." "Five days." "Two-thirds of a cup." "Two-thirds of a meter." "Square root of two meters." "Minus ten degrees."
(Five is a name for how many. Five people, axioms, or days are applications of that name.)
A number for mathematics is a named entity. It is a defined entity, which every student must confront. "Two" versus "Three." "Minus nine-sixteenths." "Fifth root of 10." "log 2." Each name is its individuality and its usefulness. We assert, in fact, that a number will exist for mathematics, upon saying its name.
Mathematical existence
Existence is a concept. We have the clearest criterion for when we may say something "exists" for mathematics in Euclid's Elements, where a figure, such as a circle or a square, will exist only when we have drawn it. Obviously, we have the idea of a square, and according to that idea we frame a definition. But we are able to present the logical steps to draw a square, and that shows it is more than just an idea. The square that exists for mathematics is the square we have actually produced. (See the commentary on the definitions of Euclid's Elements.) As with everything in life that begins as an idea, we must bring it into this world. If we cannot, then it is nothing but an idea, which is to say, a fantasy.
Moreover, 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.
As for numbers, we deal with them completely through their names. We say, then, that a number will exist at the moment we name it, whether in writing, speech, or thought. Naming will be a form of producing it, of bringing a verbal symbol for that idea into this world..
A number, for mathematics, has a potential existence. But it will not have an actual existence until we name it. Which in practice is all that is necessary.
("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 more could your possibly need?)
Expressions such as "all" natural numbers or "all" real numbers, then, will mean all that we name. Opposing that is the idea of an actual infinity of numbers that "exist" all at once, now. But it is only that, an idea, which, unlike the idea of a square, we cannot bring into this world. What is more, it is not necessary. Calculus has no need of what we cannot name or observe. If mathematics is in any sense a science, then what more scientific meaning could we give to saying that something exists? "Fourteen" That exists.
The existence of irrational numbers
A rational number acquires its name from its direct relationship to 1, which is the ground. The whole numbers are the multiples of 1, the fractions are its parts: its halves, thirds, fourths, and so on. An irrational number has no such direct relationship to 1. That is the great problem of the irrational. Language can not support it. Apart from unique irrationals such as π and e, names for the irrationals come from the categories of functions: roots, sines, arcsines, logarithms, and so on. ( See Topic 2 of Precalculus.)
An irrational number will exist, then, not only because it has a name. It must satisfy the essential property of being a number, which is that it we can place it relative to the order of other numbers. We can determine the order of rational numbers directly from their names. 8 is more than 7
| but less than 9. | 3 4 |
is more than | 2 3 |
but less than | 5 6 |
. |
But we cannot determine the order of irrational numbers from their
names. Is  more than or less than tan |
5π 7 |
? 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.
In particular we must be able to decide whether an irrational number is less than or greater than any rational number we name. Is it less than or greater than 2.71828103594612074?
For example,
1.414213562373095 < < 1.414213562373096.
We can place with respect to order in this way. That guarantees its mathematical existence as a number.
We say, then, that the sentence "This irrational number exists" 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 name. |
It should not be unwarranted to say that if a theory of irrationals does not satisfy those conditions, then it lacks a firm foundation.
An arithmetical continuum?
The idea of what is continuous comes from geometry. A straight line is the classic example. It is a continuum of length. When confronted with what is continuous, the job of arithmetic is to come up with 
the name of a number to be its measure. Thus if the side of a square were the unit of measure, then the measure of the diagonal would have the name, "Square root of 2."
Calculus is a theory of measurement. And a measurement implies a relationship, a ratio, to a unit of measure -- the centimeter, the gram, the second -- whose measure is 1. For if we are to measure a length L, say,
then we expect to name a rational or irrational number x such that, proportionally,
L is to the Unit of measure as x is to 1.
In coördinate geometry, we measure length as the distance from the
origin O along the x-axis. The length of OP, then, will be the x-coördinate of the endpoint P. Now, since length is continuous, it seemed to be necessary that the values of x reflect that by being a continuum of numbers, now defined as the real numbers -- a term that has come to mean more than not-imaginary. The project to create such a continuum was called the arithmetisation of geometry.
But will that be possible? Will it be possible to measure every length – to name the ratio that every length will have to a unit of measure?
No. It is impossible to name every element in a continuum – a continuum of names is an absurdity. Names are discrete. And nameless numbers do not exist, not even potentially. There is no arithmetical continuum.
(That simple argument is called the semantic rejection.)
The problem lies with the irrationals, because they are required to name lengths the rationals cannot. But for an irrational to exist, we must be able to place it with respect to order. Therefore there must be an algorithm to compute its decimal expansion. But to suppose there could be algorithms for computing "all" real numbers, irrational and rational, would require a continuum of algorithms, which again is absurd. Algorithms are discrete -- and they require the name of the number we are computing. π, 
, ln 2. 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.
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.
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 of the continuous while arithmetic belongs to what is discrete. A continuum of numbers that corresponds to a continuum of length does not exist.
