Appendix

THE MATHEMATICAL EXISTENCE
of
NUMBERS

The natural numbers

Names

Mathematical existence

The existence of irrational numbers

An arithmetical continuum?

Infinite decimals?

Is a line really composed of points?

Is it even necessary?

Let us begin with the natural numbers -- for the world is full of things to count.  Photons, electrons, atoms, molecules, stars, people. Each of them is composed of units; of the same, indivisible and separate ones.

That is what we mean by, and how we recognize, a natural number.

Every language has its names for colors, animals, and numbers. "Five" is the English name for this number of units: /////.  We could apply that name to a number of people, a number of fingers, or a number of axioms. Mathematics can teach how to calculate using those names, but it has no proprietary or logical claim to the names themselves.  Counting-names are part of each language.

We count units. And each one we count must have the same name. "One apple, two apples, three apples." Counting therefore depends on perceiving whether things are the same or different. ("Here's one, here's one; these two are the same.  That is not one. Those two are different.")

The mind that perceives same and different is of the same universe that creates same and different.  Like knows like.

Now, those units we perceive with our senses -- apples, people, fingers -- exist.  And we have the idea of units. It is with those that we actually count.  "2 apples + 2 apples = 4 apples" and "2 people + 2 people = 4 people" are not independent facts. We recognize each of them as an instance of  "2 + 2 = 4." That is, 2 units + 2 units = 4 units.

The one applies to the many.

Mathematics, then, must grant that natural numbers exist. For we are able to name any one of them, and with its name we know its position in the sequence. ("5" comes before "6," and after "4.") Those properties are also necessary for the numbers we need for measuring. Calculus is a theory of measuring.

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. We have the idea of a square, and according to that idea we frame a definition. But a definition does not assert that what we have defined exists. (See the commentary on the definitions of Euclid's Elements.) It is only when we present the logical steps to draw a square that shows it is more than just an idea.  The square that exists for mathematics is the square we have actually produced.  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 you 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, every one of which exists at the same time; as if what is infinite were like what is finite, and is complete.  Aside from the intelligibility of that, it is not necessary.  Calculus has no need of what we cannot name or bring into this world.  If mathematics is in any sense a science, then what more scientific meaning could we give to saying that something exists?  "Fourteen" That exists.

A rational number acquires its name and meaning from to its direct relationship to 1, now considered to be continuous, which is the source. Each integer is a number of 1's (or −1's); each fraction is a number of unit-fractions, which are the parts of 1: its halves, thirds, fourths, millionths.  But an irrational number is not a number of anything.  It has no relationship to 1.  An irrational number and 1 are incommensurable. (See Topic 2 of Precalculus.)  Apart from unique irrationals such as π and e, names and definitions of the irrationals come from the categories of functions:  roots, sines, arcsines, logarithms, and so on.

An irrational number will exist, however, not simply because it has been defined and given a name. It must satisfy an essential property of being a number, which is that it has its position in the sequence. We can determine the position of rational numbers directly from their names. 8 is

But we cannot determine the position of irrational numbers from their

names. Is more than or less 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.

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 that we actually know.  Is the irrational number 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:

It should not be unwarranted to say that if a theory of irrationals does not satisfy those conditions, then it lacks a firm foundation.

(See Kronecker's Algorithmic Mathematics.)

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, relative to a unit of 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."

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?  No, it will not.  Numbers -- the values of x -- have names. And 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.

-- 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 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 calculate each next digit of .24059165378 . . .  and thus be able to place it with respect to order, 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 nameless "points" that now composed the geometrical line. It is no wonder then that the symbols themselves are nameless.  As for the axiom that related them -- "To every point on the line there corresponds a real number, that is, a point on the line" -- it 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, which is a theory of measuring.

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 have indicated, 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, every one of which existed at the

same time. 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. Nothing would remain of the entire x-y plane.

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.

Why the obsession with a continuum of numbers? Again, it appeared to be demanded by coördinate geometry.  To "every" point on the x-axis there must correspond a number which is its coördinate. But not only does "every" point not exist, 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 calculations 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.

Finally, a "rational number" can be defined, and rational numbers exist. An "irrational number" can be defined (not rational), and they exist. It is perfectly clear, then, when we say that a "real number" is any rational number or any irrational number that exists. (It is called real to distinguish it from imaginary.) We have looked at the attempt to define a general real number, such as an infinite decimal, and we have seen how that leads to a very different meaning of the word "number." a meaning that has nothing to do with measuring. That theory is of a piece with 19th century modernism, which sought a complete break with the past and all that was usual and accessible by the many. It is an abstract creation, a kind of logical sport, and the most prominent current example of fantasy mathematics.

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Copyright © 2012 Lawrence Spector

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