Appendix

THE MATHEMATICAL EXISTENCE

OF

REAL NUMBERS

Is there an arithmetical continuum?

THE IDEA OF NUMBER is so fundamental, so much a property of the mind and how we perceive -- "These two are the same. These two are different" -- that it is an irreducible understanding.  It is not possible to define number.  If we tried, we would be going in circles, because the words we used would already contain the idea of number

Even the rules of calculation -- the rules of addition, subtraction, multiplication, and division -- contain within them the idea of number. Therefore we cannot define a number as that which obeys those rules.
(In the rule ab = ba, how many distinct symbols do you see? And which is first and which is second ?)

The following, then, while not a definition, nevertheless makes its point:

We have the clearest criterion for what we call mathematical or logical existence in Euclid's Elements, where a figure, such as a circle or a square, will exist only when we have drawn it.  A "square" for example will not exist simply because we have the idea of one and, according to that idea, are able to frame a definition. (See the commentary on the definitions of Euclid's Elements.)  Rather, it is our ability to draw a square -- to agree that what we have drawn is a square -- which shows that it is more than just an idea.  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 that we have actually drawn.

We say, then, that a number will exist at the moment we name it, whether is writing, speech, or thought.  Naming will be a form of producing it.  As for the natural numbers, the system of positional numeration based on the powers of 10, makes it possible to name any one. A natural number therefore has a potential existence.  But it will not have an actual existence until we name it.

("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)

Since the names of the fractions are based on the natural numbers, fractions also have a potential existence.  As for the existence of irrational numbers, we will address that shortly.

In any event, expressions such as "all" natural numbers, or "all" real numbers, will refer to every one that we name. In practice, that is all that is necessary.  On the philosophical level, the objects of mathematics will be those that we can actually experience, that is, experience more than just as an idea. What more scientific meaning could we give to saying that something exists?

The perimeter and area of a square  or of a circle  are not numbers. Yet we assimilate them to numbers. We measure them, which is to say, we name the ratio each has to a unit of measure.  For that purpose the real numbers were created.

The reals fall into two categories:  rational and irrational.  A rational number essentially is a nameable number. (Topic 2 of Precalculus.)  Apart from unique irrational numbers such as π and e, names for the irrationals come from the categories of functions:  roots, sines, arcsines, logarithms, and so on.

But an irrational number will exist not only because it has a name.  It must satisfy a property of any number, which is that we must know how to place it with respect to order.  Our knowledge of 8 is that it is more than 7 and less than 9.  As for an irrational number, we must be able to place it with respect to order relative to any rational number.  Is it less than or greater than 2.71828103594612074?  The verbal or symbolic name of an irrational -- π, e, -- does not answer that. Only a rational approximation will. And such an approximation will depend on the existence of a method, an algorithm, to actually produce one.

For example,

1.414213562373095 < < 1.414213562373096.

That we can "name" to as many decimal places as we please guarantees its mathematical existence as a number.

We say, then, that the sentence "This irrational number exists" means

Our idea of length is that it is continuous (Lesson 1).  And since we think of measuring length as the distance from the origin O along the x-axis,

which is a line, then since a line constitutes a continuum, the thought was that the values of x must reflect that by being a continuum of numbers. One way to express that is to say that, corresponding to any point P on the number line, there is a real number x, the coordinate of P, which is the measured length of OP.  In other words, we must be able to measure every length.

But will that be possible?  Will it be possible to name the ratio that every length will have to a unit of measure?

No.  It is impossible to name every point 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.

(This simple argument is called the semantic rejection.)

That is the tension between geometry and arithmetic.  Geometry is of the continuous, while arithmetic can deal only with what is nameable.  It will not be possible to assign a number to every length as its measure. There is no arithmetical continuum.

But a continuum of numbers is of no practical importance.  It is not just a fiction, it is not necessary.  When we do a calculation in calculus, we name a number. Or we show that in principle we could.  That is all anyone has ever done or ever will do. It is to that end that the definitions and theorems lead, even if the logical foundations were changed tomorrow.  (At one time, mathematicians explained calculus in terms of "infinitesimals." And neither Newton nor Leibniz could give an intelligible definition of the derivative)  The enunciations of the theorems and definitions can stay as they are, with the understanding that by "all" values of x, we mean all that we might name. Which is all that is necessary.

Now, a graph may be continuous because it is a line.  For the same reason, the x-axis is continuous,  When we now say that a function f(x) is "continuous" at the value x = c, then like any defined term, "continuous" will mean what we say it means, namely that when c and f(c) are numbers that exist -- and they will exist when we name them -- then as x approaches c as a limit, f(x) approaches f(c).

*

Let us not forget the second requirement for the existence of an irrational number, which is that there be a step-by-step algorithm to approximate it, so that we may place it with respect to order relative to any rational number.  A typical approximation is a decimal expansion. π is an example:

π 3.141592653589793

Here, we begin with a precisely defined number, and there is a rule by which we can compute its decimal expansion.  The name of that rule is "The rule for computing π."  Every rule, or algorithm, for computing the decimal expansion of an irrational number will have a similar name.

To suppose, then, that there could be algorithms for computing a continuum of real numbers, would require a continuum of algorithms with their names.  Again that is absurd.  Algorithms are discrete.

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, only a countable number of irrationals are computable.  Anyone who depends on a computer program must face that fact.  As far as a computer is concerned, it has no knowledge of  "most" irrational numbers, even though they have names and symbols; e.g. It can never order them relative to any rational number.  For a computer at least, such "numbers" do not exist.

In short, inasmuch as measurements -- numbers that we can know and name -- are the essence of the physical sciences, the theory of real numbers is not a theory of measurement.  In that theory, "numbers" are abstract elements not recognizable as characterized above.  Together with its associated set theory ("The set of real numbers," "The set of points on a line"), the theory of real numbers is the most prominent current example of fantasy mathematics.

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Copyright © 2009 Lawrence Spector

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