Infinite decimals?
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
*
"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."
"The question is," said Alice, "whether you can make words mean so many different things."
"The question is," said Humpty Dumpty, "which is to be master -- that’s all."
Alice in Wonderland
The symbol for the idea of an infinite decimal --
.24059165378. . .
-- was invented in the 19th century to serve as the form of every real number. The words continuum and "point," which have had their geometrical meaning since the time of Aristotle, were then given a completely different meaning. A "continuum" was defined to be a set of abstract elements also called "points." That meaning of "point" became unexplainedly linked with the geometrical meaning. And those real numbers, those infinite decimals, became identified with those newly-styled points It was then stated as an axiom, "Corresponding to every point on a line, there is a real number." It could then be claimed that those real numbers -- those infinite decimals -- constitute a "continuum."
A decimal of course 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.
It is not possible to name the sum of infinite decimals, we cannot name their difference, we cannot name their product, and we cannot name their quotient. Infinite decimals do not behave in any way we expect of numbers.
If a student in arithmetic class were to say, "Although I cannot name that sum, teacher, that sum exists; and to know that is sufficient," the student might deserve an A in metaphysics but in arithmetic she would certainly fail.
Finally, in the absence of an algorithm to complete the decimal expansion of .24059165378 . . ., it is nothing but a sequence of made-up digits followed by three dots. It is not the symbol of a number.
So, if infinite decimals do not represent numbers, then what do they represent? They represent those "points" that now composed the geometrical line. Hence the axiom that related them -- "To every point on the line there corresponds a real number, that is, a point on the line" -- is a tautology; a true statement devoid of substance.
What is more, the idea of a point in geometry is completely different from that of a number; at any rate, a number we need for measuring, as opposed to some completely different meaning for that word also; a meaning (or rather a non-meaning) that might appear in logic but has nothing to do with calculus, the theory of measuring.
Infinite decimals, if that is what real numbers are, are not numbers. They represent nameless points on a line. Hence it should come a no surprise that those infinite decimals -- those real "numbers" -- are nameless themselves.
The famous proof that the real numbers are uncountable, is actually a proof that the points on a line are uncountable. Which of course is obvious, if a line were really composed of points.
The rational numbers, on the other hand, are countable. Why? Because, like the natural numbers, they have systematic names.
Is a line really composed of points?
That is the question. Or is "point" simply the concept -- the word -- we employ to call attention to a particular place, such as the boundary of an interval or where two line meet? We indicate points one at a time, and in practice that is all that is necessary. (We may say there are an "infinite" number of points on a line, which is a brief way of saying that no matter how many we indicate, we could always indicate one more.)
Geometrical points -- like pitches on a violin string -- exist only 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. Say that the expression "every point" on a line, or "the number of points" on a line, had some meaning, in the sense that a line consisted of an actual infinity of points. Then at "every" point, at "every"
x-coördinate, there would be a vertical line that contains the corresponding y-coördinates. Now, it is possible to draw such a vertical line through any one point. Imagine such a line drawn through "every" point.
Then we would have made all possible divisions of the x-axis. And each division has zero magnitude. But that is absurd, because the x-axis, which is not zero magnitude, would then be composed of an infinite number of divisions which are. Nothing would remain of the x-axis.
We cannot divide a line at "every" point. We can divide it only at any specific point we choose. The expression "every point" -- equivalently "every value" -- has no meaning, other than every one that we choose or name.
A line is not composed of an actual infinity of points.
The idea of an actual infinity of points, then, cannot be represented by a geometrical line.
Is it even necessary?
Why the obsession with a continuum of numbers? Again, it appeared to be demanded by coördinate geometry. For "every" point on the x-axis there must be a number which is its coördinate. But not only does "every" point not exist -- only every one we indicate -- a continuum of numbers does not exist. And it is not necessary. When we let a variable approach a limit or do a calculation, we name numbers. That is all anyone has ever done or will ever need to, regardless of the theoretical justification. By expressions as such "all" values -- or "all" anything -- we mean all that we name; that is, all that will then exist.
Here we join the question of the relationship between caculations and the logical justification for those calculations. As every student of logic knows: A false hypothesis can lead to a true conclusion. (Neither Newton nor Leibniz provided an unobjectionable basis for calculating the derivative. Yet each of them showed that the derivative of x² is 2x.)
To summarize: If infinite decimals -- or however the real numbers are defined -- were numbers, then for them to be useful, whether in physics, engineering or economics, they must have names. But if every number had a name, they could not be elements in a continuum. That is the inherent contradiction in the project to create a continuum of numbers.
As for a "real number," then, it is best to retain its original meaning and say simply that it is not an imaginary number. We have looked at the attempt to define the form of a general real number, such as an infinite decimal. That leads to a very different meaning of the word "number." It is not a theory of numbers we depend on for practical purposes. It is an abstract creation, a kind of logical sport, and the most prominent current example of fantasy mathematics.
Copyright © 2012 Lawrence Spector